ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS EDITED BY G.-C. ROTA Editorial Board B. Doran, M. Ismail, T.-Y. Lam. E. Lutwak Volume 78 Design Theory 27 N. H. Bingham, C. M. Goldie, and J. L. Teugels Regular Variation 28 P. P. Petrushev and V. A. Popov Rational Approximation of Real Functions 29 N. White (ed.) Combinatorial Geometries 30 M. Pohst and H. Zassenhaus Algorithmic Algebraic Number Theory 31 J. Aczel and J. Dhombres Functional Equations in Several Variables 32 M. Kuczma, B. Choczewski, and R. Ger Iterative Flinctional Equations 33 R. V. Ambartzumian Factorization Calculus and Geometric Probability 34 G. Gripenberg, S.-O. Londen, and O. Staffans Volterra Integral and Functional Equations 35 G. Gasper and M. Rahman Basic Hypergeometric Series 36 E. Torgersen Comparison of Statistical Experiments 37 A. Neumaier Imerval Methods for Systems of Equations 38 N. Korneichuk Exact Constallls in Approximation TheOlY 39 R. Brnaldi and H. Ryser Combinatorial Matrix Theory 40 N. White (ed). Matroid Applications 41 S. Sakai Operator Algebras in Dynamical Systems 42 W. Hodges Basic Model Theory 43 H. Stahl and V. Totik General Ol1hogonal Polynomials 44 R. Schneider Convex Bodies 45 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions 46 A. Bjorner et al. Oriellfed Matroids 47 G. Edgar and L. Sucheston Stopping TImes and Directed Processes 48 C. Sims Computation with Finitely Presented Groups 49 T. Palmer Banach Algebras alld the General TheO/y of "-Algebras 50 F. Borceux Handbook of Categorical Algebra I 51 F. Borceux Handbook of Categorical Algebra II 52 F. Borceux Handbook of Categorical Algebra JlJ 54 A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems 55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics 56 V. N. Sachkov Probabilistic Methods ill Discrete Mathematics 57 P. M. Cohn Skew Fields 58 R. Gardner Geometric Topography 59 G. A. Baker Jr. and P. Graves-Morris Pade Approximants 60 J. Krajicek Bounded Arithmetic, Propositional Logic, and Complexity TheO/y 61 H. Groemer Geometric Applications of Fourier Series alld Spherical Hanl10llics 62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory 63 A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65 K. Engel Spemer TheOf)' 66 D. Cverkovic, P. Rowlinson, S. Simic Eigenspaces of Graphs 67 F. Bergeron, O. Labelle, and P. Leroux Combinatorial Species and Tree-Like Structures 68 R. Goodman and N. Wallach Representations and in1'ariants of the Classical Groups
Design Theory Second Edition
Thomas Beth Universitiit Karlsruhe Dieter Jungnickel Universitiit Augsburg Hanfried Lenz Freie Universitiit Berlin
Volume II
CAMBRIDGE UNIVERSITY PRESS
Foreword Doch am vierten Tag im FeIsgesteine Hat ein Zollner ihm den Weg verwehrt: "Kostbarkeiten zu verzollen?" - "Keine." Und der Knabe, der den Ochsen fiihrte, sprach: "Er hat gelehrt." Und so war auch das erkliirt. (Brecht)
Although many excellent papers and some specialised monographs on different aspects of design theory had been published, prior to 1985 (when the first edition of this book appeared) there was no comprehensive monograph on the field. Thus it was our plan to cover the main concepts and ideas of modem design theory without being encyclopedic, but including a deeper study of several representative topics. As it turned out, these aims (which required a rather long book) seem to have been met by the first edition which has been quoted extensively in the research literature. A first draft of this book was obtained by merging different manuscripts of the authors. The first edition then grew from an iterative process of rewriting in which all the authors contributed their ideas to each of the parts. For reasons of time constraints and shifting research interests, the major part of the revision now presented has been done by the second author, drawing on his previous update Jungnickel (1989a) and surveys Jungnickel (1990a, 1992a); in particular, this holds for Chapter VI. Of course, there has still been considerable help from his co-authors (with Chapter XIII being the first author's contribution), but nevertheless he is willing to accept most of the blame for the mistakes introduced during the process of revision. As in 1985, it is again hoped that this book may prove useful in several ways: as most of the presentation is based on classroom experience, it should be suitable as both a textbook for (advanced) classroom use and as a reference book for private study. Although we usually do not require any previous knowledge beyond basic algebra (except for some more advanced parts of Chapter VI and for some of the applications studied in Chapter XIII), this text should also be attractive to the specialist. The proofs of several fundamental theorems have been simplified vii
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©
Bibliographisches Institut, Zurich, 1985 Cambridge University Press, 1993 Second edition © Cambridge University Press, 1999
©
This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in Jb.TEX2 e [TB] A catalogue recordfor this book is availablejrom the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949Design theory I Thomas Beth, Dieter Jungnickel, Hanfried Lenz. 2nded. p. cm. Includes bibliographical references and index. ISBN 0 521 772311 (hardbound) 1. Combinatorial designs and configurations. 1. Jungnickel, D. (Dieter), 1952- . II. Lenz, Hanfried. Ill. Title. QAI66.25.B47 1999 511'.6 - dc21 98-29508 CIP ISBN 0 521772311 hardback
In memoriam REINHOLD BAER
x
Foreword
Let us give a brief outline of the contents of this book. Chapter I is a leisurely introduction to the different topics of design theory. The fonnal treatment begins with Chapter II which is devoted to the combinatorial and algebraic analysis of incidence structures. In Chapters ill, IV, and V, we develop in depth the interaction between groups and designs. This part of the book is well suited to fonn the core of a special course on this topic. The same holds for Chapter VI, where difference sets are considered in much detail; these are equivalent to studying symmetric designs with a regular automorphism group, thus providing the link to representation theory. The five subsequent chapters emphasise the constructive aspects of design theory. The connection to the preceding part is provided by Chapters VII and Vill which contain constructions based mostly on abelian groups, including the asymptotic existence of difference families in Galois fields. The recursive methods presented in Chapter IX have their foundation in the preceding direct constructions. Thus we obtain several general existence theorems for block designs with small block sizes as well as for Steiner quadruple systems, illustrating the close interlacing of design-like incidence structures and Latin squares which are subject to a more detailed study in Chapter X. In Chapter XI all these methods converge in the proof of the asymptotic existence theorem for 2-designs and related structures. Chapter XII is devoted to characterising the classical examples among the families of designs we have studied. The final Chapter Xli discusses some known and new applications of design theory. The book concludes with an appendix of several tables of designs in a range feasible for applications, and an extensive bibliography. Items are numbered and quoted consecutively within each chapter as follows: "Item x.y" refers to the y'th item in §x of the given chapter; similarly, fonnulae are numbered with reference to the item where they occur, that is, in the fonn (x.y.a), (x.y.b), etc. If an item from another chapter needs to be quoted, the number of this chapter is added, resulting in quotes of the fonn "by item Z.x.y"; a similar convention holds for fonnulae. The symbol. indicates the end (or absence) of a proof; as mentioned before, we will have to state many re~ults without proofs. Finally, we would like to acknowledge those people and institutions, who by their mental or financial support throughout the last two decades made this work possible. Particular thanks are due to Klaus Metsch and Bernhard Schmidt who each provided us with drafts for some parts of the present text (namely Sections X.7 and some of the later parts of Chapter VI, respectively). Also the engagement of Detlef Zerfowski in preparing the completely new Chapter Xli must be acknowledged. We also thank the following colleagues for their comments:
;;
Foreword ;;
xi
A. E. Brouwer, P. J. Cameron, C. Chames, C. 1. Colbourn, 1. A. Davis, M. 1. de Resmini, R. H. F. Denniston, D. A. Drake, S. Furino, W. Geiselmann, 1. W. P. Hirschfeld, D. R. Hughes, J. ledwab, I. H. van Lint, H. Luneburg, W. H. Mills, R. C. Mullin, G. Pickert, F. C. Piper, A. Pott, D. K. Ray-Chaudhuri, 1. 1. Seidel, H. Siemon, N. J. A. Sloane, S. A. Vanstone, and V. Zinoviev. The institutions we are indebted to are: Deutscher Akademischer Austauschdienst, Deutsche Forschungsgemeinschaft, Heisenberg-Programm, National Science and Engineering Research Council (Canada), Istituto di Geometria dell' Universita di Bologna, Department of Mathematics of The Ohio State University, Mathematics Department of the University of Florida, Mathematics Department of Westfield College London, Department of Combinatorics and Optimization of the University of Waterloo, Institut fUr Mathematische Maschinen und Datenverarbeitung as well as Mathematisches Institut der Universitat Erlangen, Mathematisches Institut der Universitat GieSen, II. Mathematisches Institut der Freien Universitat Berlin, Institut fUr Algorithmen und Kognitive Systeme der Universitat Karlsruhe, Institut fur Mathematik der Universitat Augsburg, and, in particular, Mathematisches Forschungsinstitut Oberwolfach both for providing a multitude of stimulating contacts and for giving us the opportunity to design the final draft of the first edition of this book in its unique atmosphere.
Thomas Beth Dieter lungnicke1 Hanfried Lenz November, 1998
Neither art nor science can be achieved without studying. (Demokrit)
viii
Foreword
and many advanced results are presented in detail. Unfortunately, the explosive growth and advancing technical difficulty of many parts of design theory have made it impossible to include proofs for most of the new results obtained after our first edition appeared. To mention just two examples, the existence theory for designs is becoming more and more involved and nowadays requires a considerable amount of computer work; and the theory of difference sets has reached a stage where major new results can only be obtained by the use of algebraic number theory and lengthy and delicate computations. On the other hand, we still wanted to provide our readers with the present state of knowledge, instead of more or less only reprinting our first edition. Therefore, several important parts of the text have been completely rewritten (for instance, the chapter on difference sets and the appendix giving a collection of tables as well as the sections concerning constructions for sets of mutually orthogonal Latin squares of small order and the section on completing Bruck nets, to mention just a few) and a fair amount of new material has been added with a full exposition (for instance, a chapter on applications of design theory as well as sections on strongly regular graphs, Cayley graphs, codes of designs and nets, embedding designs, and Steiner quadruple systems). However, most new results can only be quoted without giving proofs - otherwise we would require three volumes. An extensive bibliography of well over 2000 titles, all quoted in the text, has been included. Thus the book may also be used as a reference, though it is - as already mentioned - not intended to meet encyclopedic standards. Last, but not least, in view of the central importance of discrete mathematics for a multitude of applications, the book will provide some ofthe necessary mathematical background for anyone working in communications engineering, optimisation, statistical planning, computer science, and signal processing. With these aims in mind we have tried to select and present the material not only according to our personal points of view: a large part of our work has been devoted to presenting appropriately the results of the different schools of combinatorics concerned with design theory. We have also tried to cover the different areas of this subject in such a manner that the reader may work with other related monographs and papers. We first list (in alphabetical order) some important books on different aspects of finite geometry, design theory and closely related areas, namely Assmus and Key (1992a), Bannai and Ito (1984), Batten and Beutelspacher (1993), Baumert (1971), Brouwer, Cohen and Neumaier (1989), Cameron (1976a), Cameron and van Lint (1991), Dembowski (1968), Denes and Keedwell (1974, 1991), Evans (1992b), Geramita and Seberry (1979), Godsil (1993), Haemers (1980), Hirschfeld (1985,1998), Hirschfeld and Thas (1991), Huppert (1967), Huppert and Blackburn (1982a, b), Kallaher (19al),
Foreword
ix
Lander (1983), Ltineburg (1965b, 1969, 1980), Pott (1995), Shrikhande and Sane (1991), Tsuzuku (1982), Wallis (1997) and Zieschang (1996a). There is also a direct link to books on related applications, for instance Beker and Piper (1982), Conway and Sloane (1993), Harwit and Sloane (1979), Jungnickel (1993a), Liineburg (1979), Mac Williams and Sloane (1978), Raghavarao (1971), Simmons (1992a) and Stinson (1995). We should also mention that there are by now five other books studying combinatorial designs, all of which are intended as introductory texts emphasising various aspects of the field; in order of appearance, these are Hughes and Piper (1985), Street and Street (1987), Tonchev (1988), Wallis (1988), and Anderson (1990). Design theoretic algorithms are surveyed in a collection edited by Colbourn (1985); a recent volume on computational and constructive aspects of design theory was edited by Wallis (1996). Surveys of applications of designs are given by Colbourn and van Oorschot (1989) and in Chapter V of Colboum and Dinitz (1996a); the latterreference basically is a vast compendium of tables. Recent expositions of designs from the point of view of statistics are due to Bailey (1989, 1996); for an upto-date treatment of the optimal design of experiments, see Pukelsheim (1993). The relevance of design theory to pure, applied, and even applicable mathematics has already been indicated. Nevertheless, we would like to quote some sentences from Lovasz (1979): The mathematical world had been attracted by the successes of algebra and analysis and only in recent years it has become clear, due largely to problems arising from economics. statistics, electrical engineering and other applied sciences, that combinatorics, the study of finite sets and finite structures, has its own problems and principles. These are independent of those in algebra and analysis but match them in difficulty, practical and theoretical interest and beauty. Yet the opinion of many first-class mathematicians about combinatorics is still in the pejorative. While accepting its interest and difficulty. they deny its depth. It is often forcefully stated that combinatorics is a collection of problems, which may be interesting in themselves but are not linked and do not constitute a theory .. " The above accusations are clearly characteristic of any field of science at an early stage of its development - at the stage of collecting data .... Those techniques whose absence has been disapproved of above await their discoverers. So nnderdevelopment is not a case against, but rather for, directing young scientists towards a given field. In my opinion, combinatorics is now growing out of this early stage.
In this spirit, we think that the development of design theory over the last few decades has clearly reached a critical point at which we now see at least the beginnings of a systematic treatment of the area leading to its codification into a mathematical theory, as opposed to a collection of individual results. We hope that this development can be discerned from the present text.
Contents
IX. Recursive constructions . . . . . . § 1. §2. §3. §4. §5. §6. §7. §S. §9. § 10. § 11. §12.
Product constructions . . . . . . Use of pairwise balanced designs Applications of divisible designs Applications of Hanani's lemmas Block designs of block size three and four Solution of Kirkman's schoolgirl problem The basis of a closed set . . . . . . . . Block designs with block size five .. . Divisible designs with small block sizes Steiner quadruple systems. . . . . . . . Embedding theorems for designs and partial designs Concluding remarks . . . . . . . . . . . . . . . . .
X. Transversal designs and nets § 1. §2. §3. §4. §5. §6. §7. §S. §9.
60S 60S 617 621 627 636 641 644
651 660 664 673 6Sl 690
A recursive construction Transversal designs with A > 1 A construction of Wilson . . . Six and more mutually orthogonal Latin squares The theorem of Chowla, Erdos and Straus Further bounds for transversal designs and orthogonal arrays . . . . . . . . . . . . . . . . . . . . . . . Completion theorems for Bruck nets . . . . . . . . . . Maximal nets with large deficiency . . . . . . . . . . . Translation nets and maximal nets with small deficiency xiii
690 693 696 703 706 70S 713 725 731
Contents
xiv
§1O. Completion results for f.L > I . . . . . . . . . . . . . . .. § 11. Extending symmetric nets. . . . . . . . . . . . . . . . ., §I2. Complete mappings, difference matrices and maximal nets. §13. Tarry's theorem. . . §14. Codes of Bruck nets. . .
XI. Asymptotic existence theory § 1. §2.
§3. §4. §5. §6. §7. §8.
§2. §3. §4. §5. §6.
Projective and affine spaces as linear spaces Characterisations of projective spaces . Characterisation of affine spaces Locally projective linear spaces . Good blocks . . . . Concluding remarks. .
XIII. Applications of designs . § 1.
§2. §3. §4. §5. §6. §7. §8. §9. § 1O.
Introduction...... Design of experiments Experiments with Latin squares and orthogonal arrays Application of designs in optics . Codes and designs . . . . . . . . . . . . . . . . . Discrete tomography . . . . . . . . . . . . . . . . Designs in data structures and computer algorithms. Designs in hardware . . . . . . . . . Difference sets rule matter and waves. No waves, no rules, but security.
Appendix. Tables . . . . . § 1. §2.
781
781 Preliminaries . . . . . . The existence of Steiner systems with v in given residue classes . . . . . . . . . . . . . . . . . . . . . . 783 The main theorem for Steiner systems S(2, k; v) 787 790 The eventual periodicity of closed sets The main theorem for ).. = 1 . . . . . . . . . . . 793 The main theorem for).. > 1 . . . . . . . . . . . 796 An existence theorem for resolvable block designs 801 Some results for t ::: 3 . . . . . . . . 805
XII. Characterisations of classical designs . § 1.
749 758 761 772 778
Block designs . . . Symmetric designs
806 806 808 821
828 833 841 852 852
856 874
880 892 926
930 937 946
956 971 . 971 . 981
!
I I
Contents
xv
§3. Abelian difference sets . . . . . §4. Small Steiner systems . . . . . . §S. Infinite series of Steiner systems §6. Remark on t-designs with t 2: 3 . §7. Orthogonal Latin squares
· 990 · 997 · 999 1001
Notation and symbols Bibliography . Index . . . . . . . . .
I
1002 100S 1013
1093
Contents of Volume I
I. Examples and basic definitions §l. §2. §3. §4. §5. §6. §7. §8. §9.
........ .
Incidence structures and incidence matrices . Block designs and examples from affine and projective geometry . . . . . . . . . . . . . . . . . . . . . . . . 6 t-designs, Steiner systems and configurations. 15 20 Isomorphisms, duality and correlations . . 24 Partitions of the block set and resolvability 32 Divisible incidence structures . 37 Transversal designs and nets 44 Subspaces . . . . . 50 Hadamard designs . . . . .
II. Combinatorial analysis of designs §l. §2. §3. §4. §5. §6.
Basics . . . . . . . . . . . . . Fisher's inequality for pairwise balanced designs Symmetric designs . . . . . . . . . . . . . . . The Bruck-Ryser-Chowla theorem . . . . . . . Balanced incidence structures with balanced duals Generalisations of Fisher's inequality and intersection numbers . . . . . . . . . . . . . . . . . . . . .. §7. Extensions of designs . §8. Mfine designs. . . . . . . §9. Strongly regular graphs . . §1O. The Hall-Connor theorem §11. Designs and codes . . . xvii
62 62 64 77 89
96 10 I III 123 136 146 152
Contents
§ 11. §12. §13. § 14. §15. §16. § 17.
Extended building sets and difference sets . . . . . . Constructions for Hadamard and Chen difference sets Some applications of algebraic number theory Further non-existence results . . Characters and cyclotomic fields . . . Schmidt's exponent bound . . . . . . Difference sets with Singer parameters
VII. Difference families .
§ 1. §2. §3. §4. §5. §6. §7. §8.
Basic facts .. Multipliers .. More examples Triple systems. Some difference families in Galois fields Blocks with evenly distributed differences Some more special block designs Proof of Wilson's theorem
VIII. Further direct constructions § 1. §2. §3. §4. §5. §6. §7. §8. §9. § 10.
Pure and mixed differences . . . . . . . . . . . . . . . .. Applications to the construction of resolvable block designs A difference construction for transversal designs Further constructions for transversal designs Some constructions using projective planes . . . t-designs constructed from graphs . . . . . . . The existence of t-designs for large values of A . Higher resolvability of t-designs Infinite t-designs . . . . . . . . Cyclic Steiner quadruple systems
Notation and symbols Bibliography . Index . . . . . . . . .
xix 382 388 410 . 4 19 . 435 441 455 468 . 468 .472 476 481 488 499 502 509 520 520 528 531 544 564 584 588 595 598 600 1005 1013 1093
Contents
§ 11. §12. §13. § 14. §15. §16. § 17.
Extended building sets and difference sets . . . . . . Constructions for Hadamard and Chen difference sets Some applications of algebraic number theory Further non-existence results . . Characters and cyclotomic fields . . . Schmidt's exponent bound . . . . . . Difference sets with Singer parameters
VII. Difference families .
§ 1. §2. §3. §4. §5. §6. §7. §8.
Basic facts .. Multipliers .. More examples Triple systems. Some difference families in Galois fields Blocks with evenly distributed differences Some more special block designs Proof of Wilson's theorem
VIII. Further direct constructions § 1. §2. §3. §4. §5. §6. §7. §8. §9. § 10.
Pure and mixed differences . . . . . . . . . . . . . . . .. Applications to the construction of resolvable block designs A difference construction for transversal designs Further constructions for transversal designs Some constructions using projective planes . . . t-designs constructed from graphs . . . . . . . The existence of t-designs for large values of A . Higher resolvability of t-designs Infinite t-designs . . . . . . . . Cyclic Steiner quadruple systems
Notation and symbols Bibliography . Index . . . . . . . . .
xix 382 388 410 . 4 19 . 435 441 455 468 . 468 .472 476 481 488 499 502 509 520 520 528 531 544 564 584 588 595 598 600 1005 1013 1093
§l. Product constructions
609
over G = {YI, ... , Yg}. Then the matrix A I8i D defined by au
~D
akl
+D
AI8iD=
(
is an OAu,(kk, gg').1
if E
is a (g, k, A.)-difference matrix, then E I8i D is a (g, kk, g)...)...I)-difference matrix. In particular, the Kronecker product of two generalised Hadamard matrices (VIII.3.4) is a generalised Hadamard matrix. •
1.3 Example. The existence of (3,3; 1)- and (3, 6; 2)-difference matrices (see Theorem VIII.3.14) implies that of a (3, 18; 6)-difference matrix over &::3 x &::6. More generally, we obtain (3,3),; A.)-difference matrices for all ). = 2i 3j with j :::: i-I. All these difference matrices are generalised Hadamard matrices. 1.4 Definition. Let (A, 0), (B, *) be quasigroups (see Definition VIII.4.lO). Their direct product is defined as the set A x B with the operation 0 defined by (lA.a)
Ca, b) 0 (a', b' ) := (a
0
a', b
* b').
1.5 Lemma. Let (A, 0) and (A, 0 ' ) be orthogonal quasigroups, and letCB, *). (B, *') be orthogonal quasigroups too. Then the direct products CA, 0) x (B, *) and (A, 0') x CB, *') are orthogonal quasigroups. if CA, 0) and (B, *) are idempotent, then (A, 0) x (B, *) is also idempotent. • The proof is straightforward. Again MacNeish's theorem (1.7.7.b) follows immediately; furthermore (1.5.a)
N*(gh) :::: min{N*(g), N*(h)}.
For the sake of completeness we mention the following connection between quasigroups and Steiner triple systems.
1
Note that this would be the Kronecker product of matrices if G were written multiplicatively.
§l. Product constructions
609
over G = {YI, ... , Yg}. Then the matrix A I8i D defined by au
~D
akl
+D
AI8iD=
(
is an OAu,(kk, gg').1
if E
is a (g, k, A.)-difference matrix, then E I8i D is a (g, kk, g)...)...I)-difference matrix. In particular, the Kronecker product of two generalised Hadamard matrices (VIII.3.4) is a generalised Hadamard matrix. •
1.3 Example. The existence of (3,3; 1)- and (3, 6; 2)-difference matrices (see Theorem VIII.3.14) implies that of a (3, 18; 6)-difference matrix over &::3 x &::6. More generally, we obtain (3,3),; A.)-difference matrices for all ). = 2i 3j with j :::: i-I. All these difference matrices are generalised Hadamard matrices. 1.4 Definition. Let (A, 0), (B, *) be quasigroups (see Definition VIII.4.lO). Their direct product is defined as the set A x B with the operation 0 defined by (lA.a)
Ca, b) 0 (a', b' ) := (a
0
a', b
* b').
1.5 Lemma. Let (A, 0) and (A, 0 ' ) be orthogonal quasigroups, and letCB, *). (B, *') be orthogonal quasigroups too. Then the direct products CA, 0) x (B, *) and (A, 0') x CB, *') are orthogonal quasigroups. if CA, 0) and (B, *) are idempotent, then (A, 0) x (B, *) is also idempotent. • The proof is straightforward. Again MacNeish's theorem (1.7.7.b) follows immediately; furthermore (1.5.a)
N*(gh) :::: min{N*(g), N*(h)}.
For the sake of completeness we mention the following connection between quasigroups and Steiner triple systems.
1
Note that this would be the Kronecker product of matrices if G were written multiplicatively.
611
§l. Product constructions
distinct points a, b, c, d are on a block iff f(a, b, c) = d.lf (V, f) and (W, g) are such ternary algebras, then their direct product, defined in the obvious way, has the same properties. Hence (l.9.f)
u,
V
E S(3, 4) =}
uv
E S(3, 4).
Note that 2 E S(3, 4) (two points, no blocks), and that a ternary operation 0 on a 2-set, satisfying (1.9.b), has all the properties of 1.9. Hence (1.9.£) specializes to (lo9.g)
2· S(3, 4) S; S(3, 4).
In view of Hanani's theorem stating that S(3, 4) = 2 N \ 6 N, (1.9.£) is a very weak result; see § 10 for a proof of Hanani' s theorem.
1.10 Remark. These examples may be generalised to the idea of applying universal algebra to combinatorial structures; cf. Ganter (1976a), Evans (1975), Quackenbush (1975). A nice introductory paper on this subject was given by Evans (1979). Next we shall use difference matrices and orthogonal arrays for product constructions of difference families, as well as the other way round.
1.11 Proposition. Let k be the order of an affine plane. Furthermore, let A = (aij) be an OA(k, k) over S = Moreover, let D = (D 1 ••..• Ds) and E = (EI,"" E t ) be difference families in the groups G and G' with parameters (v, k. A) and (v', k, A'), respectively, say Di = {do, ... , did and E j = {ejl, ... , ejk} for i E N~ and j E Ni. Then there is a (vv', k, U')difference family in G EB G'.
N1.
Proof. We may assume that the last k columns of A are (1, 1, ... , ll, ... , (k, k, ... , k)T. We omit these k columns and get a k x k(k -I)-matrix B = (b ij ). Now let the desired difference family F consist of the following base blocks: A' copies of {(do, 0), ... , (dik. O)} for each i E Nl;
A copies of {CO, ejd}, ... , (0, ejk)} for each j EN;; one copy of {(do, ejb,,,), ... , (dit. ejb,,)} for each n E and each pair (i, j) E N~
X
Nf- k
NI!.
The somewhat lengthy though not difficult verification of this construction is left to the reader; cf. Jungnickel (1978) . •
611
§l. Product constructions
distinct points a, b, c, d are on a block iff f(a, b, c) = d.lf (V, f) and (W, g) are such ternary algebras, then their direct product, defined in the obvious way, has the same properties. Hence (l.9.f)
u,
V
E S(3, 4) =}
uv
E S(3, 4).
Note that 2 E S(3, 4) (two points, no blocks), and that a ternary operation 0 on a 2-set, satisfying (1.9.b), has all the properties of 1.9. Hence (1.9.£) specializes to (lo9.g)
2· S(3, 4) S; S(3, 4).
In view of Hanani's theorem stating that S(3, 4) = 2 N \ 6 N, (1.9.£) is a very weak result; see § 10 for a proof of Hanani' s theorem.
1.10 Remark. These examples may be generalised to the idea of applying universal algebra to combinatorial structures; cf. Ganter (1976a), Evans (1975), Quackenbush (1975). A nice introductory paper on this subject was given by Evans (1979). Next we shall use difference matrices and orthogonal arrays for product constructions of difference families, as well as the other way round.
1.11 Proposition. Let k be the order of an affine plane. Furthermore, let A = (aij) be an OA(k, k) over S = Moreover, let D = (D 1 ••..• Ds) and E = (EI,"" E t ) be difference families in the groups G and G' with parameters (v, k. A) and (v', k, A'), respectively, say Di = {do, ... , did and E j = {ejl, ... , ejk} for i E N~ and j E Ni. Then there is a (vv', k, U')difference family in G EB G'.
N1.
Proof. We may assume that the last k columns of A are (1, 1, ... , ll, ... , (k, k, ... , k)T. We omit these k columns and get a k x k(k -I)-matrix B = (b ij ). Now let the desired difference family F consist of the following base blocks: A' copies of {(do, 0), ... , (dik. O)} for each i E Nl;
A copies of {CO, ejd}, ... , (0, ejk)} for each j EN;; one copy of {(do, ejb,,,), ... , (dit. ejb,,)} for each n E and each pair (i, j) E N~
X
Nf- k
NI!.
The somewhat lengthy though not difficult verification of this construction is left to the reader; cf. Jungnickel (1978) . •
613
§1. Product construCtions
1.15 Examples. (a) Proposition 1.14 and Lemma VIII.3.10 yield alternative proofs for Examples 1.l3.(b), (c). An alternative proof of Example 1.13.(a) will be given a little later. (b) Using Proposition 1.14, Lemma VIII.3.10 and Wilson's Theorem VII.6.6, we obtain the existence of a (v, k, A)-difference family whenever v admits a prime power factorisation
1)
with A(qi large.
="
0 (mod k(k - 1», where the qi (i
= 1, ... , n) are sufficiently
(c) Assume that k or k - 1 divides 2A. Then there exists a (v, k, A)-difference family whenever each factor qi in the prime power factorisation of v satisfies the condition (1.15.b)
A(qi - 1)
== 0 (mod k(k
- 1».
This follows from Lemma 1.1 (with A' = 1), together with Lemma VIII.3.IO. Thus, for example, we obtain (compare with Examples VII.5.4) v E DI (3) n D2(4) n Ds(6) n D6(7),
(1.15.c)
= 1, ... , m;
whenever qi
== 1 (mod6) for i
whenever qi
== 1 (mod 10) for i
= 1, ... , n;
whenever qi
== 1 (mod 14) for i
= 1, ... , n;
== 1 (mod 18) for i
= 1, ... , n.
(1.15.f) whenever qi (d) Let t, n
E
(U5.g)
(4t - l)n
Nand 4t - 1 be a prime power. Then E
D t - l (2t - 1).
This result follows by using a Paley difference set (Theorem VI. 1.12), and is a special case of (c). (e) There is a (91, 10, I)-difference set in Z91 (Theorem VI. 1.10). Also there are (l3, 13; I)-difference matrices (Lemma VIII.3.1O), and (7, 14; 2)-difference
613
§1. Product construCtions
1.15 Examples. (a) Proposition 1.14 and Lemma VIII.3.10 yield alternative proofs for Examples 1.l3.(b), (c). An alternative proof of Example 1.13.(a) will be given a little later. (b) Using Proposition 1.14, Lemma VIII.3.10 and Wilson's Theorem VII.6.6, we obtain the existence of a (v, k, A)-difference family whenever v admits a prime power factorisation
1)
with A(qi large.
="
0 (mod k(k - 1», where the qi (i
= 1, ... , n) are sufficiently
(c) Assume that k or k - 1 divides 2A. Then there exists a (v, k, A)-difference family whenever each factor qi in the prime power factorisation of v satisfies the condition (1.15.b)
A(qi - 1)
== 0 (mod k(k
- 1».
This follows from Lemma 1.1 (with A' = 1), together with Lemma VIII.3.IO. Thus, for example, we obtain (compare with Examples VII.5.4) v E DI (3) n D2(4) n Ds(6) n D6(7),
(1.15.c)
= 1, ... , m;
whenever qi
== 1 (mod6) for i
whenever qi
== 1 (mod 10) for i
= 1, ... , n;
whenever qi
== 1 (mod 14) for i
= 1, ... , n;
== 1 (mod 18) for i
= 1, ... , n.
(1.15.f) whenever qi (d) Let t, n
E
(U5.g)
(4t - l)n
Nand 4t - 1 be a prime power. Then E
D t - l (2t - 1).
This result follows by using a Paley difference set (Theorem VI. 1.12), and is a special case of (c). (e) There is a (91, 10, I)-difference set in Z91 (Theorem VI. 1.10). Also there are (l3, 13; I)-difference matrices (Lemma VIII.3.1O), and (7, 14; 2)-difference
615
§l. Product constructions Note that s = ~ v, as it must be.
= A. g~g~\)' hence the column number of A is sg(g -
1) + A.
=
1.18 Examples. (a) Let q be a prime power and assume the existence of a TD[k; q + 1] with a parallel class. Then there exists a (q2 + q + 1, k; 1)difference matrix in Zq2+q+! by Singer's Theorem VI. 1. 10. (b) There are (15, 7; 3)-, (21,5; 1)-, (40, 13; 4)-, and (57, 8; I)-difference matrices in the cyclic groups of the respective orders. The reader is asked to check this. (c) Let q be a prime power and let G be a group of order q + 1. Then there is a (q + 1, q; q - I)-difference matrix over G. To see this, use the trivial (q + 1, q, q - I)-difference set G \ {OJ. Note that this yields non-abelian difference matrices too (though of moderate size; the maximum feasible value of k is q2 - 1 by Corollary VIII.3.7).
It is worthwile to state the following consequence of Proposition 1.17.
1.19 Corollary. Assume the existence of a (v, g, I)-difference family. Then (1.19.a)
g E TD(k) =? v E TD(k).
In particular, (1.19.b)
q
+ 1 E TD(k)
=? q2
+q + 1E
TD(k)
for prime powers q.
Thus (1.19.c)
21
(1.19.d)
57 E TD(9),
(1.19.e)
273 E TD(18).
E
TD(6),
Proof. Use Lemma 1.1 together with (I.7.15.a) and Corollary VIII.3.8. The particular examples follow from 1.18.(a). • The values given in (1.19.c, d, e) improve those following from MacNeish's Corollary 1.7.8 and give the best result for the numbers 57 and 273 known to date. For 21, the better result 21 E TD(7) is known, see Corollary VIII.3.17.
615
§l. Product constructions Note that s = ~ v, as it must be.
= A. g~g~\)' hence the column number of A is sg(g -
1) + A.
=
1.18 Examples. (a) Let q be a prime power and assume the existence of a TD[k; q + 1] with a parallel class. Then there exists a (q2 + q + 1, k; 1)difference matrix in Zq2+q+! by Singer's Theorem VI. 1. 10. (b) There are (15, 7; 3)-, (21,5; 1)-, (40, 13; 4)-, and (57, 8; I)-difference matrices in the cyclic groups of the respective orders. The reader is asked to check this. (c) Let q be a prime power and let G be a group of order q + 1. Then there is a (q + 1, q; q - I)-difference matrix over G. To see this, use the trivial (q + 1, q, q - I)-difference set G \ {OJ. Note that this yields non-abelian difference matrices too (though of moderate size; the maximum feasible value of k is q2 - 1 by Corollary VIII.3.7).
It is worthwile to state the following consequence of Proposition 1.17.
1.19 Corollary. Assume the existence of a (v, g, I)-difference family. Then (1.19.a)
g E TD(k) =? v E TD(k).
In particular, (1.19.b)
q
+ 1 E TD(k)
=? q2
+q + 1E
TD(k)
for prime powers q.
Thus (1.19.c)
21
(1.19.d)
57 E TD(9),
(1.19.e)
273 E TD(18).
E
TD(6),
Proof. Use Lemma 1.1 together with (I.7.15.a) and Corollary VIII.3.8. The particular examples follow from 1.18.(a). • The values given in (1.19.c, d, e) improve those following from MacNeish's Corollary 1.7.8 and give the best result for the numbers 57 and 273 known to date. For 21, the better result 21 E TD(7) is known, see Corollary VIII.3.17.
617
§2. Use of pairwise balanced designs
§IV.ll.4 of Colboum and Dinitz (l996a). Further recursive constructions for difference matrices and difference families are due to Buratti (1998a). In this section we have seen how closely related the concepts of difference families and difference matrices are. The corresponding interaction between pairwise balanced designs and transversal designs will be of paramount importance in the recursive constructions of designs.
§2. Use of Pairwise Balanced Designs We shall present some important recursive constructions, in particular for PBD's and TD's. We use Ranani's notation (cf. 1.2.19).
2.1 Lemma. Let g
o ::s gi ::s g for i = (2.l.a)
kg
E TD(k + n) and k, k 1, ... , n. Then
+ 1, ... , k + n
+ g, + ... + gn E GD(K, {g, g" ~
E K; also, let
... , gnD
B(K U {g, g" ... , gnD.
Proof. Delete g - g" ... , g - gn points from the last n point classes of a TD[k +n; g] . •
2.2 Lemma. Let g gn + 1 E K; 0 < gi (2.2.a)
kg
E TD(k+n) and k, k+
1, ... , k+n, g + 1, g,
+ 1, ... ,
::s g for i = 1, ... ,n. Then
+ g, + ... + gn + 1 E
B(K).
Proof. Proceed as in Lemma 2.1 and use (I.6.5.b) . •
2.3 Theorem. ifv
E
B(L) and L ~ B(K), then v
E
B(K).
Proof. Let D = (V, {B" ... , Bb}, E) be an S(2, L; v). By hypothesis, there are PBD's Di = (B i , {Ci!, ... , Cib,}, E) with ICiJ-t1 E K for all i, f.L. Then
is the desired PBD. Wilson (l972a) calls this procedure "breaking up blocks" .
•
617
§2. Use of pairwise balanced designs
§IV.ll.4 of Colboum and Dinitz (l996a). Further recursive constructions for difference matrices and difference families are due to Buratti (1998a). In this section we have seen how closely related the concepts of difference families and difference matrices are. The corresponding interaction between pairwise balanced designs and transversal designs will be of paramount importance in the recursive constructions of designs.
§2. Use of Pairwise Balanced Designs We shall present some important recursive constructions, in particular for PBD's and TD's. We use Ranani's notation (cf. 1.2.19).
2.1 Lemma. Let g
o ::s gi ::s g for i = (2.l.a)
kg
E TD(k + n) and k, k 1, ... , n. Then
+ 1, ... , k + n
+ g, + ... + gn E GD(K, {g, g" ~
E K; also, let
... , gnD
B(K U {g, g" ... , gnD.
Proof. Delete g - g" ... , g - gn points from the last n point classes of a TD[k +n; g] . •
2.2 Lemma. Let g gn + 1 E K; 0 < gi (2.2.a)
kg
E TD(k+n) and k, k+
1, ... , k+n, g + 1, g,
+ 1, ... ,
::s g for i = 1, ... ,n. Then
+ g, + ... + gn + 1 E
B(K).
Proof. Proceed as in Lemma 2.1 and use (I.6.5.b) . •
2.3 Theorem. ifv
E
B(L) and L ~ B(K), then v
E
B(K).
Proof. Let D = (V, {B" ... , Bb}, E) be an S(2, L; v). By hypothesis, there are PBD's Di = (B i , {Ci!, ... , Cib,}, E) with ICiJ-t1 E K for all i, f.L. Then
is the desired PBD. Wilson (l972a) calls this procedure "breaking up blocks" .
•
§2. Use ojpairwise balanced designs
619
is a closure operator; i.e. it has the properties (2.7.b)
K
~
B(K),
(2.7.c)
K
~
B(L) :::} B(K)
~
B(L),
and hence (2.7.d)
= B(L)
B(B(L»
for all K, L ~ N. A subset K
~
N is called closed if B(K) = K or, in case of ambiguity, B-closed.
The concept of closed subsets of N was introduced by Wilson (1 972a, b). It is a very important tool which considerably simplified previous constructions of Ranani and other authors.
2.8 Examples. (a) For each K
~
N and A E N, the set B(K, A) is closed.
(b) The sets 6N + {I, 3}, 12N + {I, 4}, 20N + {I, 5} are closed. The next lemma generalises this example. ~
2.9 Lemma. Let K
N be given and define a, f3 by
(2.9.a)
a := gcd{k - 1 : k E K},
(2.9.b)
f3
:= gcd{k(k - 1) : k E K}.
Then
== 0 (mod a), I) == 0 (modf3)
(2.9.c)
A(V - 1)
(2.9.d)
AV(V -
are necessary conditions jor the existence ojan SA (2, K; v). Let L be the set (!f all v E N satisfying these two conditions. Then L is closed. Proof. The necessity of (2.9.c) and (2.9.d)follows by counting the flags (p, B) with P =F C and c, p I B for a given point c, and the triples (x, y, B) with x =F y and x, y lB. Note that Corollary I.2.11 is the special case K = {k}.
§2. Use ojpairwise balanced designs
619
is a closure operator; i.e. it has the properties (2.7.b)
K
~
B(K),
(2.7.c)
K
~
B(L) :::} B(K)
~
B(L),
and hence (2.7.d)
= B(L)
B(B(L»
for all K, L ~ N. A subset K
~
N is called closed if B(K) = K or, in case of ambiguity, B-closed.
The concept of closed subsets of N was introduced by Wilson (1 972a, b). It is a very important tool which considerably simplified previous constructions of Ranani and other authors.
2.8 Examples. (a) For each K
~
N and A E N, the set B(K, A) is closed.
(b) The sets 6N + {I, 3}, 12N + {I, 4}, 20N + {I, 5} are closed. The next lemma generalises this example. ~
2.9 Lemma. Let K
N be given and define a, f3 by
(2.9.a)
a := gcd{k - 1 : k E K},
(2.9.b)
f3
:= gcd{k(k - 1) : k E K}.
Then
== 0 (mod a), I) == 0 (modf3)
(2.9.c)
A(V - 1)
(2.9.d)
AV(V -
are necessary conditions jor the existence ojan SA (2, K; v). Let L be the set (!f all v E N satisfying these two conditions. Then L is closed. Proof. The necessity of (2.9.c) and (2.9.d)follows by counting the flags (p, B) with P =F C and c, p I B for a given point c, and the triples (x, y, B) with x =F y and x, y lB. Note that Corollary I.2.11 is the special case K = {k}.
§3. Applications of divisible designs
621
2.12 Examples. We use Theorem 2.11 and Lemma 2.1 with n = 1, 2. Thus 50 = 7·7 + 1
E
54 = 7·7 + 5
E B({5,
70 = 7·9 + 7
E
57 253
E
B({7, 8}) ~ TD*(7),
7, 8}) ~ TD*(5),
B({7, 8, 9}) ~ TD*(7),
B(8) ~ TD*(8),
E B({16,
13})
~ TD*(11),
cf. (VIII.5.6.b) with q = 4.
2.13 Proposition. (2.13.a) Proof. 6
B({3,4}) = (3 No
+ {l, 3}) \
{6}.
'1. B({3, 4}) follows from (I.8.4.a). "~" now follows from (2.9.c, d).
If g E K := B({3, 4}) and g E TD(4), gl E K U {O}, and gl :::0 g, then 3g + gl E K. This follows with Lemma 2.1, since K is closed. Note that 7 E B(3) c K, and that g E TD(4) if g =1= 2 (mod 4), by (I.7.8.a). The table
yields some small values of K, but leaves the gaps 18, 19, and33. But 19,33 E K by (I.6.11.a), and 18 E K by (1.5. 13.a). Now it is easily seen that each x == 0 or 1 (mod 3) with x > 52 has a representation x = 3g + gl with g, gl == 0 or 1 (mod 3), g =1= 2 (mod 4) and 7 :::0 gl :::0 g. Hence the assertion follows by induction. • The proof would be even shorter if one used result (1.7.9.b).
§3. Applications of Divisible Designs 3.1 Definition. A GD)..[K, G] is a divisible design D with parameter A, block sizes in K ~ N and point class sizes in G, and a GD)..[K] is a GD)..[K, NJ. Without loss of generality, we may assume that the point set is {I, 2, ... , v} c N, and that the (non-empty) point classes are G 1 , G2, ... , G s , where (3.1.a)
x
E
Gi ,
Y E Gj
and
i < j
=}
x < y.
§3. Applications of divisible designs
621
2.12 Examples. We use Theorem 2.11 and Lemma 2.1 with n = 1, 2. Thus 50 = 7·7 + 1
E
54 = 7·7 + 5
E B({5,
70 = 7·9 + 7
E
57 253
E
B({7, 8}) ~ TD*(7),
7, 8}) ~ TD*(5),
B({7, 8, 9}) ~ TD*(7),
B(8) ~ TD*(8),
E B({16,
13})
~ TD*(11),
cf. (VIII.5.6.b) with q = 4.
2.13 Proposition. (2.13.a) Proof. 6
B({3,4}) = (3 No
+ {l, 3}) \
{6}.
'1. B({3, 4}) follows from (I.8.4.a). "~" now follows from (2.9.c, d).
If g E K := B({3, 4}) and g E TD(4), gl E K U {O}, and gl :::0 g, then 3g + gl E K. This follows with Lemma 2.1, since K is closed. Note that 7 E B(3) c K, and that g E TD(4) if g =1= 2 (mod 4), by (I.7.8.a). The table
yields some small values of K, but leaves the gaps 18, 19, and33. But 19,33 E K by (I.6.11.a), and 18 E K by (1.5. 13.a). Now it is easily seen that each x == 0 or 1 (mod 3) with x > 52 has a representation x = 3g + gl with g, gl == 0 or 1 (mod 3), g =1= 2 (mod 4) and 7 :::0 gl :::0 g. Hence the assertion follows by induction. • The proof would be even shorter if one used result (1.7.9.b).
§3. Applications of Divisible Designs 3.1 Definition. A GD)..[K, G] is a divisible design D with parameter A, block sizes in K ~ N and point class sizes in G, and a GD)..[K] is a GD)..[K, NJ. Without loss of generality, we may assume that the point set is {I, 2, ... , v} c N, and that the (non-empty) point classes are G 1 , G2, ... , G s , where (3.1.a)
x
E
Gi ,
Y E Gj
and
i < j
=}
x < y.
623
§3. Applications of divisible designs
where each M(Xi, B) is a W(Xi) x bB-matrix such that M(Xi, B)M(Xi, B)T is diagonal and M (Xi, B)M (X j, Bl = J11 fori =I' j. If W(Xi) = 0, then the matrix M (Xi, B) is omitted. Now replace each I = axB in the incidence matrix (ax, B) of D by the auxiliary matrix M(x, B) defined by (3.2.b), and each 0= axB by a w(x) x bB-zero-matrix. In this case write 00 .. M(x,B)=
(
,0)
........
.
00 ... 0
Then the matrix
describes the desired GD).p.[K]. • 3.3 Remark. We may w.l.o.g. assume that each M (x, B) =I' 0 has a I in the upper left comer. If A. = J1 = 1, then E has a subspace which is isomorphic to D. (Consider only the upper left comers of all auxiliary matrices.) 3.4 Corollary. IfD is an S)..(2, H; v), thenE has class type (W(XI), ... , w(xv»· •
3.5 Corollary. Let D be a GD)..[H, G; v] (see I.6.1) and suppose that mk E GD/L(K, m) for each k E H. Then there is a GD)../L[K, mG; mv], say E. Here mG := {mx : X E G) ~ N.
Proof. Use Theorem 3.2 with w(x) = m for each x E V. Or directly: replace the l's in each column of the incidence matrix of D by the partial matrices Ml, ... , Mk (see Proposition 1.6.2) of the incidence matrix of the respective GD/L[K, m; mk] . •
3.6 Corollary. Let D be a GD)..[K, G; v] and suppose that (3.6.a)
m E TD/L(l)
for I := max K.
Then there is a GD)../L[k, mG; mv].
Proof. Condition (3.6.a) is equivalent to requiring that km GD/L(K, m) for each k E K. Now apply Corollary 3.5 . •
E
GD/L(k, m)
~
623
§3. Applications of divisible designs
where each M(Xi, B) is a W(Xi) x bB-matrix such that M(Xi, B)M(Xi, B)T is diagonal and M (Xi, B)M (X j, Bl = J11 fori =I' j. If W(Xi) = 0, then the matrix M (Xi, B) is omitted. Now replace each I = axB in the incidence matrix (ax, B) of D by the auxiliary matrix M(x, B) defined by (3.2.b), and each 0= axB by a w(x) x bB-zero-matrix. In this case write 00 .. M(x,B)=
(
,0)
........
.
00 ... 0
Then the matrix
describes the desired GD).p.[K]. • 3.3 Remark. We may w.l.o.g. assume that each M (x, B) =I' 0 has a I in the upper left comer. If A. = J1 = 1, then E has a subspace which is isomorphic to D. (Consider only the upper left comers of all auxiliary matrices.) 3.4 Corollary. IfD is an S)..(2, H; v), thenE has class type (W(XI), ... , w(xv»· •
3.5 Corollary. Let D be a GD)..[H, G; v] (see I.6.1) and suppose that mk E GD/L(K, m) for each k E H. Then there is a GD)../L[K, mG; mv], say E. Here mG := {mx : X E G) ~ N.
Proof. Use Theorem 3.2 with w(x) = m for each x E V. Or directly: replace the l's in each column of the incidence matrix of D by the partial matrices Ml, ... , Mk (see Proposition 1.6.2) of the incidence matrix of the respective GD/L[K, m; mk] . •
3.6 Corollary. Let D be a GD)..[K, G; v] and suppose that (3.6.a)
m E TD/L(l)
for I := max K.
Then there is a GD)../L[k, mG; mv].
Proof. Condition (3.6.a) is equivalent to requiring that km GD/L(K, m) for each k E K. Now apply Corollary 3.5 . •
E
GD/L(k, m)
~
§3. Applications of divisible designs
625
Proof. Put H:= R'lI:. w By hypothesis, mH = GD1.JK, m). Now apply Corollary 3.5 . • These theorems and lemmas are fairly general, and hence they need some explanation by examples. The following results are due to Wilson (1972b).
3.10 Corollary. The sets R'lI:,).. are closed. In particular the sets B(K, A) and Rk are closed. Proof. In the Main Lemma take G = {I}, A = 1. Then
By (3.7.b),
3.11 Lemma. Let K '" {I} be a non-empty subset of N. Put (3.1 1. a)
L := B(K, A),
and suppose that the positive integer m has the property
(3.11.b)
x=l(modm)
forallxEL.
Define Y S; No by
(3.11.c)
L
=
B(K, A)
= mY + 1.
Then, with the convention 0 E GD)..(X, Y)for all X, YEN,
(3.11.d)
GD(R'lI:.).., Y)
= Y.
Proof. By the Main Lemma 3.9, m GD(R~.).., Y) = GD)..(K, my) = GD)..(K, B(K, A) - 1).
With (2.6.b) and (3.11.c) the assertion follows, as trivially Y S; GD(R'K.).., Y) .
•
Note that the case m
= 1 is already contained in (2.6.b) and (2.S.a).
§3. Applications of divisible designs
625
Proof. Put H:= R'lI:. w By hypothesis, mH = GD1.JK, m). Now apply Corollary 3.5 . • These theorems and lemmas are fairly general, and hence they need some explanation by examples. The following results are due to Wilson (1972b).
3.10 Corollary. The sets R'lI:,).. are closed. In particular the sets B(K, A) and Rk are closed. Proof. In the Main Lemma take G = {I}, A = 1. Then
By (3.7.b),
3.11 Lemma. Let K '" {I} be a non-empty subset of N. Put (3.1 1. a)
L := B(K, A),
and suppose that the positive integer m has the property
(3.11.b)
x=l(modm)
forallxEL.
Define Y S; No by
(3.11.c)
L
=
B(K, A)
= mY + 1.
Then, with the convention 0 E GD)..(X, Y)for all X, YEN,
(3.11.d)
GD(R'lI:.).., Y)
= Y.
Proof. By the Main Lemma 3.9, m GD(R~.).., Y) = GD)..(K, my) = GD)..(K, B(K, A) - 1).
With (2.6.b) and (3.11.c) the assertion follows, as trivially Y S; GD(R'K.).., Y) .
•
Note that the case m
= 1 is already contained in (2.6.b) and (2.S.a).
627
§4. Applications of Hanani's lemmas
3.15 Notation. Let F K(U) denote the set of positive integers for which there exists an S(2, K; v) with a subspace of order u. For instance h(O) = FK(1) = B(K), Fk(k) = B(k),
h(u)=C/J
ifu
GD(K, u), GD(K,
U -
1) + 1 E h(u).
Doyen and Wilson (1973) proved that (3.15.a)
F3(U) = {u} U {[61'1 + {I, 3}]ru+l
for u
E
61'1 + {I, 3}.
We shall prove this result in Theorem 11.3. Wilson (1972b) calls the following theorem the "Adjunction Theorem".
3.16 Theorem. Let D (3. 16.a)
IG I + d
E
= (V, B, G) be a GD[K] and dEN.
If
h (d) for each point class G of D,
then, for each G E G,
(3.16.b)
IVI + dE h(IGI + d)
<;;; B(K).
Proof. Let U be a set of d new points. If G E G, construct an S (2, K; IG I + d) on G U U, with block set BG, such that U is the point set of a fixed subspace (independent of G). Note that this is always possible, since any subspace of a PBD may be replaced by any other subspace of the same order. Then B + LGEGBG is the block set of the desired S(2, K; IVI + d) . • 3.17 Examples.
+ 4 E B({4, 25)) = B(4), cf. Example VII.3.2.(a), 366 = 6·60 + 6 E F6(66) <;;; B(6), cf. Exercise VIIl.l.l0, 85 = 5·16 + 5 E F 5 (21) <;;; B(5), cf. (1.2. 19.b). 88 = 4·21
§4. Applications of Hanani's Lemmas Most of the following examples are due to Hanani and Wilson.
627
§4. Applications of Hanani's lemmas
3.15 Notation. Let F K(U) denote the set of positive integers for which there exists an S(2, K; v) with a subspace of order u. For instance h(O) = FK(1) = B(K), Fk(k) = B(k),
h(u)=C/J
ifu
GD(K, u), GD(K,
U -
1) + 1 E h(u).
Doyen and Wilson (1973) proved that (3.15.a)
F3(U) = {u} U {[61'1 + {I, 3}]ru+l
for u
E
61'1 + {I, 3}.
We shall prove this result in Theorem 11.3. Wilson (1972b) calls the following theorem the "Adjunction Theorem".
3.16 Theorem. Let D (3. 16.a)
IG I + d
E
= (V, B, G) be a GD[K] and dEN.
If
h (d) for each point class G of D,
then, for each G E G,
(3.16.b)
IVI + dE h(IGI + d)
<;;; B(K).
Proof. Let U be a set of d new points. If G E G, construct an S (2, K; IG I + d) on G U U, with block set BG, such that U is the point set of a fixed subspace (independent of G). Note that this is always possible, since any subspace of a PBD may be replaced by any other subspace of the same order. Then B + LGEGBG is the block set of the desired S(2, K; IVI + d) . • 3.17 Examples.
+ 4 E B({4, 25)) = B(4), cf. Example VII.3.2.(a), 366 = 6·60 + 6 E F6(66) <;;; B(6), cf. Exercise VIIl.l.l0, 85 = 5·16 + 5 E F 5 (21) <;;; B(5), cf. (1.2. 19.b). 88 = 4·21
§4. Applications of Hanani's Lemmas Most of the following examples are due to Hanani and Wilson.
§4. Applications of Hanani 's lemmas
629
(b) From the projective plane of order 8 or the affine plane of order 9 delete at most eight points of an oval (cf. Corollary VIII.5.13) and get (4.3.b) (c) Let g
E
(4.3.c)
TD(9) n Land u,
W E
L U {OJ with u,
W
.::0 g; then, by Lemma 2.1,
7g+U+WEL.
In particular, 58 = 7 . 8 + 1 + I (d) For each m
E N~o
E
L.
there is a decomposition
such that 0::; u, W ::; 63 and u, WE (L U {O}) vention 0 E TD(n) for all n EN.
n TD(8). Here we use the con-
Proof" For even m it is easy to find two odd prime powers u, W ::: 7 with m = u + w, except for m::; 13 where W = 1. For all odd m =1= 71 with 15 ::; m ::; 80 one of the numbers m - 8, m - 16, m - 32, and m - 56 turns out to be an odd prime power W ::: 7. Of course 9 = 8 + 1, 11 = 11 + 0, 13 = 13 + O. Finally 71 = 63 + 8. We give the decomposition up to m = 36. m 7 8 9 10 11 12 13 14 15 16
W 7 0 7 8 9 11 0 u
11 13 7 8 8
0 7 7 8
m 17 18 19 20 21 22 23 24 25 26
W 9 8 9 9 11 8 11 9 13 8 13 9 16 7 13 11 16 9 13 13 u
m 27 28 29 30 31 32 33 34 35 36
u
16 17 16 17 23 16 17 17 19 19
W 11 11 13 13 8 16 16 17 16 17
(e) If gEL
n TD(9), then with m = min{8g, 7g + 80},
(4.3.d)
N~~+7 ~ L,
by Lemma 2.1 and part (d).
§4. Applications of Hanani 's lemmas
629
(b) From the projective plane of order 8 or the affine plane of order 9 delete at most eight points of an oval (cf. Corollary VIII.5.13) and get (4.3.b) (c) Let g
E
(4.3.c)
TD(9) n Land u,
W E
L U {OJ with u,
W
.::0 g; then, by Lemma 2.1,
7g+U+WEL.
In particular, 58 = 7 . 8 + 1 + I (d) For each m
E N~o
E
L.
there is a decomposition
such that 0::; u, W ::; 63 and u, WE (L U {O}) vention 0 E TD(n) for all n EN.
n TD(8). Here we use the con-
Proof" For even m it is easy to find two odd prime powers u, W ::: 7 with m = u + w, except for m::; 13 where W = 1. For all odd m =1= 71 with 15 ::; m ::; 80 one of the numbers m - 8, m - 16, m - 32, and m - 56 turns out to be an odd prime power W ::: 7. Of course 9 = 8 + 1, 11 = 11 + 0, 13 = 13 + O. Finally 71 = 63 + 8. We give the decomposition up to m = 36. m 7 8 9 10 11 12 13 14 15 16
W 7 0 7 8 9 11 0 u
11 13 7 8 8
0 7 7 8
m 17 18 19 20 21 22 23 24 25 26
W 9 8 9 9 11 8 11 9 13 8 13 9 16 7 13 11 16 9 13 13 u
m 27 28 29 30 31 32 33 34 35 36
u
16 17 16 17 23 16 17 17 19 19
W 11 11 13 13 8 16 16 17 16 17
(e) If gEL
n TD(9), then with m = min{8g, 7g + 80},
(4.3.d)
N~~+7 ~ L,
by Lemma 2.1 and part (d).
§4. Applications of Hanani 's lemmas
631
(h) Now assume that (4.3.a) is false and put
n := min(N~ \
L).
Then n > 632, by (4.3.h). Choose g, mEN such that
n=7g+m, 7:::0 m :::0 76, and g E B(PIT), which is possible by (f). Since n :::: 608 we also have g :::: 76 :::: m,
n
1M7 g+ 76 E 1'I7g+7 '
and n E L by (4.3.d),
a contradiction. • As a corollary, we obtain the following theorem on the existence of five MOLS which is essentially due to Hanani (1970). For the small values not covered by the preceding arguments, see Table 7.1 in the appendix; with the exception of 52 and 55, constructions for all these values were given in Chapter VIII.
4.4 Theorem. One has (4.4.a)
P7 U{12, 21, 24, 28, 33, 35, 36}UN:gUN~~ UNnUN~ S; TD(7). 6I
The following result is due to Wilson (1974b) and Brouwer (1979b).
4.5 Proposition. One has (4.5.a)
B({4, 7)) ;2 (3No
+ 1) \
{lO, 19, 31}.
and 10,19 r:f. B{(4, 7)).
Proof. Use notation as in 3.11: K 3Y + 1. The assertion is (4.5.b)
Y
=
{4, 7),
J....
=
I
== No \ {3, 6, IOJ.
By (3.11.d) and (3.7.c), we have (4.5.c)
Ri,
=
GD(Ri, 1)
S;
GD(Ri, Y)
=
Y.
1, m
=
3, L
= B(K) =
§4. Applications of Hanani 's lemmas
631
(h) Now assume that (4.3.a) is false and put
n := min(N~ \
L).
Then n > 632, by (4.3.h). Choose g, mEN such that
n=7g+m, 7:::0 m :::0 76, and g E B(PIT), which is possible by (f). Since n :::: 608 we also have g :::: 76 :::: m,
n
1M7 g+ 76 E 1'I7g+7 '
and n E L by (4.3.d),
a contradiction. • As a corollary, we obtain the following theorem on the existence of five MOLS which is essentially due to Hanani (1970). For the small values not covered by the preceding arguments, see Table 7.1 in the appendix; with the exception of 52 and 55, constructions for all these values were given in Chapter VIII.
4.4 Theorem. One has (4.4.a)
P7 U{12, 21, 24, 28, 33, 35, 36}UN:gUN~~ UNnUN~ S; TD(7). 6I
The following result is due to Wilson (1974b) and Brouwer (1979b).
4.5 Proposition. One has (4.5.a)
B({4, 7)) ;2 (3No
+ 1) \
{lO, 19, 31}.
and 10,19 r:f. B{(4, 7)).
Proof. Use notation as in 3.11: K 3Y + 1. The assertion is (4.5.b)
Y
=
{4, 7),
J....
=
I
== No \ {3, 6, IOJ.
By (3.11.d) and (3.7.c), we have (4.5.c)
Ri,
=
GD(Ri, 1)
S;
GD(Ri, Y)
=
Y.
1, m
=
3, L
= B(K) =
633
§4. Applications of Hanani's lemmas
Using this for g = 4, 5, 7, and 8, together with the previous results, we obtain (4.5.n)
NjQ \ {3, 6, 1O}
c
Y.
Attaching two or seven "infinite" points to a resolvable S(2, 4; 40) yields (4.5.0)
42,47 E GD({4, 5}, {1, 2, 7}) S; Y,
cf. (VII.7.5.a). Similarly, 54,66,78,90 (4.5.p)
E
Y. By (4.5.g),
58,67,70,82 E Y.
Using (4.S.m) with g = 9,11,12,13,16,17, 19,21,23,25,27, 6n± l(n :::: 5) yields the assertion. • Brouwer (1979b) used refined methods to get the following sharper result. If v = 12n + 7 or v = 12n - 2 (n > 1), then there is an S(2, {4, 7}; v) with exactly one line of size 7. In particular, 31 E B({4, 7}).
4.6 Proposition. (4.6.a)
L := B({4, 5, 8, 9, 12}) = 4N + {I, 4}.
Proof. First we note 13, 28 E B(4) S; L; cf. (VII.3.2.a). Also, 29 = 28 + 1 E GD(4, 7)
+ 1 S; B({4, 8}) c
L.
Again we use Lemma 2.1; here (4.6.b)
4g
+ u E L,
if gEL n TD(5), U E L U {O}, and u :'S g. Using this for g = 4,5,8,9, 12, 12n + 1, 12n + 4, 12n + 5, 12n + 8, and 12n + 13 = 12(n + 1) + 1 proves the assertion. •
4.7 Lemma. Let P be the set of prime powers, and put K := Pt. Then (4.7.a)
B(K) = N \ ({2, 3,6, 10, 12, 14, 15, 18} U X),
where X S; {26, 30, 38, 42}.
Proof. Put L := B(K). Again, (4.6.b) holds. Since we already know (4.3.a), only the values in N \ (Pt! U Nti}) are in doubt. By (4.6.b), we get
633
§4. Applications of Hanani's lemmas
Using this for g = 4, 5, 7, and 8, together with the previous results, we obtain (4.5.n)
NjQ \ {3, 6, 1O}
c
Y.
Attaching two or seven "infinite" points to a resolvable S(2, 4; 40) yields (4.5.0)
42,47 E GD({4, 5}, {1, 2, 7}) S; Y,
cf. (VII.7.5.a). Similarly, 54,66,78,90 (4.5.p)
E
Y. By (4.5.g),
58,67,70,82 E Y.
Using (4.S.m) with g = 9,11,12,13,16,17, 19,21,23,25,27, 6n± l(n :::: 5) yields the assertion. • Brouwer (1979b) used refined methods to get the following sharper result. If v = 12n + 7 or v = 12n - 2 (n > 1), then there is an S(2, {4, 7}; v) with exactly one line of size 7. In particular, 31 E B({4, 7}).
4.6 Proposition. (4.6.a)
L := B({4, 5, 8, 9, 12}) = 4N + {I, 4}.
Proof. First we note 13, 28 E B(4) S; L; cf. (VII.3.2.a). Also, 29 = 28 + 1 E GD(4, 7)
+ 1 S; B({4, 8}) c
L.
Again we use Lemma 2.1; here (4.6.b)
4g
+ u E L,
if gEL n TD(5), U E L U {O}, and u :'S g. Using this for g = 4,5,8,9, 12, 12n + 1, 12n + 4, 12n + 5, 12n + 8, and 12n + 13 = 12(n + 1) + 1 proves the assertion. •
4.7 Lemma. Let P be the set of prime powers, and put K := Pt. Then (4.7.a)
B(K) = N \ ({2, 3,6, 10, 12, 14, 15, 18} U X),
where X S; {26, 30, 38, 42}.
Proof. Put L := B(K). Again, (4.6.b) holds. Since we already know (4.3.a), only the values in N \ (Pt! U Nti}) are in doubt. By (4.6.b), we get
§4. Applications of Hanani 's lemmas
635
4.9 Theorem. For every n 9'. {2, 3, 6}, there are two idempotent MOLS of order n, andfor n 9'. {2, 6}, there are two MOLS of order n . • 4.10 Remark. In order to prove Theorem 4.9, one may avoid the cumbersome constructions of self-orthogonal Latin squares given in Example VIll.4.9 by using the MOLS constructed in Examples Vlll.4.17.(b) and (d).
4.11 Remark. There is a wealth of results similar to Propositions 4.1 and 4.2. For instance, Gronau, Mullin and Pietsch (1995) determined the closure of all subsets of (3, 4, ... , lO} which include 3. Other important references include Drake and Larson (1983), Lenz (1984), Colbourn and Ling (1997), Ling, Zhu et a1. (1997) and Mullin, Linget a1. (1997). A big list of closures can be found in Table 1ll.3.18 of Colbourn and Dinitz (1996a). In what follows, we report on some results related to Lemma 4.3 and Theorem 4.4, where we considered the PBD-closure B(L) of the set L = {q: q :::: 7 a prime power}. Lemma 4.3 was rediscovered by Bennett (1987a); it is not too difficult to see that one actually has equality in this result. We now list some results concerning the closure of certain interesting subsets of the set of all prime powers. 4.12 Theorem. Let Pq and OPq denote the set ofprime powers ::::q and of odd prime powers ::::q, respectively. Then one has: (4. 12.a)
B(P3) = N \ {2, 6}
(4.12.b)
B(P4 ) ;2 N \ {2, 3, 6,10, 12, 14, 15, 18,26, 30}
(4.l2.c)
B(Ps) ;2 N \ {2, 3, 6,10, 12, 14, 15, 18,20,22,24,26,28,
(4. 12.d)
B(P7) = p~l U {50, 56, 57, 58} U N6'3
(4. 12.e)
B(OPs ) S; (2N + 1) \ {IS, 33, 39, 51, 75,87,183, 195} . •
30,33,34,38, 39,42,44,46,51, 52, 60}
Here (4.12.a) is immediate from Proposition 4.2, while (4.12.b, d) just restate Lemmas 4.3 and 4.7. Result (4.12.e) essentially is due to Mullin and Stinson (1984), with some improvements by Bennett (1990a) who also proved (4.12.c). Similarly, Bennett (1987b) proved that B(Ps) contains every v :::: 2207, with 675 possible exceptions below this bound. Mullin and Stinson (1990) studied B (0 P7) and proved that this set contains all odd v :::: 2129, with 103 possible exceptions below this bound. These results can be somewhat improved using work of Greig (1998c), see Table III.3.18 in Colbourn and Dinitz (1996a).
§4. Applications of Hanani 's lemmas
635
4.9 Theorem. For every n 9'. {2, 3, 6}, there are two idempotent MOLS of order n, andfor n 9'. {2, 6}, there are two MOLS of order n . • 4.10 Remark. In order to prove Theorem 4.9, one may avoid the cumbersome constructions of self-orthogonal Latin squares given in Example VIll.4.9 by using the MOLS constructed in Examples Vlll.4.17.(b) and (d).
4.11 Remark. There is a wealth of results similar to Propositions 4.1 and 4.2. For instance, Gronau, Mullin and Pietsch (1995) determined the closure of all subsets of (3, 4, ... , lO} which include 3. Other important references include Drake and Larson (1983), Lenz (1984), Colbourn and Ling (1997), Ling, Zhu et a1. (1997) and Mullin, Linget a1. (1997). A big list of closures can be found in Table 1ll.3.18 of Colbourn and Dinitz (1996a). In what follows, we report on some results related to Lemma 4.3 and Theorem 4.4, where we considered the PBD-closure B(L) of the set L = {q: q :::: 7 a prime power}. Lemma 4.3 was rediscovered by Bennett (1987a); it is not too difficult to see that one actually has equality in this result. We now list some results concerning the closure of certain interesting subsets of the set of all prime powers. 4.12 Theorem. Let Pq and OPq denote the set ofprime powers ::::q and of odd prime powers ::::q, respectively. Then one has: (4. 12.a)
B(P3) = N \ {2, 6}
(4.12.b)
B(P4 ) ;2 N \ {2, 3, 6,10, 12, 14, 15, 18,26, 30}
(4.l2.c)
B(Ps) ;2 N \ {2, 3, 6,10, 12, 14, 15, 18,20,22,24,26,28,
(4. 12.d)
B(P7) = p~l U {50, 56, 57, 58} U N6'3
(4. 12.e)
B(OPs ) S; (2N + 1) \ {IS, 33, 39, 51, 75,87,183, 195} . •
30,33,34,38, 39,42,44,46,51, 52, 60}
Here (4.12.a) is immediate from Proposition 4.2, while (4.12.b, d) just restate Lemmas 4.3 and 4.7. Result (4.12.e) essentially is due to Mullin and Stinson (1984), with some improvements by Bennett (1990a) who also proved (4.12.c). Similarly, Bennett (1987b) proved that B(Ps) contains every v :::: 2207, with 675 possible exceptions below this bound. Mullin and Stinson (1990) studied B (0 P7) and proved that this set contains all odd v :::: 2129, with 103 possible exceptions below this bound. These results can be somewhat improved using work of Greig (1998c), see Table III.3.18 in Colbourn and Dinitz (1996a).
§5. Block designs of block size three and four
637
5.3 Lemma. (5.3.a)
B(3,2) = 3No + to, I}.
Proof. Trivially 3, 4
6
E
B(3, 2). By (I.5.14.a),
E B(3, 2).
Since B(3, 2) is closed, the assertion follows with (2.l3.iIl) •
5.4 Lemma. (5.4.a)
B(3, 6) = N \ {2}.
Proof. Obviously B(3, 6) ::2 B(3, 2) U B(3, 3). Hence (5.4.b)
3,4,5,6
E
B(3, 6).
Since B(3, 6) is closed, (4.2.a) yields B(3, 6) ::2 N \ {2, 8}, and the trivial S6(2, 3; 8) takes care of the value 8. •
5.5 Theorem. The necessary existence conditions for an S).(2, 3; v) (Ire sufficient. Proof. As in VII.4.l2. • 5.6 Lemma. (S.6.a)
R4 = 4No + {I, 4}
(S.6.b)
B(4) = 12No + {I, 4}.
or, equivalently,
Proof. We know that 13, 16, 2S, 28, 37 E B(4); cf. (VII.3.2.a). Hence (S.6.c)
4, S, 8, 9, 12 E R4 .
Since R4 is closed, the assertion follows by (4.6.a) . •
5.7 Lemma. (S.7.a)
B(4, 2) = 3No + l.
§5. Block designs of block size three and four
637
5.3 Lemma. (5.3.a)
B(3,2) = 3No + to, I}.
Proof. Trivially 3, 4
6
E
B(3, 2). By (I.5.14.a),
E B(3, 2).
Since B(3, 2) is closed, the assertion follows with (2.l3.iIl) •
5.4 Lemma. (5.4.a)
B(3, 6) = N \ {2}.
Proof. Obviously B(3, 6) ::2 B(3, 2) U B(3, 3). Hence (5.4.b)
3,4,5,6
E
B(3, 6).
Since B(3, 6) is closed, (4.2.a) yields B(3, 6) ::2 N \ {2, 8}, and the trivial S6(2, 3; 8) takes care of the value 8. •
5.5 Theorem. The necessary existence conditions for an S).(2, 3; v) (Ire sufficient. Proof. As in VII.4.l2. • 5.6 Lemma. (S.6.a)
R4 = 4No + {I, 4}
(S.6.b)
B(4) = 12No + {I, 4}.
or, equivalently,
Proof. We know that 13, 16, 2S, 28, 37 E B(4); cf. (VII.3.2.a). Hence (S.6.c)
4, S, 8, 9, 12 E R4 .
Since R4 is closed, the assertion follows by (4.6.a) . •
5.7 Lemma. (S.7.a)
B(4, 2) = 3No + l.
639
§5. Block designs of block size three and four
in Zv-l U (oo} determine an S6(2, 4; v). In particular 6, 12, 14,18,26,30,38,42 By (5.7.b), we know that 10
E
E
B(4, 6).
B(4, 6).
(d) In Z15 put A := to, 2, 3, ll}, B := to, 1,5, 1O}, C := (A, A, A, A, A, B, C) is a (15, 4; 6)-difference family. •
to, 3, 5,
1O}. Then
The preceding results allow us to prove the following theorem due to Hanani (1961, 1975).
5.10 Theorem. The necessary existence conditions v i= 2,3, J..(v -1) =
o(mod 3),
and
J..v(v -1) =
o(mod 12)
for an S')., (2, 4; v) are sufficient.
Proof. Note that the necessary existence conditions for J.. and gcd(J.., 6) are identical. For J.. I 6 the theorem is true by the previous lemmas. Hence it follows in general by (1.5.3.b) . • 5.11 Remark. We emphasise that Hanani's existence theorems 5.5 and 5.10 allow repeated blocks. The existence problem for simple designs is more difficult. We ask the reader to show that the necessary conditions for the existence of a simple design S').,(2, k; v) are as follows: (5.Il.a)
J..(v - 1) =0 (modk -1), J..v(v-I)=O(modk(k-I»
and
J..::sG=~).
By a result of Dehon (1983), these conditions are also sufficient for k = 3.
5.12 Theorem. A simple triple system S),,(2, 3; v) exists if and only if (5.I2.a)
J..(v -1) =0(mod2),
J..v(v -1) =0(mod6)
and
J..::sv - 2. II
Another proof of Theorem 5.12 was given by Sarvate (1986). A simple direct construction (similar to the one given in Theorem VIIA.9, but using four infinite points) for simple S2 (2,3, v)' s was provided by Bhat-Nayak and Kane (1988). We also have the following strengthening of Theorem 5.12, namely an analogue
639
§5. Block designs of block size three and four
in Zv-l U (oo} determine an S6(2, 4; v). In particular 6, 12, 14,18,26,30,38,42 By (5.7.b), we know that 10
E
E
B(4, 6).
B(4, 6).
(d) In Z15 put A := to, 2, 3, ll}, B := to, 1,5, 1O}, C := (A, A, A, A, A, B, C) is a (15, 4; 6)-difference family. •
to, 3, 5,
1O}. Then
The preceding results allow us to prove the following theorem due to Hanani (1961, 1975).
5.10 Theorem. The necessary existence conditions v i= 2,3, J..(v -1) =
o(mod 3),
and
J..v(v -1) =
o(mod 12)
for an S')., (2, 4; v) are sufficient.
Proof. Note that the necessary existence conditions for J.. and gcd(J.., 6) are identical. For J.. I 6 the theorem is true by the previous lemmas. Hence it follows in general by (1.5.3.b) . • 5.11 Remark. We emphasise that Hanani's existence theorems 5.5 and 5.10 allow repeated blocks. The existence problem for simple designs is more difficult. We ask the reader to show that the necessary conditions for the existence of a simple design S').,(2, k; v) are as follows: (5.Il.a)
J..(v - 1) =0 (modk -1), J..v(v-I)=O(modk(k-I»
and
J..::sG=~).
By a result of Dehon (1983), these conditions are also sufficient for k = 3.
5.12 Theorem. A simple triple system S),,(2, 3; v) exists if and only if (5.I2.a)
J..(v -1) =0(mod2),
J..v(v -1) =0(mod6)
and
J..::sv - 2. II
Another proof of Theorem 5.12 was given by Sarvate (1986). A simple direct construction (similar to the one given in Theorem VIIA.9, but using four infinite points) for simple S2 (2,3, v)' s was provided by Bhat-Nayak and Kane (1988). We also have the following strengthening of Theorem 5.12, namely an analogue
§6. Solution of Kirkman's schoolgirl problem
641
(b) A topic which is somewhat related to the construction of simple designs is the existence question for indecomposable designs; here an SA (2, k; v) is called indecomposable if it is not the union of two designs SA (2, k; v) and Sv(2, k; v) with A = J1- + v. Many of the references cited above actually consider indecomposable simple designs. For a construction of indecomposable simple triple systems SA (2, 3; v) for all values.of Aand for all sufficiently large values ofv (given A), see Archdeacon and Dinitz (1993). Griittmiiller (1996) gave computational results on the numbers of indecomposable block designs with small parameters.
§6. Solution of Kirkman's Schoolgirl Problem Ray-Chaudhuri and Wilson (1971) proved RB(3) = 6N - 3, i.e. for every v == 3 (mod 6) there is a Kirkman design, i.e. a resolvable STS. The proof will need several steps.
6.1 Observation. If q (6. 1. a)
2q
==
1 (mod 6) is a prime power, then
+ 1, 3q E RB(3).
This was proved in Theorem VIII.2.1 and Corollary VIll.2.3. In particular, (6.l.b)
15,21,39,63 E RB(3)
and, of course, (6.l.c)
9 E RB(3).
6.2 Notation. (6.2.a)
RZ := {x EN: (k - l)x
+ 1 E RB(k)}.
Ray-Chaudhuri and Wilson (1971) proved that the sets R;; are closed. This enabled them to determine RB(3) and later RB(4) completely; cf. Ranani, RayChaudhuri and Wilson (1972). The sets R;; and RB(k) for k > 4 are unknown. If we know that Rj is closed, then the assertion (6.2.b)
RB(3)
= 6N -
3
follows immediately with (6.1.b, c), which imply 4,7, 10, 19,31 and (4.5.a).
E
Rj,
§6. Solution of Kirkman's schoolgirl problem
641
(b) A topic which is somewhat related to the construction of simple designs is the existence question for indecomposable designs; here an SA (2, k; v) is called indecomposable if it is not the union of two designs SA (2, k; v) and Sv(2, k; v) with A = J1- + v. Many of the references cited above actually consider indecomposable simple designs. For a construction of indecomposable simple triple systems SA (2, 3; v) for all values.of Aand for all sufficiently large values ofv (given A), see Archdeacon and Dinitz (1993). Griittmiiller (1996) gave computational results on the numbers of indecomposable block designs with small parameters.
§6. Solution of Kirkman's Schoolgirl Problem Ray-Chaudhuri and Wilson (1971) proved RB(3) = 6N - 3, i.e. for every v == 3 (mod 6) there is a Kirkman design, i.e. a resolvable STS. The proof will need several steps.
6.1 Observation. If q (6. 1. a)
2q
==
1 (mod 6) is a prime power, then
+ 1, 3q E RB(3).
This was proved in Theorem VIII.2.1 and Corollary VIll.2.3. In particular, (6.l.b)
15,21,39,63 E RB(3)
and, of course, (6.l.c)
9 E RB(3).
6.2 Notation. (6.2.a)
RZ := {x EN: (k - l)x
+ 1 E RB(k)}.
Ray-Chaudhuri and Wilson (1971) proved that the sets R;; are closed. This enabled them to determine RB(3) and later RB(4) completely; cf. Ranani, RayChaudhuri and Wilson (1972). The sets R;; and RB(k) for k > 4 are unknown. If we know that Rj is closed, then the assertion (6.2.b)
RB(3)
= 6N -
3
follows immediately with (6.1.b, c), which imply 4,7, 10, 19,31 and (4.5.a).
E
Rj,
§6. Solution of Kirkman's schoolgirl problem
643
Here the k-row submatrices 1 (6.4.b)
o
(i = 1, ... , r)
M' I
o satisfy the equations Mi MJ = J for i =I j, see 1.6.2.
= (aij) be the incidence matrix of an S(2, Rk;v) with block sizes (j = 1, ... , b). For any column with rj =: r entries 1 we take the submatrices M; of (6.4.a) as auxiliary matrices, that is we replace these l's by
Now let A rj E
Rk
the matrices M; in any order, and the O's in the same column by zero matrices of the same size. We get a vk-row large matrix N' which splits into k-row matrices N;, ... , N~; that is
....~.; ..... N='~ . .. ...... ... .
N'v Now put the column (1 0 0 ...
of in front of each submatrix N; and get
Ni=UQ~ :
N[
fori=I, ... ,v,
o
N~6d Then N is the incidence matrix of a GD[{k + 1, v}, k; kv] which actually contains lines of sizes v and k + 1. The reader is asked to check this. Hence, by Lemma 6.3, v E Rk, and Lemma 6.4 is proved. • As noted before, this finishes the proof of the following theorem of RayChaudhuri and Wilson (1971).
§6. Solution of Kirkman's schoolgirl problem
643
Here the k-row submatrices 1 (6.4.b)
o
(i = 1, ... , r)
M' I
o satisfy the equations Mi MJ = J for i =I j, see 1.6.2.
= (aij) be the incidence matrix of an S(2, Rk;v) with block sizes (j = 1, ... , b). For any column with rj =: r entries 1 we take the submatrices M; of (6.4.a) as auxiliary matrices, that is we replace these l's by
Now let A rj E
Rk
the matrices M; in any order, and the O's in the same column by zero matrices of the same size. We get a vk-row large matrix N' which splits into k-row matrices N;, ... , N~; that is
....~.; ..... N='~ . .. ...... ... .
N'v Now put the column (1 0 0 ...
of in front of each submatrix N; and get
Ni=UQ~ :
N[
fori=I, ... ,v,
o
N~6d Then N is the incidence matrix of a GD[{k + 1, v}, k; kv] which actually contains lines of sizes v and k + 1. The reader is asked to check this. Hence, by Lemma 6.3, v E Rk, and Lemma 6.4 is proved. • As noted before, this finishes the proof of the following theorem of RayChaudhuri and Wilson (1971).
§7. The basis of a closed set
645
7.1 Problem and Examples. For a given closed set L <; 1"1 with L \ {I} # 0, find a minimal subset K <; L such that B(K) = L. As examples we already know: (a) L = 2N - 1, K = {3, 5l. by (4.l.a). (b) L = N \ {2}, K = {3, 4,5,6, 8), by (4.2.a), L = N \ {2, 6}, K = {3, 4, 5, 8l. (c) L = 3 No + 1, K = {4, 7, 10, 19}, by (4.5.a). L = (31"10 + 1) \ (10, 19l, K = {4, 7}. (d) L = 41"10 + {l, 4l, K = {4, 5, 8, 9, 12}, by (4.6.a). (e) L = N~, K = N!2 U (14, 15, 18, 19, 23} [S.9.(c) and exercise]. (f) L = 3No + {I, 3l, K = {3,4, 6} or L = (31"10 + (1, 3}) \ {6l, K = {3, 4l, by (2. 13.a).
In most cases it is difficult to determine K exactly but nevertheless the knowledge of a comparatively small finite set K' with K C K' C L will be useful. Even this may be difficult, e.g. for L = [21"1 + 1]5". 7.2 Lemma. Let L be a closed subset of N. Then there exists a unique minimal subset K of L satisfying B(K) = L. Proof. We put kJ := min(L \ {I}) and, recursively, kn+l := min(L \ B({k 1 ,
••• ,
knD,
provided that the set on the right-hand side is not empty; otherwise the sequence ends with kn.1t is easily seen that K = {k J , ••• , kn } is the desired set. •
7.3 Remark. The set K described in Lemma 7.2 is called the basis (or the minimal generating set) of L. The subsets M of 1"1 with B(M) = L are precisely the supersets of K contained in L; each such set is called a generating set for L. The elements of K are also called the essential elements in any generating set. Note that v rf. K, whenever v E B(L \ {v}). A famous (and difficult) result due to Wilson (1972c) shows that every closed set L is infinite and that the basis of L is always finite; this will be proved in §XI.5.
In what follows, L will always denote a closed set with basis K. We shall present a method of Hanani and Wilson which either allows the determination of the basis K or at least of a "small" generating set in practical cases. In the following examples, we will be satisfied with the latter (for the sake of simplicity); some stronger results will be discussed at the end of this section.
§7. The basis of a closed set
645
7.1 Problem and Examples. For a given closed set L <; 1"1 with L \ {I} # 0, find a minimal subset K <; L such that B(K) = L. As examples we already know: (a) L = 2N - 1, K = {3, 5l. by (4.l.a). (b) L = N \ {2}, K = {3, 4,5,6, 8), by (4.2.a), L = N \ {2, 6}, K = {3, 4, 5, 8l. (c) L = 3 No + 1, K = {4, 7, 10, 19}, by (4.5.a). L = (31"10 + 1) \ (10, 19l, K = {4, 7}. (d) L = 41"10 + {l, 4l, K = {4, 5, 8, 9, 12}, by (4.6.a). (e) L = N~, K = N!2 U (14, 15, 18, 19, 23} [S.9.(c) and exercise]. (f) L = 3No + {I, 3l, K = {3,4, 6} or L = (31"10 + (1, 3}) \ {6l, K = {3, 4l, by (2. 13.a).
In most cases it is difficult to determine K exactly but nevertheless the knowledge of a comparatively small finite set K' with K C K' C L will be useful. Even this may be difficult, e.g. for L = [21"1 + 1]5". 7.2 Lemma. Let L be a closed subset of N. Then there exists a unique minimal subset K of L satisfying B(K) = L. Proof. We put kJ := min(L \ {I}) and, recursively, kn+l := min(L \ B({k 1 ,
••• ,
knD,
provided that the set on the right-hand side is not empty; otherwise the sequence ends with kn.1t is easily seen that K = {k J , ••• , kn } is the desired set. •
7.3 Remark. The set K described in Lemma 7.2 is called the basis (or the minimal generating set) of L. The subsets M of 1"1 with B(M) = L are precisely the supersets of K contained in L; each such set is called a generating set for L. The elements of K are also called the essential elements in any generating set. Note that v rf. K, whenever v E B(L \ {v}). A famous (and difficult) result due to Wilson (1972c) shows that every closed set L is infinite and that the basis of L is always finite; this will be proved in §XI.5.
In what follows, L will always denote a closed set with basis K. We shall present a method of Hanani and Wilson which either allows the determination of the basis K or at least of a "small" generating set in practical cases. In the following examples, we will be satisfied with the latter (for the sake of simplicity); some stronger results will be discussed at the end of this section.
647
§7. The basis of a closed set
Completing a resolvable S(2, 4; v), i.e. introducing a new point on each parallel class and joining the new points by a line, yields an S(2, {S, (v - 1)/3}; (4v - 1)/3);
hence 37, S3, 69,133
f/ K and 9,13,17,33 f/ N.
By Proposition VIII.S.4 with g = 4, we have 77 f/ K, 19 f/ N. Hence N Nt U {7, 8,12,14,22,23,24,32, 34}. Finally 24 = 2S - 1 E GD(S, 4)
~
C
GD(R5, N) f/ N,
by Lemma 7.4. Thus we have obtained the following result due to Wilson (1974b): (7.S.f)
4No +l=B(K)
with (7.S.g)
K
~
{S, 9,13,17,29,33,49,57,89,93,129, 137}.
We do not assert that equality holds in (7.5.g) if K is defined as in Problem 7.1, see (7.I4.c). The next example is likewise due to Wilson (l974b).
7.6 Example. (7.6.a)
L
= S No + {I, 5}.
Here Lemma 7.4 is useless; only Remark 7.3 is helpful. By Lemma 2.1, (7.6.b)
Sg
+ u f/ K
if g E TD(6) n L, 0 :::: u :::: g and u E L U {OJ. In particular, using previous knowledge on 4 MOLS (check this!),
30 L66 L 90 L96 L I26 L I50 LI86 L210 L 246 L270 L 25' 55' 75' 80' 105' 125' 155' 175' 205' 225'
L~~g, Ljgg, L~: ~ L \ K Similarly, using Lemma 2.2, (7.6.c)
Sg+u+lf/K,
for
all g
E
Lrs.
647
§7. The basis of a closed set
Completing a resolvable S(2, 4; v), i.e. introducing a new point on each parallel class and joining the new points by a line, yields an S(2, {S, (v - 1)/3}; (4v - 1)/3);
hence 37, S3, 69,133
f/ K and 9,13,17,33 f/ N.
By Proposition VIII.S.4 with g = 4, we have 77 f/ K, 19 f/ N. Hence N Nt U {7, 8,12,14,22,23,24,32, 34}. Finally 24 = 2S - 1 E GD(S, 4)
~
C
GD(R5, N) f/ N,
by Lemma 7.4. Thus we have obtained the following result due to Wilson (1974b): (7.S.f)
4No +l=B(K)
with (7.S.g)
K
~
{S, 9,13,17,29,33,49,57,89,93,129, 137}.
We do not assert that equality holds in (7.5.g) if K is defined as in Problem 7.1, see (7.I4.c). The next example is likewise due to Wilson (l974b).
7.6 Example. (7.6.a)
L
= S No + {I, 5}.
Here Lemma 7.4 is useless; only Remark 7.3 is helpful. By Lemma 2.1, (7.6.b)
Sg
+ u f/ K
if g E TD(6) n L, 0 :::: u :::: g and u E L U {OJ. In particular, using previous knowledge on 4 MOLS (check this!),
30 L66 L 90 L96 L I26 L I50 LI86 L210 L 246 L270 L 25' 55' 75' 80' 105' 125' 155' 175' 205' 225'
L~~g, Ljgg, L~: ~ L \ K Similarly, using Lemma 2.2, (7.6.c)
Sg+u+lf/K,
for
all g
E
Lrs.
§7. The basis of a closed set
649
This implies the following result due to Hanani, Ray-Chaudhuri and Wilson (1972). 7.10 Corollary. RB(4) :2 (12No + 4) \ {172, 280, 388}. Proof.
R4 is closed by Lemma 6.4. Use Remark VII.7.5 and (VIII.2.4.a) . •
In fact, Hanani, Ray-Chaudhuri and Wilson also constructed resolvable designs 8(2,4; v) for v = 172,280,388. This also follows in a rather simple way from the results of Lamken, Mills and Wilson (1991) who exhibited PBD's on 57, 93 and 129 points with block sizes 5, 9, 13 and 29, thus removing the exceptions in (7.9.b). (We note in passing that the purported construction of an S(2, (5, 29); 129) given by Wang (1978) is incorrect.) Hence we actually have the following result. 7.11 Theorem. RB(4)
=
12No + 4. •
7.12 Exercises. (a) In GF(97) with CtJ = 14, CtJ2 = 2 consider the base block A = {t, 16,25, 58}. Check that the differences in t.A are evenly distributed over H3, Hf,··., Ht (see Definition VII.6.2), and that 1, 16,25,58 are in four distinct co sets of H6 = H3. Hence the pairwise disjoint eight base blocks A . 8; (i = 0, ... , 7) form a (97, 4, I)-difference family. Now Theorem VIII.2.2 yields (7J2.a)
4·97 = 388
E
RB(4).
(b) Use (4.3.a) and Lemma 2.1 to show (7. 12.b)
B({6}
UPs"") :2 Nf \ {lO, 12, 14, 15, 18,20,22,24, 28, 33, 34, 39}.
By now, many results regarding the bases (or at least small generating sets) of some interesting closed sets L have been obtained. Of course, every result giving the closure of some set can be interpreted in this way; for a wealth of such results, see §III.3.2 in Colbourn and Dinitz (1996a). We will close this section by listing some particularly interesting results concerning sets which are either of the form N \ S for a "small" set 8 or of the form eN + 1 or eN + to, I} for some positive integer e. The results in the following theorem are due to Hanani (1975) and to Ling and Colbourn (1996).
§7. The basis of a closed set
649
This implies the following result due to Hanani, Ray-Chaudhuri and Wilson (1972). 7.10 Corollary. RB(4) :2 (12No + 4) \ {172, 280, 388}. Proof.
R4 is closed by Lemma 6.4. Use Remark VII.7.5 and (VIII.2.4.a) . •
In fact, Hanani, Ray-Chaudhuri and Wilson also constructed resolvable designs 8(2,4; v) for v = 172,280,388. This also follows in a rather simple way from the results of Lamken, Mills and Wilson (1991) who exhibited PBD's on 57, 93 and 129 points with block sizes 5, 9, 13 and 29, thus removing the exceptions in (7.9.b). (We note in passing that the purported construction of an S(2, (5, 29); 129) given by Wang (1978) is incorrect.) Hence we actually have the following result. 7.11 Theorem. RB(4)
=
12No + 4. •
7.12 Exercises. (a) In GF(97) with CtJ = 14, CtJ2 = 2 consider the base block A = {t, 16,25, 58}. Check that the differences in t.A are evenly distributed over H3, Hf,··., Ht (see Definition VII.6.2), and that 1, 16,25,58 are in four distinct co sets of H6 = H3. Hence the pairwise disjoint eight base blocks A . 8; (i = 0, ... , 7) form a (97, 4, I)-difference family. Now Theorem VIII.2.2 yields (7J2.a)
4·97 = 388
E
RB(4).
(b) Use (4.3.a) and Lemma 2.1 to show (7. 12.b)
B({6}
UPs"") :2 Nf \ {lO, 12, 14, 15, 18,20,22,24, 28, 33, 34, 39}.
By now, many results regarding the bases (or at least small generating sets) of some interesting closed sets L have been obtained. Of course, every result giving the closure of some set can be interpreted in this way; for a wealth of such results, see §III.3.2 in Colbourn and Dinitz (1996a). We will close this section by listing some particularly interesting results concerning sets which are either of the form N \ S for a "small" set 8 or of the form eN + 1 or eN + to, I} for some positive integer e. The results in the following theorem are due to Hanani (1975) and to Ling and Colbourn (1996).
§8. Block designs with block size five
651
the reader to Table 111.3.19 of Co1boum and Dinitz (1996a, 1998). For the case k = 5, an improvement to the result stated there was obtained by Abel and Greig (1997) who showed that the following positive integers comprise a generating set for N 1 mod 5; see also Co1boum and Dinitz (1997). 611 15 162126364146515661 71 86 101116 191 236286291311321
§8. Block Designs with Block Size Five We shall prove Hanani's result that the necessary conditions for the existence of an S)..(2, 5; v) with (v, A) i= (15, 2) are sufficient, see Hanani (1972, 1975). The proof needs special constructions and refined recursive techniques. The case A = 1 is covered by Corollary 7.8.
8.1 Observation. If q particular (8.l.a)
==
1 (mod 4) is a prime power, then q E B(5, 5). In
5,9, 13, 17,29,49, 89, 137 E B(5, 5).
Proof. See Theorem VII.5.2 . •
8.2 Lemma. If q 2: 5 is an odd prime power, then (8.2.a)
4q E GDs(5, 4),
(8.2.b)
4q
+ 1 E B(5, 5).
In particular, (8.2.c)
93 E B(5, 5).
Proof. Let GF(4) = {O, 1, s, w q - 2 }. In V = GF(4)
S2}
with
S2
= s + 1 and GF(q) = {O, 1, w, ... ,
ED GF(q) take the following base blocks (with d =
(q - 1)/2):
An := {CO, 0), (1, wn), (1, _wn), (s, wn+l), (s, _wn+ 1)}, Bn := {(O, 0), (0, wn), (0, _wn), (1, wn+l), (s, _wn+ 1)},
for n = 0, ... , d - 1. It is easily checked that all differences (x, y) with y i= 0 occur equally often; hence the 2d base blocks determine a GD}.[5, 4; 4q] with b = 2d· 4q, v = 4q, and class size g = 4. Hence A = 5, by (I.6.13.b) . •
§8. Block designs with block size five
651
the reader to Table 111.3.19 of Co1boum and Dinitz (1996a, 1998). For the case k = 5, an improvement to the result stated there was obtained by Abel and Greig (1997) who showed that the following positive integers comprise a generating set for N 1 mod 5; see also Co1boum and Dinitz (1997). 611 15 162126364146515661 71 86 101116 191 236286291311321
§8. Block Designs with Block Size Five We shall prove Hanani's result that the necessary conditions for the existence of an S)..(2, 5; v) with (v, A) i= (15, 2) are sufficient, see Hanani (1972, 1975). The proof needs special constructions and refined recursive techniques. The case A = 1 is covered by Corollary 7.8.
8.1 Observation. If q particular (8.l.a)
==
1 (mod 4) is a prime power, then q E B(5, 5). In
5,9, 13, 17,29,49, 89, 137 E B(5, 5).
Proof. See Theorem VII.5.2 . •
8.2 Lemma. If q 2: 5 is an odd prime power, then (8.2.a)
4q E GDs(5, 4),
(8.2.b)
4q
+ 1 E B(5, 5).
In particular, (8.2.c)
93 E B(5, 5).
Proof. Let GF(4) = {O, 1, s, w q - 2 }. In V = GF(4)
S2}
with
S2
= s + 1 and GF(q) = {O, 1, w, ... ,
ED GF(q) take the following base blocks (with d =
(q - 1)/2):
An := {CO, 0), (1, wn), (1, _wn), (s, wn+l), (s, _wn+ 1)}, Bn := {(O, 0), (0, wn), (0, _wn), (1, wn+l), (s, _wn+ 1)},
for n = 0, ... , d - 1. It is easily checked that all differences (x, y) with y i= 0 occur equally often; hence the 2d base blocks determine a GD}.[5, 4; 4q] with b = 2d· 4q, v = 4q, and class size g = 4. Hence A = 5, by (I.6.13.b) . •
653
§8. Block designs with block size five
8.6 Examples. (a) The trivial S4(2, 5; 5) and S4(2, 5; 6) yield (S.6.a)
5,6
E
B(5, 4).
By Theorems VII.5.2 and VII.7.3, (S.6.b)
11,16,31
(S.6.c)
35 E GD2 (5, 5) ~ GD4 (5, 5) ~ B(5, 4).
E
B(5, 4),
(b) The residual design (see Exercise IT.S.13) of the Paley-Hadamard design S4(2,9; 19) (1.9.11) yields (S.6.d)
10 E B(5, 4).
(c) In V:= 219 U {(X)} the base blocks A={oo,0,1,7,11},
B={1,4,7,9, 16},7B,
and
7 2 =llB
determine an S4(2, 5; 20). Hence (S.6.e)
20 E B(5,4).
(d) In order to prove (S.6.f)
15
E
B(5,4),
Hanani (1975) exhibited thefollowing ten base blocks mod 5 in Zs x {O, 1, 2} : {OJ, 2j , 3j , 0j+1, 4j +2}, j mod3, {OJ, I j , 0j+1, 2j+1' OJ+2}, j mod3, {Oo, 21, 3 1, 4 1, 12},
{Oo, 10, 21 , 32 , 42 }, {Oi,l i ,2i ,3i ,4i }, i
E
{0,2}.
Check the pure and mixed differences.
8.7 Lemma. (S.7.a)
40 E GD4 (5, 5)
~
B(5,4).
Proof. Letev be a generator of (GF(S)*, .) with ev3 = ev + 1 and V
= GF(5) EB
GF(S). Hanani's seven base blocks of a GDD in (V, +) are
{CO, 0), (1, eva), (1, eva+ I ), (4, eva+2 ), (4, eva+3 )}
with IX ENg . •
653
§8. Block designs with block size five
8.6 Examples. (a) The trivial S4(2, 5; 5) and S4(2, 5; 6) yield (S.6.a)
5,6
E
B(5, 4).
By Theorems VII.5.2 and VII.7.3, (S.6.b)
11,16,31
(S.6.c)
35 E GD2 (5, 5) ~ GD4 (5, 5) ~ B(5, 4).
E
B(5, 4),
(b) The residual design (see Exercise IT.S.13) of the Paley-Hadamard design S4(2,9; 19) (1.9.11) yields (S.6.d)
10 E B(5, 4).
(c) In V:= 219 U {(X)} the base blocks A={oo,0,1,7,11},
B={1,4,7,9, 16},7B,
and
7 2 =llB
determine an S4(2, 5; 20). Hence (S.6.e)
20 E B(5,4).
(d) In order to prove (S.6.f)
15
E
B(5,4),
Hanani (1975) exhibited thefollowing ten base blocks mod 5 in Zs x {O, 1, 2} : {OJ, 2j , 3j , 0j+1, 4j +2}, j mod3, {OJ, I j , 0j+1, 2j+1' OJ+2}, j mod3, {Oo, 21, 3 1, 4 1, 12},
{Oo, 10, 21 , 32 , 42 }, {Oi,l i ,2i ,3i ,4i }, i
E
{0,2}.
Check the pure and mixed differences.
8.7 Lemma. (S.7.a)
40 E GD4 (5, 5)
~
B(5,4).
Proof. Letev be a generator of (GF(S)*, .) with ev3 = ev + 1 and V
= GF(5) EB
GF(S). Hanani's seven base blocks of a GDD in (V, +) are
{CO, 0), (1, eva), (1, eva+ I ), (4, eva+2 ), (4, eva+3 )}
with IX ENg . •
655
§8. Block designs with block size five Then, by (3.ll.d), (8.lO.c)
GD(R;.IO' Y) = Y.
(d) Ifv == 1 or 5 (mod 6), then vEL, cf. Lemma VII.7.LMoreover, by Lemma 8.5, 4N + 1 <; L. Hence (8.1O.d)
Y ;:2 No \ (6No + 1).
8.11 Lemma. Ilu, (8.1 1. a)
If g
E
5g
W,
+u E
g
E
Y such that u,
W
:S g and g
E
TD(6) \ {I}, then
Y.
TD(7), then
(8.11.b)
5g + u
+ wE Y.
Proof. By Lemma 2.1, we obtain 5g
+ u + W E GD({5, 6, 7}, y) <; GD(R~.to, Y)
= Y. •
8.12 Lemma. (8.12.a)
B(5, 10)
= [2N + l]f.
Proof. (a) By (8.1O.d), is suffices to show (8.12.b)
6n + 1 E Y for all n
E
N.
First we consider some small values of n. (b) In ZI5 the base blocks to, 3, 6, 9, 12}, to, 1,2,4, 8}, to, 1,2,9, 13}, to, 1,4,5, 1O} and to, 3, 5, 10, 12} generate an S6(2, 5; 15). Hence 15 E B(5, 4) n B(5, 6) <; B(5, 10).
In GF(27) choose A = to, a, b, c, d} arbitrarily. Then the 13 base blocks A . x 2 (x =j:. 0) form a (27, 5, lO)-difference family. In the ring ZI3 EB Z3 the 19 base blocks {lo, 30 , 90 , 4 1 , 12d . Yt {lo, 30 , 90, 01, ~} . Yl too, 10, 120, It, 12d . Yl
(y
= 1, ... , 12)
(Y = 1,2,4,8), (y = 1,3,4)
655
§8. Block designs with block size five Then, by (3.ll.d), (8.lO.c)
GD(R;.IO' Y) = Y.
(d) Ifv == 1 or 5 (mod 6), then vEL, cf. Lemma VII.7.LMoreover, by Lemma 8.5, 4N + 1 <; L. Hence (8.1O.d)
Y ;:2 No \ (6No + 1).
8.11 Lemma. Ilu, (8.1 1. a)
If g
E
5g
W,
+u E
g
E
Y such that u,
W
:S g and g
E
TD(6) \ {I}, then
Y.
TD(7), then
(8.11.b)
5g + u
+ wE Y.
Proof. By Lemma 2.1, we obtain 5g
+ u + W E GD({5, 6, 7}, y) <; GD(R~.to, Y)
= Y. •
8.12 Lemma. (8.12.a)
B(5, 10)
= [2N + l]f.
Proof. (a) By (8.1O.d), is suffices to show (8.12.b)
6n + 1 E Y for all n
E
N.
First we consider some small values of n. (b) In ZI5 the base blocks to, 3, 6, 9, 12}, to, 1,2,4, 8}, to, 1,2,9, 13}, to, 1,4,5, 1O} and to, 3, 5, 10, 12} generate an S6(2, 5; 15). Hence 15 E B(5, 4) n B(5, 6) <; B(5, 10).
In GF(27) choose A = to, a, b, c, d} arbitrarily. Then the 13 base blocks A . x 2 (x =j:. 0) form a (27, 5, lO)-difference family. In the ring ZI3 EB Z3 the 19 base blocks {lo, 30 , 90 , 4 1 , 12d . Yt {lo, 30 , 90, 01, ~} . Yl too, 10, 120, It, 12d . Yl
(y
= 1, ... , 12)
(Y = 1,2,4,8), (y = 1,3,4)
657
§8. Block designs with block size five Hence we already know the following initial values of Y. (S.13.h)
2,5,10,12,15,17,20,22,27,30,32,35,40,47,50, 52,57,60,62,65,67.
(b) By Example (VTII.3.5.b) and Corollary VIII.3.S, we obtain 2 hence (8.13.i)
5
E
Rl
E
TD2(5),
2•
Moreover, 6 E R~.2' cf. (S.lO.b). By (3.11.d), we conclude that (S.ll.a) holds again, and we obtain more initial values of Y, e.g. 25 = 5 . 5 + 0,
70
= 5 . 12 + 10,
SO = 5 . 15 + 5.
Thus the present proof does not need the knowledge of 141,161
E
R(5).
(c) By (3.ll.d), (S.13.i), and Theorem VII.7.2, we obtain 45
E
GD(5, 5)
c
Y.
(d) Now let y E (5No U {O, 2}) \ {7, 37, 55}.
By (8.13.d, e), it suffices to assume that (S.13.j)
y
== 5 (mod 10) or y == 7, 37 (mod 90).
(e) Case 1, y == 5 (mod 10). Then there is an m Y - 50m E {5, 15,25,35, 45}.
E
Z with y - 50m - 10,
If 45 < 10m + 2 E TD(6) or 45 < 10m E TD(6), then (S.ll.a) implies y E Y.
There remain a few exceptional values y = IOn + 5,0 < n < 36. Now choose g= 12,15,17,25,27,32,35,40,45,50,60,70
in (S.ll.a). This settles all the exceptional values of case 1, except for 105 and 115. By (S.13.g), 105 E Y. In order to show that 115 E Y, 231 E B(5, 2), note that 55 = 5·11 E GD(5, 11), i.e. 5 E RJI. As RJI is closed, 21 E RJI, i.e. 11 ·21 E GD(5, 11),231 E R({5, ll}) ~ R(5, 2).
657
§8. Block designs with block size five Hence we already know the following initial values of Y. (S.13.h)
2,5,10,12,15,17,20,22,27,30,32,35,40,47,50, 52,57,60,62,65,67.
(b) By Example (VTII.3.5.b) and Corollary VIII.3.S, we obtain 2 hence (8.13.i)
5
E
Rl
E
TD2(5),
2•
Moreover, 6 E R~.2' cf. (S.lO.b). By (3.11.d), we conclude that (S.ll.a) holds again, and we obtain more initial values of Y, e.g. 25 = 5 . 5 + 0,
70
= 5 . 12 + 10,
SO = 5 . 15 + 5.
Thus the present proof does not need the knowledge of 141,161
E
R(5).
(c) By (3.ll.d), (S.13.i), and Theorem VII.7.2, we obtain 45
E
GD(5, 5)
c
Y.
(d) Now let y E (5No U {O, 2}) \ {7, 37, 55}.
By (8.13.d, e), it suffices to assume that (S.13.j)
y
== 5 (mod 10) or y == 7, 37 (mod 90).
(e) Case 1, y == 5 (mod 10). Then there is an m Y - 50m E {5, 15,25,35, 45}.
E
Z with y - 50m - 10,
If 45 < 10m + 2 E TD(6) or 45 < 10m E TD(6), then (S.ll.a) implies y E Y.
There remain a few exceptional values y = IOn + 5,0 < n < 36. Now choose g= 12,15,17,25,27,32,35,40,45,50,60,70
in (S.ll.a). This settles all the exceptional values of case 1, except for 105 and 115. By (S.13.g), 105 E Y. In order to show that 115 E Y, 231 E B(5, 2), note that 55 = 5·11 E GD(5, 11), i.e. 5 E RJI. As RJI is closed, 21 E RJI, i.e. 11 ·21 E GD(5, 11),231 E R({5, ll}) ~ R(5, 2).
659
§8. Block designs with block size five
matrix form a (111,5, 2)-difference family. 00 00 30 60 10 70 140 320 28 1 191 33 0 282 192 35 1
[%
00 100 130 23 0 301
00 00 00 00 00 I, 81 101 6 1 110 180 36 1 29, 271 31 1 290 82 102 62 112 331 29 2 272 31 2 262
00 111 141 26 1 23 1 . 142 12 23 2 362
%)
The following result is due to Mullin and Stanton (1968).
8.16 Lemma. Ifv then (8.I6.a)
v
+1E
E B(k,,l.,)
n B(k + 1, r
-,l.,) with r = ,l.,(v - I)/(k - 1),
B(k + 1, r).
Proof. Let(V,A) bean SA (2, k; v) and (V, B) anSr - A (2, k + 1; v). On V U too} define the list of blocks C by AU too} E C if A EA and BEe if B E B . • 8.17 Examples. (a) 11 Hence (8.I7.a)
E B(4,
12 E B(5, 20).
(b) 13 E B(4, 1) S; B(4, 5), r Hence (8.I7.b)
6), r = 20, 11 E B(5, 2) S; B(5, 14).
= 20, 13 E
B(5, 5) S; B(5, 15).
14 E B(5, 20).
(c) 21 E B(4, 3), r = 20,21 E B(5, 1) S; B(5, 17); hence (8.I7.c)
22
E
(5,20).
8.18 Constructions. (a) In Z17 U too} the base blocks
too, 0, 9,14, 16}, {oo,O, 1,3, 8}, too, 0, 4, 9, 1O}, too, 0, 2, 5, I3}, {00,0, 1, 11, I5}, to, 7,11,13, I6} generate an incidence structure A with exactly five blocks through any two points of Z17 and with 20 blocks through 00 and any point of Z17. Let B be an SI5(2, 5; 17) on Z17, which exists by (S.l.a). Then A + B is the desired S20(2, 5; 18). The next three examples are exercises for the reader.
659
§8. Block designs with block size five
matrix form a (111,5, 2)-difference family. 00 00 30 60 10 70 140 320 28 1 191 33 0 282 192 35 1
[%
00 100 130 23 0 301
00 00 00 00 00 I, 81 101 6 1 110 180 36 1 29, 271 31 1 290 82 102 62 112 331 29 2 272 31 2 262
00 111 141 26 1 23 1 . 142 12 23 2 362
%)
The following result is due to Mullin and Stanton (1968).
8.16 Lemma. Ifv then (8.I6.a)
v
+1E
E B(k,,l.,)
n B(k + 1, r
-,l.,) with r = ,l.,(v - I)/(k - 1),
B(k + 1, r).
Proof. Let(V,A) bean SA (2, k; v) and (V, B) anSr - A (2, k + 1; v). On V U too} define the list of blocks C by AU too} E C if A EA and BEe if B E B . • 8.17 Examples. (a) 11 Hence (8.I7.a)
E B(4,
12 E B(5, 20).
(b) 13 E B(4, 1) S; B(4, 5), r Hence (8.I7.b)
6), r = 20, 11 E B(5, 2) S; B(5, 14).
= 20, 13 E
B(5, 5) S; B(5, 15).
14 E B(5, 20).
(c) 21 E B(4, 3), r = 20,21 E B(5, 1) S; B(5, 17); hence (8.I7.c)
22
E
(5,20).
8.18 Constructions. (a) In Z17 U too} the base blocks
too, 0, 9,14, 16}, {oo,O, 1,3, 8}, too, 0, 4, 9, 1O}, too, 0, 2, 5, I3}, {00,0, 1, 11, I5}, to, 7,11,13, I6} generate an incidence structure A with exactly five blocks through any two points of Z17 and with 20 blocks through 00 and any point of Z17. Let B be an SI5(2, 5; 17) on Z17, which exists by (S.l.a). Then A + B is the desired S20(2, 5; 18). The next three examples are exercises for the reader.
§9. Divisible designs with small block sizes
661
For K = {3} this is fairly easy. Of course
(9. 1. a) for KeN and n, g, A EN.
9.2 Lemma. (9.2.a)
R~.). ~ R;'f
for all mEN.
Proof. Use (I.7.4.a) and Corollary 3.6. •
In what follows, 9.1 and 9.2 will be used tacitly.
9.3 Observations. (a) The necessary conditions for s E (9.3.a) (9.3.b)
Ag(s - 1) 2
Ag s(s -
RL are s ll'N~-l and
== 0 (mod(k - 1», 1) == 0 (mod k(k - 1».
(b) Let k = 3, A' := gcd(A, 6), g' := gcd(g, 6). Then the necessary conditions (9.3.a) and (9.3.b) are the same for (g, A) and (g', A').
Proof. (a) Use v = gs in (1.6.14). (b) Trivially (9.3.a, b) for g', A' imply the same congruences for g, A, since g' I g, A'I A. On the other hand, there are integers a, b, c, d with A' = Aa + 6b, g' = gc+6d. Hence the conditions (9.3.a, b) for (g, A) imply those for (g', A') .
•
In the remainder of this section, we shall prove the following result of Hanani (1975).
9.4 Theorem.
If the conditions
(9.3.a), (9.3.b), and s =F 2 are fulfilled, then
sERh· 9.5 Remark. It suffices to prove the theorem for the feasible pairs (g', A') with g' I 6 and A' I 6. To see this, let Nf). be the set of conditions of (9.3.a). By 9.3.(b),
S
E No \ {2} which satisfy the necessary
§9. Divisible designs with small block sizes
661
For K = {3} this is fairly easy. Of course
(9. 1. a) for KeN and n, g, A EN.
9.2 Lemma. (9.2.a)
R~.). ~ R;'f
for all mEN.
Proof. Use (I.7.4.a) and Corollary 3.6. •
In what follows, 9.1 and 9.2 will be used tacitly.
9.3 Observations. (a) The necessary conditions for s E (9.3.a) (9.3.b)
Ag(s - 1) 2
Ag s(s -
RL are s ll'N~-l and
== 0 (mod(k - 1», 1) == 0 (mod k(k - 1».
(b) Let k = 3, A' := gcd(A, 6), g' := gcd(g, 6). Then the necessary conditions (9.3.a) and (9.3.b) are the same for (g, A) and (g', A').
Proof. (a) Use v = gs in (1.6.14). (b) Trivially (9.3.a, b) for g', A' imply the same congruences for g, A, since g' I g, A'I A. On the other hand, there are integers a, b, c, d with A' = Aa + 6b, g' = gc+6d. Hence the conditions (9.3.a, b) for (g, A) imply those for (g', A') .
•
In the remainder of this section, we shall prove the following result of Hanani (1975).
9.4 Theorem.
If the conditions
(9.3.a), (9.3.b), and s =F 2 are fulfilled, then
sERh· 9.5 Remark. It suffices to prove the theorem for the feasible pairs (g', A') with g' I 6 and A' I 6. To see this, let Nf). be the set of conditions of (9.3.a). By 9.3.(b),
S
E No \ {2} which satisfy the necessary
§9. Divisible designs with small block sizes
Proof. By (9.7.a), we have 3, 5 E show (9.9.b)
8 E Rj.2;
663
Rj.2 and 4,6 £: Rj.2 £: Rlz, by (9.6.b). We
i.e. 24 E GD2(3, 3).
Indeed, by Example VII.3.2.(a), 25 E B(4); hence 24 E GD(4, 3). Since also 4 E B(3, 2), the assertion 24 E GD2(3, 3) follows by breaking up blocks; cf. 2.3 to 2.5. Now (4.2.a) implies the assertion. • 9.10 Lemma. (9.10.a)
.
R~ 3 = No \ {2} =
Nl.3 •
Proof. By (9.6.c) and (9.6.e) we get (9.1O.b)
3,4,5,6
E
R~,3'
Rl
Moreover, 8 E 3 • This is shown by the seven difference triples (see Definition Vll.4.2) (1,1,2), (3,3,6), (5,5,6), (7,7,2), (1, 3, 4), (2,4,6), (4,5,7) in Z16. Again (4.2.a) implies the assertion. • 9.11 Corollary. (9.1 1. a)
Rf,6 = R~'A = No \ {2} for g, A EN.
(9.11.b)
Rj,z = RL = 3No + {O, I}.
Proof. (9.1 1. a) follows from (9.6.d) and (9.8.a), and (9.Il.b) follows from NI,z = NI,l' Furthermore (9.11.c) since
Nj,3
R~.3 = R~,l = 2No + 1, =
Nj,l' •
Now the assertion R~... = Nf, .. is proved in all cases where g 16 and A 16. Hence, by Remark 9.5, the proof of Theorem 9.4 is finished. Resolvable GDD's with k = 3 were studied by Rees and Stinson (1987), Assaf and Hartman (1989) and Rees (1993); their spectrum is as follows. 9.12 Theorem. A resolvable GD.. [3, g; sg] exists (9. 12.a)
s::::3,
and (A, g, s)
j
Ag(s-l)
= o(mod 2),
if and only if
sg=O(mod3)
rt {(I, 2, 6), (1,6,3), (2j + 1,2,3), (4j + 2,1,6):
j :::: O}. •
§9. Divisible designs with small block sizes
Proof. By (9.7.a), we have 3, 5 E show (9.9.b)
8 E Rj.2;
663
Rj.2 and 4,6 £: Rj.2 £: Rlz, by (9.6.b). We
i.e. 24 E GD2(3, 3).
Indeed, by Example VII.3.2.(a), 25 E B(4); hence 24 E GD(4, 3). Since also 4 E B(3, 2), the assertion 24 E GD2(3, 3) follows by breaking up blocks; cf. 2.3 to 2.5. Now (4.2.a) implies the assertion. • 9.10 Lemma. (9.10.a)
.
R~ 3 = No \ {2} =
Nl.3 •
Proof. By (9.6.c) and (9.6.e) we get (9.1O.b)
3,4,5,6
E
R~,3'
Rl
Moreover, 8 E 3 • This is shown by the seven difference triples (see Definition Vll.4.2) (1,1,2), (3,3,6), (5,5,6), (7,7,2), (1, 3, 4), (2,4,6), (4,5,7) in Z16. Again (4.2.a) implies the assertion. • 9.11 Corollary. (9.1 1. a)
Rf,6 = R~'A = No \ {2} for g, A EN.
(9.11.b)
Rj,z = RL = 3No + {O, I}.
Proof. (9.1 1. a) follows from (9.6.d) and (9.8.a), and (9.Il.b) follows from NI,z = NI,l' Furthermore (9.11.c) since
Nj,3
R~.3 = R~,l = 2No + 1, =
Nj,l' •
Now the assertion R~... = Nf, .. is proved in all cases where g 16 and A 16. Hence, by Remark 9.5, the proof of Theorem 9.4 is finished. Resolvable GDD's with k = 3 were studied by Rees and Stinson (1987), Assaf and Hartman (1989) and Rees (1993); their spectrum is as follows. 9.12 Theorem. A resolvable GD.. [3, g; sg] exists (9. 12.a)
s::::3,
and (A, g, s)
j
Ag(s-l)
= o(mod 2),
if and only if
sg=O(mod3)
rt {(I, 2, 6), (1,6,3), (2j + 1,2,3), (4j + 2,1,6):
j :::: O}. •
§1O. Steiner quadruple systems
665
Proof. We now give an outline of the proof, by reducing the assertion to the following series of lemmas; the real work lies in the subsequent proofs of these lemmas. (1O.2.a)
2SQS4 C SQSg,
(1O.2.b)
3SQSg - 2
(1O.2.c)
3SQSg - a C SQSg for a E {4, 8, 16},
(1O.2.d)
{22, 26,34, 38} C SQSg.
c
SQSg,
Now assume that the theorem does not hold. From previous chapters we know that 8, 10, 14 E SQS, see (I.3.4.a), Theorem III.6.9, Example 111.1.3 and Example VIII.6.1. Hence there would exist a smallest Vo rf. SQSg with Vo E;; 2 or 4 (mod 6) and Vo ~ 16. Using (l0.2.a) and (1O.2.d), we have Vo E;; 2 (mod 4) and Vo ~ 46. Hence there is a positive integer n with (1O.2.e)
Vo = 36n
+ a,
a
E
{2, 10, 14,22,26, 34}.
But all of the cases in (l0.2.e) lead to a contradiction. For instance, using (1O.2.b), we get 36n
+ 10 = 3(l2n + 4) -
2 E SQSg
36n
+ 22 = 3(l2n + 8) -
2 E SQSg,
and
as 12n + 4, 12n + 8 < Vo. Similarly, (l0.2.c) rules out the remaining cases in (1O.2.e):
+ 2 = 3(l2n + 2) - 4, 36n + 26 = 3(12n + 10) - 4
36n
36n + 14 = 3(12n and
+ 10) - 16, 36n + 34 = 3(l2n + 14) -
8. •
It remains to prove the auxiliary assertions (l0.2.a)-(1O.2.d). The Steiner quadruple systems required in (l0.2.d) will be obtained via particular applications of the constructions giving the validity of (1O.2.a, b, c). The first two of these assertions only require rather simple constructions, the doubling construction and the easy tripling construction.
10.3 Lemma. One has 2 SQS4 C SQSg.
§1O. Steiner quadruple systems
665
Proof. We now give an outline of the proof, by reducing the assertion to the following series of lemmas; the real work lies in the subsequent proofs of these lemmas. (1O.2.a)
2SQS4 C SQSg,
(1O.2.b)
3SQSg - 2
(1O.2.c)
3SQSg - a C SQSg for a E {4, 8, 16},
(1O.2.d)
{22, 26,34, 38} C SQSg.
c
SQSg,
Now assume that the theorem does not hold. From previous chapters we know that 8, 10, 14 E SQS, see (I.3.4.a), Theorem III.6.9, Example 111.1.3 and Example VIII.6.1. Hence there would exist a smallest Vo rf. SQSg with Vo E;; 2 or 4 (mod 6) and Vo ~ 16. Using (l0.2.a) and (1O.2.d), we have Vo E;; 2 (mod 4) and Vo ~ 46. Hence there is a positive integer n with (1O.2.e)
Vo = 36n
+ a,
a
E
{2, 10, 14,22,26, 34}.
But all of the cases in (l0.2.e) lead to a contradiction. For instance, using (1O.2.b), we get 36n
+ 10 = 3(l2n + 4) -
2 E SQSg
36n
+ 22 = 3(l2n + 8) -
2 E SQSg,
and
as 12n + 4, 12n + 8 < Vo. Similarly, (l0.2.c) rules out the remaining cases in (1O.2.e):
+ 2 = 3(l2n + 2) - 4, 36n + 26 = 3(12n + 10) - 4
36n
36n + 14 = 3(12n and
+ 10) - 16, 36n + 34 = 3(l2n + 14) -
8. •
It remains to prove the auxiliary assertions (l0.2.a)-(1O.2.d). The Steiner quadruple systems required in (l0.2.d) will be obtained via particular applications of the constructions giving the validity of (1O.2.a, b, c). The first two of these assertions only require rather simple constructions, the doubling construction and the easy tripling construction.
10.3 Lemma. One has 2 SQS4 C SQSg.
§10. Steiner quadruple systems
667
on the point set
U x {OJ = {uo: u E U} . •
10.5 Example. In particular, applying Lemma lOA with u = 8 and u = 16 yields (lO.5.a)
22,46 E SQSg.
The proof of (1 O.2.c) requires a considerably more involved method, the difficult tripling construction. It may be obtained from the following general result due to Hartman (1980,1982).
10.6 Theorem. Assume the existence of an SQS(u), say D, with a proper subSQS(w), where 1 < w ::: u/2. Then there exists an SQS(3u - 2w) with three sub-SQS(u) 's isomorphic to D (and hence also with a sub-SQS(w»). • We will not prove Theorem 10.6 in full generality, but only in the following two special cases: (1Q.6.a)
g = u- w
(1Q.6.b)
g
=/= 0 (mod 6);
= u - w == 0 (mod 6)
and
w = 2.
The reader should check that these two special cases of Theorem 10.6, applied with w = 2,4, and 8, indeed suffice to finish the proof Theorem 10.2, since we have applied (1Q.2.c) only under these restrictions. We remark that Theorem 10.2 then yields the validity of (l0.2.c) in full generality. We prepare the proof of the cases (1Q.6.a, b) of Theorem 10.6 with some notation and with the construction of some partial quadruple classes.
10.7 Notation. Let D = (U,A) be the given SQS(u) and (W, C) the given sub-SQS(w), say W={OOl, ... ,OOw}
and
U':=U\W=Zg={O,I, ... ,g-I}(modg),
where g = u - w. As new point set we take (1O.7.a)
v=
WU(Zg x {O, I,2}) = WU{Xi:X modg,i mod3}.
We partition the triples of points in V into four classes Tl, T2, T3 and T4 as follows:
§10. Steiner quadruple systems
667
on the point set
U x {OJ = {uo: u E U} . •
10.5 Example. In particular, applying Lemma lOA with u = 8 and u = 16 yields (lO.5.a)
22,46 E SQSg.
The proof of (1 O.2.c) requires a considerably more involved method, the difficult tripling construction. It may be obtained from the following general result due to Hartman (1980,1982).
10.6 Theorem. Assume the existence of an SQS(u), say D, with a proper subSQS(w), where 1 < w ::: u/2. Then there exists an SQS(3u - 2w) with three sub-SQS(u) 's isomorphic to D (and hence also with a sub-SQS(w»). • We will not prove Theorem 10.6 in full generality, but only in the following two special cases: (1Q.6.a)
g = u- w
(1Q.6.b)
g
=/= 0 (mod 6);
= u - w == 0 (mod 6)
and
w = 2.
The reader should check that these two special cases of Theorem 10.6, applied with w = 2,4, and 8, indeed suffice to finish the proof Theorem 10.2, since we have applied (1Q.2.c) only under these restrictions. We remark that Theorem 10.2 then yields the validity of (l0.2.c) in full generality. We prepare the proof of the cases (1Q.6.a, b) of Theorem 10.6 with some notation and with the construction of some partial quadruple classes.
10.7 Notation. Let D = (U,A) be the given SQS(u) and (W, C) the given sub-SQS(w), say W={OOl, ... ,OOw}
and
U':=U\W=Zg={O,I, ... ,g-I}(modg),
where g = u - w. As new point set we take (1O.7.a)
v=
WU(Zg x {O, I,2}) = WU{Xi:X modg,i mod3}.
We partition the triples of points in V into four classes Tl, T2, T3 and T4 as follows:
669
§1O. Steiner quadruple systems
are distinct modulo g. For abbreviation, we write Idl := min{d, g - d}. Then the quadruples in Q3(a, b, c; d) cover each of the triples in the classes
exactly once, and no other triples. The reader should check this. (d) Class Q4(D) is defined for subsets D of {I, 2, ... , g12} which contain at least one element of even order. Then the Stem-Lenz Lemma III.IO.S guarantees that the cyclic graph eGg (D) admits a factorisation F. We let the class Q4(D) consist of all quadruples of the fonn {Xi, (X
+ d)i, Yj, (y + e)j}
with d, e ED
and
i
#j
for which the edges {x, x + d} and {y, Y + e} are in the same factor of F. Then the quadruples in Q4(D) cover each of the triples in the classes T4(d) with d E D exactly once, and no other triples. If condition (*) in §III.lO is satisfied for the factorisation F, we write Q4(D) instead of Q4(D). Now assume that (*) holds, and that {a, b, c, d} is a block in A. Then QI and Q:(D) contain the following 14 quadruples (for simplicity, written without commas and brackets): ao bo bo Co ao bo Co do ao bo
bl
bl CI
CI
dl
al
bl
ao bo ao bo
Co
do
al
al
Co do Co do bl
al
CI
ao
bl
d[
Co
bl al
dl CI
ao bo
CI
d[
do do do
al
Co
aJ
CI
bl
dl dl C1 d[
NotethatthesefourteenblocksfonnanSQS(S) on the point set {a, b, c, d} x {O, I}. The analogous result holds for {a, b, c, d} x {I, 2} and {a, b, c, d} x {2,O}. (e) The following quadruple classes will only be used if (lO.6.b) holds. We define classes Q5(a, d), where (a, d) = (-1,2) or (-2,4), as unions Q5(a, d) := Q~(a, d) U Q~(d)
with a
=1=
0 (mod 3) and d
== a (mod 3).
Here the subclass Q;(a, d) is defined as the set of all quadruples
(1O.8.a)
{Xi, (X
+ d)i, Yi+I, Zi+2)
with x, Y, Z E Zg,X
+ y +Z == a (mod g) andy == z(mod 3).
669
§1O. Steiner quadruple systems
are distinct modulo g. For abbreviation, we write Idl := min{d, g - d}. Then the quadruples in Q3(a, b, c; d) cover each of the triples in the classes
exactly once, and no other triples. The reader should check this. (d) Class Q4(D) is defined for subsets D of {I, 2, ... , g12} which contain at least one element of even order. Then the Stem-Lenz Lemma III.IO.S guarantees that the cyclic graph eGg (D) admits a factorisation F. We let the class Q4(D) consist of all quadruples of the fonn {Xi, (X
+ d)i, Yj, (y + e)j}
with d, e ED
and
i
#j
for which the edges {x, x + d} and {y, Y + e} are in the same factor of F. Then the quadruples in Q4(D) cover each of the triples in the classes T4(d) with d E D exactly once, and no other triples. If condition (*) in §III.lO is satisfied for the factorisation F, we write Q4(D) instead of Q4(D). Now assume that (*) holds, and that {a, b, c, d} is a block in A. Then QI and Q:(D) contain the following 14 quadruples (for simplicity, written without commas and brackets): ao bo bo Co ao bo Co do ao bo
bl
bl CI
CI
dl
al
bl
ao bo ao bo
Co
do
al
al
Co do Co do bl
al
CI
ao
bl
d[
Co
bl al
dl CI
ao bo
CI
d[
do do do
al
Co
aJ
CI
bl
dl dl C1 d[
NotethatthesefourteenblocksfonnanSQS(S) on the point set {a, b, c, d} x {O, I}. The analogous result holds for {a, b, c, d} x {I, 2} and {a, b, c, d} x {2,O}. (e) The following quadruple classes will only be used if (lO.6.b) holds. We define classes Q5(a, d), where (a, d) = (-1,2) or (-2,4), as unions Q5(a, d) := Q~(a, d) U Q~(d)
with a
=1=
0 (mod 3) and d
== a (mod 3).
Here the subclass Q;(a, d) is defined as the set of all quadruples
(1O.8.a)
{Xi, (X
+ d)i, Yi+I, Zi+2)
with x, Y, Z E Zg,X
+ y +Z == a (mod g) andy == z(mod 3).
670
IX. Recursive constructions
The subclass
Q~(a)
consists of the quadruples with i #j
{Xi, (X +d)i, Yj, (y +d)j)
and
x
== Y
(mod 3).
10.9 Lemma. The quadruple class Qs(a, d) defined in 1O.8.(e) covers each of the triples in the classes T4(d), T3(a) and T3(a + d) exactly once, and no other triples. Proof. We only consider the case (a, d) = (-1,2), as the other case (a, d) = ( - 2, 4) is analogous. By definition, Q~ ( -1, 2) consists of all quadruples {Xi, (X + 2)i, Yi+I, Zi+z)
Y
with X + Y + Z == -1 (mod g)
and
== z (mod 3),
hencex+2y == -1(mod3). Note that any triple which is covered by Q;(-I, 2) must be in one of the classes T3(-I), T3(1) and T4(2) and that any triple which is covered by Q~(2) has to belong to T4(2). Now let T be any triple in T4(2) U T3(-I) U T3(1); we shall show that T is indeed covered by a unique quadruple in Qs( -1,2). Case 1: T = {Xi, Yi+I, Zi+2} E T3(-I). As X + Y + Z ¥= 0 (mod 3), we may assume Y == Z covered by the quadruple {Xi, (X + 2)i, Yi+l, Zi+Z).
¥= X (mod 3). Then T is
Case 2: T = {x;, Yi+l, Zi+2} E T3(l). Again, we may assume Y == Z ¥= X (mod 3). Now T is covered by the quadruple {(x - 2)i, Xi, Yi+l, Zi+2). Case 3: T = {Xi, (X + 2)i, Yj) E T4(2) with j Subcase 3a: X + 2y X
Then X
+ Y+
# i.
== -1 (mod 3). We determine Z from the equation z == -1 (mod g).
¥= Y (mod 3), and T is covered by exactly one of the quadruples {Xi, (X + 2)i' Yi+I, Zi+2)
as X + Y +
and
{Xi, (X + 2)i, Zi+l, Yi+2},
z == -1 (mod 3) implies Y == Z (mod 3).
Subcase 3b: X + 2y == 0 (mod 3). Then X quadruple {Xi, (X + 2)i, Yj, (y + 2)j).
== Y (mod 3), and T
is covered by the
§10. Steiner quadruple systems
671
Subcase 3c: x + 2y == 1 (mod 3), that is x == y + 1 == Y - 2 (mod 3). Then T is covered by the quadruple {Xi, (x - 2)i, Yj, (y - 2)j} . •
In what follows, the block set B of the desired SQS(3u - 2w) will be obtained as the union of appropriate quadruple classes defined in Construction 10.8. We begin with the case (l0.6.a).
10.10 Construction. We now prove Theorem 10.6 in the case g =1= 0 (mod 6). Then we have u == 2w (mod 6), as u, W == 2 or 4 (mod 6), and thus (lO.IO.a)
g
=u-
W
= 6n + w
for some positive integer n.
We define B as the union of the following quadruple classes: (a) Class Qt. This class covers the triples of class Tj • (b) Classes Q2(6n - 1 + A, A) for A = 1,2, ... , w. These classes cover the triples in classes T2(A) and T3(6n - 1 + A) for A = 1,2, ... , w. (c) Classes Q3(a, a + 1, a + 2; d), where a = 3(n - 1 - h) and d = 6h + 3, for h = 0, 1, ... , n - 1. These quadruple classes cover the triples in classes T4(i6h
+ 31)
for h = 0, 1, ... , n - 1
and T3(O), T3(1), T3(2), T3(6n - 3), T3(6n - 2), T3(6n - 1), T3(3), 13(4), T3(5), ... , T3(6n - 6), T3(6n - 5), T3 (6n - 4),
(d) Class Q4(D) with D = {I, 2, ... , g/2} \ {16h + 31 : h = 0, 1, ... , n - J}. This class covers the triples in the classes T4 (d), dE D. Hence the union of the above quadruple classes yields the desired SQS(3u - 2w) . • 10.11 Examples. (a) If g == 0 (mod 4), say g = 4m, then the set D defined in Construction 10.10 contains m, 2m and at least one other element d of even order, say D = {m, 2m} U D'. Hence the Stem-Lenz lemma may be applied to factorise CGg({m, 2m}) and CGg(D') separately, and by Example III. 10.9 we may achieve the validity of condition (*). Without loss of generality, we may
IX. Recursive constructions
672
assumethat{a, b, c, d} = to, m, 2m, 3m} is a block inA. As we have shown in Construction 1O.8.(d), (V, B) contains three subsystems SQS(8). In particular, using u = 10 and w = 2, we obtain
(10.1 La)
26
E
SQSg.
A direct construction for a strictly cyclic SQS(26) was given in Example
VIII. 10.10. (b) For u
= 14 and w = 4, Construction 10.10 gives D = {l, 2, 3,4, 5} \ {3} = {I, 2, 4, 5}.
By Example III. 10. 10, we may again assume the validity of condition (*). The same reasoning as in (a) now yields. (1O.l1.b)
34
E
SQSg.
A direct construction for a strictly cyclic SQS(34) was given in Example VIII. 10.21.
10.12 Construction. We now prove Theorem 10.6 in the case (1O.6.b), that is g == (0 mod 6) and w = 2, say g = 6n + 6. We define B as the union of the following quadruple classes: ClassQj. Classes Q2(0, 1) and Q2(5, 2). Class Q3(3, 3n + 3, 3n + 5; 1). Classes Q3(6n -3h+ 1, 6n -3h + 2, 6n -3h 1, ... , n -2. (e) Classes Qs(-1,2)andQ5(-2,4). (f) Class Q4(D), where
(a) (b) (c) (d)
D
= {I, 2, ... , g/2} \
({1, 2, 4} U (Ill
+ 6hl
:h
+ 3; 11 +6h)
= 0,1, ... , n -
for h=O,
2}).
As in Construction 10.10, it is not difficult to check that (V, B) is the desired SQS(3u - 2w) . •
10.13 Example. In particular, Construction 10.12 works for n = 1. In this case, there are no classes under (d), and B is the union of the classes Qj,
Q2(0,1),
Q5(-2,4),
and
Q2(5,2),
Q3(3, 6, 8; 1),
Q4({3, 5, 6}).
Q5(-1,2),
673
§11. Embedding theorems
As in Example 1O.11.a, thefactorisation of CG 12 ({3, S, 6}) can be chosen such that condition (*) in §III.1O holds. Hence we may assume that the constructed SQS(38) contains three subsystems SQS(8). This shows (1O.13.a)
38 E SQS8.
Together with Examples 10.5 and 10.11 we have now also obtained the validity of assertion (IO.2.d). To sum up, all the auxiliary assertions in Theorem 10.2 are now verified, and the proof of Theorem 10.2 is complete.
10.14 Remarks. Lenz (198Sa) gave a bound on the number of non-isomorphic SQS(v). For cyclic quadruple systems, see §VIII.1O and the references cited there. Good surveys on Steiner quadruple systems are given by Lindner and Rosa (I978) and, as an update to the older survey, by Hartman and Phelps (1992).
§11. Embedding Theorems for Designs and Partial Designs
11.1 Observation. In Notation 3.1S, the symbol Fk(U) was introduced to denote the set of all v for which an S(2, k; v) with a subspace of size u exists. Since the subspace has to be an S(2, k; u) itself, necessary conditions are (I 1. La)
== 0 (mod k - 1), 1) == 0 (mod k(k -1))
u - 1, v-I u(u - 1), v(v -
and
v::: (k - l)u + 1.
The fact that (ll.l.a) is sufficient for k = 3 is equivalent to the celebrated Doyen-Wilson theorem already mentioned in Remark 1.8.12 and in (3.1S.a); alternative proofs of this result were given by Stern and Lenz (1980) and by Stinson (1989). We shall present the simple and elegant proof of Stem and Lenz (1980) which makes essential use of the Stern-Lenz Lemma III. 10.8. We first introduce an auxiliary concept. 11.2 Definition. A partial triple system (for short, an PTS) is a linear space with line sizes 2 and 3. A PTS is called regular (or an RPTS) if each point is on equally many lines of size 2, and cyclic (or a CPTS) if it admits a cyclic automorphism group that acts regularly on the point set. Note that any CPTS is regular. The graph of a PTS has as vertices the points of D and as edges the lines of size 2.
IX. Recursive constructions
674
11.3 Theorem (The Doyen-Wilson Theorem). The necessary conditions (l1.3.a)
u, v
== 1 or 3 (mod 6)
and
v:::: 2u
+1
for the existence of an STS(v) with a sub-STS(u) are also sufficient.
Proof. We first outline the idea of the proof. For u = 1 and u = 3 the assertion is trivial, hence we may assume u :::: 7. As point set V of an RPTS we choose the cyclic group Zw with w := v - u. Then the desired STS(v) is constructed via the following steps. Step 1: Construct two auxiliary CPTS's with point set Zw, see Examples 1 and 2 below. Step 2: Construct an RPTS, say D, on w = v - u points such that each point x is on u lines of size 2, one of which is {x, x + w /2}, and such that the graph r (D) of this RPTS is cyclic. Step 3: By Corollary m.1O.8, the cyclic graph reD) whose edges are the lines of size 2 admits a I-factorisation, say F. Step 4: Introduce a new point on each factor F E F. Step 5: Form an STS(u) on the new points. Thus the real work lies in the proof of the Stem-Lenz Lemma Ill. 10.8 and the constructions outlined in Steps 1 and 2; after this, Steps 3, 4 and 5 are obvious. In order to perform the required constructions, we will define lines of size 3 by giving difference triples, as in §VII.4. We begin with the examples announced in Step 1. '
Example 1. Let w :::: 12t - 1 with t > 0, and put s = 2t - 1 (this parameter will be used later). As difference triples take (1
(3
3t - 1 3t) 3t - 2 3t + 1)
(2 (4
St - 1 St St - 2 St
+ 1) + 2)
and
(2t - 1 2t
4t - 1)
(2t - 2
4t + 1 6t - 1).
In these difference triples every x E {t, 2, ... , 6t - I} except for 4t and 5t occurs exactly once. Example 2. Let w :::: 12t + 5 with t > 0, and put s = 2t. As difference triples
675
§11. Embedding theorems take (1 (3
5t + 2 5t + I
St St
+ 3) + 4) and
(2t - 1
4t
Each x E {I, 2, ... , 6t
+3
6t
+ 2)
(2 3t 3t + 2) (4 3t - I 3t + 3) ................... . (2t
2t
+1
4t
+ 1).
+ 2} except for 3t + 1 and 4t + 2 occurs exactly once.
Now we proceed with the construction of the desired RPTS's of Step 2. Case 1: v = 2u + 1, W = U + l. Here D has no lines of size 3, and reD) is the complete graph on W vertices. Thus this case is trivial. Case 2: W = 12t + 2 or 12t + 8, U = 6h + 7 (with h = 0, 1, ... ,s). In Zw we take the difference triples of Example 1 or 2, respectively, and delete h of these difference triples. We obtain a CPTS, say D, whose graph is cyclic with edges of the form {i, i + d}. If w = 12t + 2, then d runs over the values 4t, 5t, 6t and w 12 = 6t + 1 as well as 6h further values arising from the difference triples that were deleted. If w = 12t + 8, then d runs over the values 3t + 1, 4t + 2, 6t + 3, and w 12 = 6t + 4 as well as 6h further values arising from the difference triples that were deleted. In either case, the cyclic graph r(D) has degree 6h + 7. In particular, reD) contains the edge {O, w12} of the even order two. Moreover, each u == I (mod 6) with 7 ::s u ::s w - I occurs for the appropriate value of h, as desired. Case 3: w = I2t + 6 or I2t + 10, U = 6h + 9 (with h = 0, 1, ... , s). Again we choose the triples of Example 1 or 2, respectively, omit h of them and get the desired CPTS. The details of this case are left to the reader. Case 4: w = 6m = 12t or 12t + 6 with t > 0. If we again use the difference triples of Example 1 or 2, respectively, and proceed exactly as before we fail, since the resulting graph reD) has degree 6h + 5 which cannot be the order of an STS. We deal with this problem by adding further lines to D in such a way that the new (smaller) graph remains cyclic but has a suitable (smaller) degree. This requires another case distinction, depending on the congruence class of u modulo 6. Case 4a: u
== 3 (mod 6), say u = {i, 2m
6h
+ i, 4m + i}
+ 3. We introduce 2m new lines (i = 0,1, ... , 2m - 1).
IX. Recursive constructions
676
This removes all edges of the form {i, 2m +i} from reD). Hence the resulting graph is cyclic of degree u = 6h + 3 with 0 < h :::: s. Case 4b: u == 1 (mod 6), say u = 6h + 1. In this case, we select (2t -1, 2t, 4t - 1) or (2t -1, 2t + 1, 4t + 1), respectively, as one of the h difference triples which are deleted in the construction of D. Then we introduce the following 4m new lines (with i
= 0, 1, ... , m -
1):
+ i, 2m + i}; {4m + i, 5m + i, i};
{i, m
+ i, 3m + i, 4m + i}; {m + i, 3m + i, 5m + i}.
{2m
The resulting RPTS is not cyclic, but its graph is cyclic, as any pair of the form {x, m + x} or {x, 2m + x} is contained in exactly one of the new lines. The degree of the new graph is u = 6h + 1 with 0< h:::: s. This finishes Step 2. (The reader should check that our case distinction is indeed complete.) As already noted, Steps 3, 4, and 5 are now obvious . • The Doyen-Wilson theorem is a seminal result which prompted much research on the analogous problem for larger values of k and on related questions. We now discuss the most important results in this direction, beginning with the classical subspace problem already mentioned in Remark 1.8.12. The combined efforts of Rees and Stinson (1989a, b) and Wei and Zhu (1987,1989) settle the case k = 4. Thus we have the following result.
11.4 Theorem. The necessary conditions (1 1. 1. a) for the existence of an S(2,4; v) with a sub-S(2, 4; u) are sufficient. • Some results on the number of maximal subsystems in an S (2, 4; v) were given by Zeitler (1993). Bennett and Yin (1995) considered the existence problem for S(2, 5; v) with a sub-S(2, 5; u) and proved that the necessary conditions are sufficient in the case u == v (mod 20) whenever v 2: 5w - 4, with the possible exception of the pair (425, 65). We now discuss a problem related to the subspace problem, namely the existence oflinear spaces where all lines but one (which is of size u) have size k; we shall denote such a PBD by S(2, {k, u*}; v). In the special case where there exists an S(2, k; u), this problem is clearly equivalent to the existence problem for an S(2, k; v) with a sub-S(2, k; u) just discussed. We also note that the existence
§11. Embedding theorems
677
of a resolvable design RS(2, k; v) is equivalent to that of an
S(2, {k + 1, r*}; v + r),
where r = (v - l)/(k - 1),
by Proposition 1.8.7. Rees and Stinson (1989b) and Mills (1989) considered the case k = 4 and proved the following result.
11.5 Theorem. An S(2, {4, u*}; v) exists if and only if either (1l.1.a) holds (with k = 4) or ifv :::: 3u + 1 and u, v == 7 or 10 (mod 12). • The preceding problem has also been studied for k = 5, though considerably less is known in this case. Assaf (1994), Hamel, Mills et al. (1993) and Bennett and Yin (1995) prove that the necessary conditions for the existence of an S(2, {5, u*}; v) are sufficient for the values u' E {9, 13, 17,21,25,29, 33}, with 26 possible exceptions. Hartman and Heinrich (1993) and Chee, Colbourn et al. (1997) completely settle the existence question for PBDs with one block of prescribed size u and all other blocks of size at least 3, that is (in self-explanatory notation) for S(2, {[3, 00], u*}; v). Note that there may be more than one block of size u in such a linear space. Colboum, Haddad and Linek (1996) show the existence of S(2, {{3, 5, 7, ... }, u*}; v) for all odd integers v and u* satisfying v:::: 2k + 1. Bennett, Du andZhu (1990) settle the existence problem for S(2, {4, 5, 11 *}; v) up to 33 possible exceptions. Theorem 11.5 deals with PBD's with two block sizes where both these values have to occur. More generally, a mandatory representation design MR(K; v) is a PBD on v points with line sizes from K in which for each k E K there actually is a line of size k in the design. The study of these designs was initiated by Mendelsohn and Rees (1988) who considered the special case where 3 E K. In particular, these authors determined the spectrum for MR({3, k}; v) for any odd integer k. Similar questions have been studied for designs with arbitrary A. Here the analogue of the Doyen-Wilson theorem for arbitrary triple systems is due to Stem (1979); see also Bigelow and Colboum (1993) for a generalisation of this result where "embeddings" of triple systems into systems with a larger A-value are considered.
11.6 Theorem. An SA (2,3; v) with a sub-SA (2, 3; v) exists if and only if both u and v are admissible (that is, satisfy the necessary arithmetic conditions for the existence of the respective triple system) and v :::: 2u + 1. •
678
IX. Recursive constructions
The analogous result for the case k = 4 is due to Wei (1991), Rees and Rodger (1993) and Kong and Zhu (1994), and an analogue of Theorem 11.5 for the case A. > 1 was given by Kong and Wei (1993). We only state the first of these results.
11.7 Theorem. An S).(2, 4; v) with a sub-S).(2, 4; v) exists ifand only ifboth u and v are admissible (that is, satisfy the necessary arithmetic conditions for the existence of the respective design) and v 2: 3u + 1. • The analogous problem for Steiner quadruple systems if far from being solved; the best known results are due to Granville and Hartman (1991). Of course, one may also consider subdesigns while imposing extra structural conditions. For example, cyclic STS with cyclic sub-STS were considered by Phelps, Rosa and Mendelsohn (1989). Some results on Kirkman triple systems with sub-KTS's were already discussed at the end of §6. Some general existence results for resolvable Steiner systems RS(2, 4; v) with resolvable sub-systems RS(2, 4; u) were obtained by Cai (1990). Another related problem concerns the "embedding" of a "partial" design. Here a partial design PS).(t, k; v) is defined in the obvious way as an incidence structure on v points with blocks of size k for which any t points are contained in at most A. blocks. Note that this definition is not essentially different from the one we gave for the special case of partial triple systems in Definition 11.2 above: we just have to omit all lines of size 2. The problem of embedding a PSl.(t, k; v) into some design S).(t, k; v*) by adding further points and blocks has been studied extensively for various types of designs. We emphasise that v* may be much larger than the original v, though of course embeddings with small values of v* are of particular interest. We refer the reader to Lindner (1980) for a good survey for the special case of partial triple systems and only state a few major theorems. The seminal result in this area is the theorem of Treash (1971) which states that every PTS can be embedded into a finite STS. This is a special case of the following more general result due to Ganter (1976a).
11.8 Theorem. Any partial Steiner system PS(2, k; u) can always be embedded into an S(2, k; v) (jor a suitable value of v) . • Regarding Steiner systems with t 2: 3, one has the following analogous result of Ganter (1974) for quadruple systems.
11.9 Theorem. Any partial Steiner quadruple system can be embedded into a Steiner quadruple system SQS( v) (jor a suitable value of v). •
§1l. Embedding theorems
679
As mentioned above, embeddings with small values of v are of particular interest. For partial Steiner triple systems, the best known lower bound is due to Andersen, Hilton and Mendelsohn (1980). The corresponding result for partial triple systems with arbitrary index A is due to Rodger and Stubbs (1987) and Hilton and Rodger (1991).ltis easy to see that v :::: 2u + 1 is always a necessary condition; thus the known results we quote now are quite good.
11.10 Theorem. Any partial Steiner triple system on u points can be embedded into an STS( v) for all v satisfying v :::: 4u + I and v == 1 or 3 (mod 6). • 11.11 Theorem. Any partial design S).'<2, 3, u) on u points can be embedded into a triple system S>-(2, 3, v) for all admissible v satisfying v :::: 4u + I.lf A is a multiple of 4, the bound on v can be improved to v :::: 2u + 1. • There has also been considerable interest in the problem of embedding designs into designs with a larger block size. We now report on some results in this direction, beginning with the rather special (but popular) problem of "nesting". Here we let D = (V, B) bean S>-(2, 3; v) and assume the existence ofa mapping f:B -+ V such that (V, {B U feB): B E B}) is an SJl.(2, 4, v); then fis called a nesting of D. The combined efforts of Colboum and Colboum (1983) and Stinson (1985) have settled this problem. The reader may check as an exercise that the condition given for this to be parametrically feasible below is indeed necessary. For a similar construction in the case of Steiner triple systems with v == 3 (mod 6), see Lindner and Rodger (1987).
11.12 Theorem. An S>-(2, 3; v) which is nested into an SJl.(2, 4, v) exists ifand only if
(11.12.a)
A(V - 1)
== 0 (mod 6)
and
/h = 2A . •
11.13 Example. As an example for nestings, we give a particularly simple construction due to Stinson (1985) which covers the case A = 6. Thus let v :::: 4 and assume v # 6. (The case v = 6 needs special treatment and will not be considered here.) By Theorem 4.9, there exists a pair of orthogonal Latin squares of order v and hence an orthogonal array OA(4, v), say A, which we assume to be in normal form, see Observation VIll.4.5. Discard the v columns of the form (i, i, i, i) from A, and use the remaining columns as blocks. Clearly, the result is an Sd2, 4, v), say D. Now omit from every block of D the symbol that came from the last row of A. This yields a triple system S6(2, 3; v) which is nested into D.
680
IX. Recursive constructions
The final topic which We will discuss in this section, namely embedding Steiner systems into projective planes, has generated much interest for geometric reasons. The most important case concerns a subset S in a projective plane P such that the lines of P induce on S a Steiner system. Obvious examples include AG(2, q) in PG(2, q), PG1(d, q) in PG1(e, q) or AGl(d, q) in AG1 (e, q) for e > d as well as subplanes and subgeometries of projective or affine planes and spaces, respectively. A series ofless obvious examples is provided by AG(2, 3) in PG(2, q) for q == 1 (mod 3), a result due to Ostrom and Sherk (1964); see also Rigby (1965), Vedder (1981) and Abduhl-Elah, Al-Dhahir and Jungnicke1 (1987), where the number of affine planes of order 3 in PG(2, q) is determined. Two further large classes of examples are given by the maximal arcs and by the unitals in PG(2, q2) (and other projective planes) which were discussed in §VIII.5. It is conjectured that every Steiner system (and, more generally, every linear space) admits an embedding into a finite projective plane. For some very general results in this direction, we refer the reader to Beutelspacher and Metsch (1986, 1987), Metsch (1991a) and the monographs Batten and Beutelspacher (1993) and Metsch (1991b). As an example of what type of results can be obtained, we state a theorem due to Metsch (1991a).
11.14 Theorem. Let D be a linear space with b lines, and assume the existence of an integer n 2:: 2 such that every point is on exactly n + I lines. Denote the smallest and the largest line size in D by n + 1 - A and n + 1 - a, respectively. Then there is an absolute constant c such that D can be extended to a projective plane of order n whenever n 2:: ca 3 • • An explicit value for the constant c occurring in Theorem 11.14 may be determined in every particular case. As an example, we mention the following consequence of Theorem 11.14 which characterises the complements of maximal k-arcs in projective planes; this is also due to Metsch (1991a).
11.15 Theorem. Let D be a linear space, and let nand k be positive integers for whichk divides n. Assume that every pointofD lies on n/ k lines of degree n+ 1 andonn + I-(n/ k) linesofsizen + 1 - k.lfonehas3n > (k-I)(8k 2 -2k-4), then L can be obtained as the complement of a maximal k-arc in a projective plane of order n. • Finally, we mention a completely different approach to embedding Steiner systems into a projective plane P. Jungnickel and Vanstone (1991) considered
§12. Concluding remarks
681
the existence of a Steiner triple system on the point set of P such that each block of the STS is a triangle of P. They proved the following result.
11.16 Theorem. Let G be a cyclic Singer group for the Desarguesian projective plane PG(2, q), as in Theorem Ill.6.2. Then there exists an STS(q2 +q + 1) on the point set ofPG(2, q) which is invariant under G and only contains blocks which are triangles in PG(2, q). • More generally, one might ask to embed an S(2, k; v), say D, with v q2 + q + 1 in such a way into PG(2, q) that the blocks of D are arcs of PG(2, q) (and, possibly, such that D is invariant under a Singer group or some other nice type of automorphism group of PG(2, If one allows "non-collinear embeddings" in the sense just discussed, one might even find interesting embeddings of designs with A#-1 into a projectiveplane. We mention the following example due to Bruen (1989). A similar result concerning the spherical 3-designs was already mentioned in Remark III.6.lD.
q».
11.17 Theorem. Let q ::: 5 be an odd prime power. Then there exist a set S of16 points and a set C of 16 conics in PG(2, q) such that the 16 intersections S n C (with C E C) form a biplane, that is an S2(2, 6; 16). •
§12. Concluding Remarks
In this final section of Chapter IX, we discuss some further general existence results beyond those dealt with in more detail in the preceding sections. In particular, we consider 2-designs, resolvable 2-designs, quadruple systems and some related structures, for instance near-resolvable designs, coverings and packings.
Let us begin by discussing the existence problem for (resolvable) 2-designs. As we shall see in Chapter XI, the necessary existence conditions are asymptotically sufficient in this case. However, the resulting explicit bounds are astronomic, leaving a lot of difficult work if one wants to completely settle the existence question for a particular pair k and A. We have already experienced this phenomenon when we settled the cases with k :::: 5, cf. Theorems 5.5, 5.lD and 8.14, Similarly, for resolvable 2-designs, the cases with k :::: 4 and A = I have also been settled completely, see Theorems 6.5 and 7.11, respectively. Beyond these ranges, the existence problem becomes more and more involved and requires a considerable amount of computer work; in fact, the existence problem for Steiner systems S(2, k; v) with k :::: 6 and for resolvable Steiner
682
IX. Recursive constructions
systems RS(2, k; v) with k ::: 5 has not been settled completely for any value of k. Presenting proofs for the present state of knowledge is clearly outside the scope of this book, even though the methods are more or less the same as those we used in the previous sections. We first summarise the state of the art regarding 2-designs with k = 6. As usual, the most difficult subcase is the case A = 1. In fact, for A ::: 2, the complete answer has been known for a long time, by the following result of Hanani (1975).
12.1 Theorem. The necessary conditions for the existence of an S)..(2, 6; v) with A::: 2 are sufficient, except for the non-existent S2(2, 6; 21). • The case A = 1 is indeed much more difficult and is, in spite of considerable effort, still open. A series of papers by Mills (1984), Mullin, Hoffman and Lindner (1987),Zhu, Du and Yin (1987), Mullin (1989a), Abel and Mills (1995), Abel (1996), Abel and Greig (1997) and Mills (1997) reduce the open cases to the 33 values of v listed in Table 5.2 of the appendix; here the most recent updates are taken from Colbourn and Dinitz (1998). Summarising, we have the following result.
12.2 Theorem. The necessary conditions v == 1 or 6 (mod 15) for the existence of a Steiner system S (2, 6; v) are sufficient for v ::: 811. • An outline of the proof of Theorem 12.2 (still with 55 open values) was given by Mills (1996). We next discuss the case k = 7. The most difficult subcase A = 1 was first systematically studied by Zhang (1990); by recent work of Greig (1998a), Abel and Greig (1998) and Janko and Tonchev (1998), the necessary conditions are now known to be sufficient except for the non-existent S(2, 7, 43) and the 22 open cases listed in Table 5.3 in the appendix; again, the most recent updates are taken from Colbourn and Dinitz (1998). For the subcase A ::: 2, the pioneering work was done by Hanani (1989); subsequently, considerable progress was made in papers of Greig (1990), Zhu and Wu (1990), Zhu (1992), Yin and Wu (1993), Abel (1994, 1996), Greig (1998b) and Abel and Greig (1998) so that this case is now almost settled, with only six open cases for A = 2. We summarise the known results as follows:
12.3 Theorem. The necessary conditions for the existence of an S)..(2, 7; v) are sufficient, except for the non-existent designs S(2, 7, 43) and S2(2, 7; 22)
§12. Concluding remarks
683
and, possibly, the following undecided cases: A = 1 and 22 values of v with v ::::: 2605;
A = 2 and v E {274, 358, 574, 694, 988, 994} . •
Greig (1990), Du and Zhu (1990), Du (1994), Greig (1998a) and Abel (1996) obtained similar, powerful results for Steiner systems S(2, k; v) with k = 8 or 9; there are just 38 and 94 undecided cases, respectively. The open cases for k = 8 are listed in Table 5.4 in the appendix. It should be noted that the construction of an S(2, 9; 153) given in Hall (1986) contains an error which apparently cannot be corrected; this also invalidates some recursive constructions in the liteni.ture making use of this design. For this and for a listing of the undecided cases with k = 9, see Section 1.2.1 of Colboum and Dinitz (1996a) and Colboum and Dinitz (1998).
12.4 Theorem. The necessary conditions for the existence of an S(2, k; v) are sufficient for k = 8 and v 2:: 3760 as well asfor k = 9 and v 2:: 16561. • Regarding higher values of A, one knows that the necessary existence conditions for S),.(2, k; v) are sufficient for the pair (k, A) = (8,7), see Yin (1994). With a few possible exceptions, they are also sufficient for the pairs (k, A) = (8,4) and (9, 8); this follows from unpublished work of Malcolm Greig, see Section 1.2.1 of Colbourn and Dinitz (1996a). The existence problem for simple designs S)"(2, k; v) with k ::::: 4 has been discussed in §5. Similar results for k = 5 and various values of A were obtained by Ding and Shen (1993) and Ding (1994). We next consider the status of the existence problem for resolvable 2-designs, beginning with Kirkman systems, that is with A= 1. The existence of RS(2, 5; v) is considered by Chen and Zhu (1987), Zhu, Chen and Du (1987) and Zhu, Du and Zhang (1991) and Abel and Greig (1997) who made considerable progress in settling this problem; there are only five undecided cases which are listed below cf. Colboum and Dinitz (1998). Similarly, the existence of RS(2, 8; v) is considered by Greig and Abel (1997); there are just 66 undecided cases which are listed in Section 1.6.6 of Colboum and Dinitz (1996a).
12.5 Theorem. The necessary conditions for the existence of an RS(2, k; v) are sufficient for k = 5, except possibly for v E {45, 225, 345, 465, 645}, and for k = 8 and v 2:: 24536. •
684
IX. Recursive constructions
Regarding higher values of A, no really general result (in the sense that it would apply to all values of A for a specific value of k) has been established up to now. However, there are a few pairs (k, A) for which the necessary conditions are known to be sufficient; the following results are due to Hanani (1974a), Baker (1975), Rokowska (1981), Baker and Wilson (1983) and Baker (1983).
12.6 Theorem. The necessary conditions for the existence of a RS).(2, k; v) are sufficient for the pairs (k, A) E {(3, 2), (4, 3), (6, 1O)} except for the nonexistent RS2 (2, 3; 6). • Moreover, there are a few more pairs (k, A) for which the necessary conditions are known to be almost sufficient; the following results are due to Rokowska (1984), Miao (1994), Miao and Zhu (1995), Furino, Greig et al' (1994), Lee (1995), Abel (1996) and Greig and Abel (1997), see also Section 1.6 ofColboum and Dinitz (1996a, 1998).
12.7 Theorem. The necessary conditions for the existence of an RS).(2, k; v) are sufficient for the following pairs (k, A) as soon as v > vo(k, A), with the number of possible exceptions for v listed: k }..
vo(k, }..)
exceptions
5 2 405
5 4 195
5 594
7 6 33936
13
10
10
?
6
8 7 1488 22
12 11 108768 see Lee (1995)
The existence question for simple resolvable designs RS).(2, k; v) with k = 3, A = 2 and k = 4, A ::: 5 is considered and almost completely settled by Shen (1988) and Shen and Wu (1990); there are only two undecided cases for k = 4, A = 11.
12.8 Remark. We note that a Kirkman system RS(2, k; v) can also be defined in terms of divisible designs as either an RGD[k, 1; v] (if we consider the singletons as trivial point classes) or an RGD[k, k; v] (if we consider the blocks in a fixed parallel class as point classes). In this context, we report briefly on nearly Kirkman systems which may be defined in terms of GDD's as RGD[k, k - I; w)'s. However, we prefer the following interpretation: an RGD[k, k - I; v-I] is, of course, equivalent to an S(2, k; v) where one omits a fixed point and considers the blocks through this point as point classes and where the remaining blocks are partitioned into parallel classes; accordingly, we
§12. Concluding remarks
685
shall call such an S(2, k; v) nearly resolvable and denote it as an NRS(2, k; v). The necessary condition for the existence of anNRS(2, k; v) is easily seen to be (12.8.a)
v == 1 (mod k(k - 1».
We have the following existence results for k = 3 due to Kotzig and Rosa (1974), Baker and Wilson (1977), Brouwer (1978a) and Guy, see Huang, Mendelsohn and Rosa (1983), and for k = 4 due to Shen (1992b):
12.9 Theorem. The necessary condition v == 1 (mod 6) for the existence of an NRS(2, 3; v) is sufficient for v ::: 18. The necessary condition v == 1 (mod 12) for the existence ofanNRS(2, 4; v) is sufficientfor v ::: 901 (with 16 undecided cases below this bound). • 12.10 Remark and Definition. There is an different notion of nearly resolvable designs in the literature: An NRB( v, k) is a design D whose block set can be partitioned into v classes such that each class contains every point of D but one, and each point is missing from exactly one class. It is easily seen that D has to have "A = k - 1 and that the necessary existence condition is v == I (mod k). We refer the interested reader to Yin and Miao (1993), Furino (1990, 1995), Furino and Vanstone (1993), Lee (1995) and the monograph by Furino, Miao and Yin (1996) for the following result. For the cases 6 ::; k ::; 8, see Table 6.26 in Colbourn and Dinitz (1996a, 1998). 12.11 Theorem. The necessary condition v == 1 (mod k) for the existence of an NRB(v, k) is always sufficient for k ::; 5. • One may also consider a-resolvable designs for arbitrary a. The necessary conditions here are easily seen to be"A(v-l) == 0 (moda(k-l» and"Av(v-l) == o (mod k(k - 1». The case k = 3 has been completely settled by Jungnickel, Mullin and Vanstone (1991); Furino (1990) and Furino and Mullin (1993) considered the case k = 4 and solved it almost completely.
12.12 Theorem. The necessary conditions (12.12.a)
"A(v - 1)
== 0 (mod a(k-I»
and "Av(v-I)
== 0 (mod k(k-l»
for the existence of an a-resolvable SA (2, k; v) are sufficient for k = 3, v oF 6 and for k = 4, except possibly for 11 undecided cases. •
§ 12. Concluding remarks
687
12.15 Theorem. The necessary conditions for the existence of an S),. (3, 4; v) are sufficient. • Regarding simplicity, the only case that has been settled is J.... = 3, where the necessary condition is sufficient, cf. Theorem VIII.6.5. Hanani (1979) gave the following recursive constructions for 3-designs. Only recently progress towards a general theory of recursive constructions of 3-designs (similar to Wilson's methods discussed in the present chapter) was made by Hartman (1994). 12.16 Theorem. Letq be a prime power. The existence of S),.(3, q + 1; v + 1) implies that of S),.(3, q + 1; qv + 1). Moreover, the existence of S),.(3, 6; v + 1) implies that of S),. (3,6; 4v + 2). • Hanani (1979) also obtained the only other known general existence theorem for t-designs with t :::: 3: 12.17 Theorem. An S30(3, 5; v) exists for every v :::: 5. • Regarding resolvable designs .with t :::: 3, Hartman (1987) almost settled the existence problem for resolvable Steiner quadruple systems; there are just 23 undecided cases listed in Table 5.5 of the appendix. 12.18 Theorem. The necessary condition v ==40r8 (mod 12)fortheexistence ofanRS(3,4; v) issufficientforv:::: 6364. • The necessary condition v == 0 (mod 4) for designs RS3b(3, 4, v)'s is also sufficient, see Theorem Vill.6.6. We conclude this section with a brief discussion of some further generalisations of t-designs. 12.19 Definition. Let t, k, v and J.... be positive integers with 2 ::: t < k < v. A t-(v, k, J....) covering is an incidence structure (V, B) on v points with blocks of size k such that any t-subset of points is contained in at least J.... blocks in B; the covering problem asks for the determination of the covering number C),.(v, k, t), that is, of the minimal number of blocks in a t-(v, k, J....) covering. Similarly, a t-(v, k, J....) packing is an incidence structure (V, B) on v points with blocks of size k such that any t -subset of points is contained in at most J.... blocks in B; the packing problem asks for the determination of the packing number D),.(v, k, t), that is, of the maximal number of blocks in a t-(v, k, J....) packing.
§ 12. Concluding remarks
687
12.15 Theorem. The necessary conditions for the existence of an S),. (3, 4; v) are sufficient. • Regarding simplicity, the only case that has been settled is J.... = 3, where the necessary condition is sufficient, cf. Theorem VIII.6.5. Hanani (1979) gave the following recursive constructions for 3-designs. Only recently progress towards a general theory of recursive constructions of 3-designs (similar to Wilson's methods discussed in the present chapter) was made by Hartman (1994). 12.16 Theorem. Letq be a prime power. The existence of S),.(3, q + 1; v + 1) implies that of S),.(3, q + 1; qv + 1). Moreover, the existence of S),.(3, 6; v + 1) implies that of S),. (3,6; 4v + 2). • Hanani (1979) also obtained the only other known general existence theorem for t-designs with t :::: 3: 12.17 Theorem. An S30(3, 5; v) exists for every v :::: 5. • Regarding resolvable designs .with t :::: 3, Hartman (1987) almost settled the existence problem for resolvable Steiner quadruple systems; there are just 23 undecided cases listed in Table 5.5 of the appendix. 12.18 Theorem. The necessary condition v ==40r8 (mod 12)fortheexistence ofanRS(3,4; v) issufficientforv:::: 6364. • The necessary condition v == 0 (mod 4) for designs RS3b(3, 4, v)'s is also sufficient, see Theorem Vill.6.6. We conclude this section with a brief discussion of some further generalisations of t-designs. 12.19 Definition. Let t, k, v and J.... be positive integers with 2 ::: t < k < v. A t-(v, k, J....) covering is an incidence structure (V, B) on v points with blocks of size k such that any t-subset of points is contained in at least J.... blocks in B; the covering problem asks for the determination of the covering number C),.(v, k, t), that is, of the minimal number of blocks in a t-(v, k, J....) covering. Similarly, a t-(v, k, J....) packing is an incidence structure (V, B) on v points with blocks of size k such that any t -subset of points is contained in at most J.... blocks in B; the packing problem asks for the determination of the packing number D),.(v, k, t), that is, of the maximal number of blocks in a t-(v, k, J....) packing.
§12. Concluding remarks
689
quoted there. The best studied case is that of triple systems; here we recommend the survey by Colbourn and Rosa (1992). Finally, we mention ternary and, more generally, n-ary designs, where -loosely speaking - points can have multiplicities; ordinary designs are, in this terminology, "binary". See Francel and Sarvate (1994) for possible definitions and references.
§12. Concluding remarks
689
quoted there. The best studied case is that of triple systems; here we recommend the survey by Colbourn and Rosa (1992). Finally, we mention ternary and, more generally, n-ary designs, where -loosely speaking - points can have multiplicities; ordinary designs are, in this terminology, "binary". See Francel and Sarvate (1994) for possible definitions and references.
§1. A recursive construction
691
Proof. Let D be a GD}..[TD*(k), TD},.(k); u] with point set N~, block set A, and point class set G. We shall construct a TD).[k; u], say E, with point set V := N~ x Nj' and point classes {i} x Nj, i E N1. By hypothesis, the following TD's exist. For each G E G there is a TD).[k; IGI], say D G , with
point set point classes
N~ x G, {i} x G, i E N7,
and block set
BG•
For each A EA there is a TD[k; IAIl, say D A, with point set point classes
N1 x (A), {i} x (A), i E N7,
and block set CA , w.l.o.g. with the parallel class P A = {N7 x {x} : x E (A)}. Now set
Let the block list B of E be defined by
B= LB GeG
G
+ LC~, AeA
We claim that E with block list B and point classes {i} x Ni, i EN~, is the desired TD).[k; u]. Let (i, x), (j, y) be two points. If i = j they are in a point class of E, but not in a block of E.lndeed, if x and yare in a point class G (resp. a block A) of D, then (i, x) and (i, y) are in a point class of DG (resp. DA)' If i # j and x = y, then the two points x, y of D are joined by exactly).. blocks of B G , where G E G is the point class containing x. Note that (i, x), (j, x) EPA if x E (A); hence (i, x) and (j, x) are not joined by a block in any C~. Now assume i # j and x # y. If x and yare in a point class G E G, then (i, x) and (j, y) are in exactly ).. blocks of B G , but not in any block of aBA, A EA. Otherwise x and y are in exactly).. blocks AI, ... , A). of D, and for each h E {I, ... , )..} the points (i, x) and (j, y) are in exactly one block of C Ah • Hence E is the desired TD).[k; u].
§1. A recursive construction
691
Proof. Let D be a GD}..[TD*(k), TD},.(k); u] with point set N~, block set A, and point class set G. We shall construct a TD).[k; u], say E, with point set V := N~ x Nj' and point classes {i} x Nj, i E N1. By hypothesis, the following TD's exist. For each G E G there is a TD).[k; IGI], say D G , with
point set point classes
N~ x G, {i} x G, i E N7,
and block set
BG•
For each A EA there is a TD[k; IAIl, say D A, with point set point classes
N1 x (A), {i} x (A), i E N7,
and block set CA , w.l.o.g. with the parallel class P A = {N7 x {x} : x E (A)}. Now set
Let the block list B of E be defined by
B= LB GeG
G
+ LC~, AeA
We claim that E with block list B and point classes {i} x Ni, i EN~, is the desired TD).[k; u]. Let (i, x), (j, y) be two points. If i = j they are in a point class of E, but not in a block of E.lndeed, if x and yare in a point class G (resp. a block A) of D, then (i, x) and (i, y) are in a point class of DG (resp. DA)' If i # j and x = y, then the two points x, y of D are joined by exactly).. blocks of B G , where G E G is the point class containing x. Note that (i, x), (j, x) EPA if x E (A); hence (i, x) and (j, x) are not joined by a block in any C~. Now assume i # j and x # y. If x and yare in a point class G E G, then (i, x) and (j, y) are in exactly ).. blocks of B G , but not in any block of aBA, A EA. Otherwise x and y are in exactly).. blocks AI, ... , A). of D, and for each h E {I, ... , )..} the points (i, x) and (j, y) are in exactly one block of C Ah • Hence E is the desired TD).[k; u].
§2. Transversal designs with A > 1
693
(e) Removing one point from an affine plane of order q yields q2 -1 EGD(q,q -1),
and thus (1.3.f)
N(q2 - 1)::: N(q - 1)
for prime powers q;
in particular (1.3.g)
N(24)::: 3, N(80)::: 7.
(f) Removing three collinear points from a projective plane of order 7 yields
54E GD({7, 8}, {I, 5}) S; TD(6); i.e. (1.3.h)
N(54) :::4.
1.4 Exercise. Give an alternative proof of 30 E TD 2 (7), using an S2(2, 9; 37); cf. Example VI.2.13. §2. Transversal Designs with oX > 1 We shall give a slightly simplified proof of the following theorem of Ranani (1974b, 1975). 2.1 Theorem. (2. La)
TDJ,.(7)=f'
for all A > 1.
Proof. The proof needs several steps. Trivially 1 E TDJ,. (7). For g > 1 we shall show g E TD2(7) n TD3(7). Then, reasoning as in Proposition 1.5.3, we get g E TDJ,.(7) for all A > 1. Also we use MacNeish's Theorem 1.7.7 as well as 12 E TD(7) (see Lemma Vill.3.l3) and Theorem IXAA. Hence it suffices to prove Theorem 2.1 for A = 2, 3 and for g = 2, 3,4,5,6,10, 15,20,30. This will be done below. Recall that the existence of a (g, k; A)-difference matrix implies that of a TDJ,.[k + 1; g]; see Corollary VIll.3.8 . • 2.2 Lemma. (2.2.a)
2 E TD)..(7) for all A > 1.
§2. Transversal designs with A > 1
693
(e) Removing one point from an affine plane of order q yields q2 -1 EGD(q,q -1),
and thus (1.3.f)
N(q2 - 1)::: N(q - 1)
for prime powers q;
in particular (1.3.g)
N(24)::: 3, N(80)::: 7.
(f) Removing three collinear points from a projective plane of order 7 yields
54E GD({7, 8}, {I, 5}) S; TD(6); i.e. (1.3.h)
N(54) :::4.
1.4 Exercise. Give an alternative proof of 30 E TD 2 (7), using an S2(2, 9; 37); cf. Example VI.2.13. §2. Transversal Designs with oX > 1 We shall give a slightly simplified proof of the following theorem of Ranani (1974b, 1975). 2.1 Theorem. (2. La)
TDJ,.(7)=f'
for all A > 1.
Proof. The proof needs several steps. Trivially 1 E TDJ,. (7). For g > 1 we shall show g E TD2(7) n TD3(7). Then, reasoning as in Proposition 1.5.3, we get g E TDJ,.(7) for all A > 1. Also we use MacNeish's Theorem 1.7.7 as well as 12 E TD(7) (see Lemma Vill.3.l3) and Theorem IXAA. Hence it suffices to prove Theorem 2.1 for A = 2, 3 and for g = 2, 3,4,5,6,10, 15,20,30. This will be done below. Recall that the existence of a (g, k; A)-difference matrix implies that of a TDJ,.[k + 1; g]; see Corollary VIll.3.8 . • 2.2 Lemma. (2.2.a)
2 E TD)..(7) for all A > 1.
§2. Transversal designs with A > 1
695
Now the exceptional values 10, 15, 20, 30 are covered by (VillA.33.c, d, e, f), (X.1.3.c, e), and Exercise X.1.4, proving Theorem 2.1. 2.8 Corollary. (a) If A > 1, k (b) If A > 1, k ~7, and r
~ 7,
and r E B(k, A), then kr E B(k, A).
+ 1 E B(k, A), then kr + 1 E B(k, A).
Proof. (a) k E GD(k, 1) and r E TD;.(k) imply kr E GD)..(k, r) S; B(k, A), by Corollary IX.3.6 and (IX.2.6.a). (b) k E GD(k, 1) andr E TD)..(k) imply kr E GD)..(k, r), kr + 1 E B(k, A), using (IX.2.6.b) . • 2.9 Examples.
(2.9.a)
36 E B(6, A)
for all A > 1,
(2.9.b)
43 E B(7, A)
for all A > 1.
(2.9.c)
91,96 E B(6, 2),
using Corollary II.3.16.
(2.9.d)
99, 105 E B(7, 3),
using Proposition 1.2.16.
2.10 Corollary. For k
(2.1O.a)
B(k) S;
~ 7,
A > 1, and all g EN
Rf)..·
Proof. Exercise. • 2.11 Exercise and Remark. (a) Show (2.11.a)
2ETD)..(11)
(2.11.b)
2u E TD)..(11)
for all 1..:::3,
for A::: 3 and U E B(P{f) (with the notation of Lemma IXA.3). (b) The question whether 2 E TD)..(4A - 1) for all A EN depends on the existence conjecture for Hadamard matrices. Colbourn (1996b) studied the spectrum of transversal designs with A > 2 and k 8 or 9. He obtained most of the following results, the proofs of which are considerably more involved than that of Hanani's Theorem 2.1; a few further
=
§2. Transversal designs with A > 1
695
Now the exceptional values 10, 15, 20, 30 are covered by (VillA.33.c, d, e, f), (X.1.3.c, e), and Exercise X.1.4, proving Theorem 2.1. 2.8 Corollary. (a) If A > 1, k (b) If A > 1, k ~7, and r
~ 7,
and r E B(k, A), then kr E B(k, A).
+ 1 E B(k, A), then kr + 1 E B(k, A).
Proof. (a) k E GD(k, 1) and r E TD;.(k) imply kr E GD)..(k, r) S; B(k, A), by Corollary IX.3.6 and (IX.2.6.a). (b) k E GD(k, 1) andr E TD)..(k) imply kr E GD)..(k, r), kr + 1 E B(k, A), using (IX.2.6.b) . • 2.9 Examples.
(2.9.a)
36 E B(6, A)
for all A > 1,
(2.9.b)
43 E B(7, A)
for all A > 1.
(2.9.c)
91,96 E B(6, 2),
using Corollary II.3.16.
(2.9.d)
99, 105 E B(7, 3),
using Proposition 1.2.16.
2.10 Corollary. For k
(2.1O.a)
B(k) S;
~ 7,
A > 1, and all g EN
Rf)..·
Proof. Exercise. • 2.11 Exercise and Remark. (a) Show (2.11.a)
2ETD)..(11)
(2.11.b)
2u E TD)..(11)
for all 1..:::3,
for A::: 3 and U E B(P{f) (with the notation of Lemma IXA.3). (b) The question whether 2 E TD)..(4A - 1) for all A EN depends on the existence conjecture for Hadamard matrices. Colbourn (1996b) studied the spectrum of transversal designs with A > 2 and k 8 or 9. He obtained most of the following results, the proofs of which are considerably more involved than that of Hanani's Theorem 2.1; a few further
=
§3. A construction of Wilson
697
As point classes in the point set V*, define the subsets (3. 1.d)
G1 := (Gi
x M) U ({i} x S)
for i E 1, ... , k,
and put G* := {Gi, .. ·, Gk}. Next we construct blocks in V* from any given block A EA in V. By (3.l.b) there is a TD[k; m + UA], say T A , with point set PA := (Ao x M) U (Nt x A'), point classes [(Ao n Gi) x M] U ({i} x A'), i = 1, ... , k, and block
setB A . Since T A has
UA
disjoint blocks, we may assume that, w.l.o.g.,
N1 x {Z},
ZEA',
are blocks of B A. (The case A' = 0 is not excluded.) Let B~ denote the set of the remaining (m + UA)2 - UA blocks of B A , and put (3.1.e)
B:=
U B~. AeA
Finally, we use (3.l.a) to construct further blocks. For each j E N~ there is a TD[k; h j ], say T j , with
:=N1
Pj x (SnHj ), {i} x (SnHj ),iE N
point set point classes
1,
and block set
Cj
•
Put (3.1.f)
A* :=BUC1 U ... UC/.
Note that the case S n Hj = 0, Cj = 0 is not excluded. We claim that (V*, A *, G*) is the desired TD[k; mg + s].
(b) The verification First we show that two points of a point class Gi are not joined by a block. Let x, y E G i and m, m' E M. The points (x, m) and (y, m') are not in a Pj , and hence not in a block of C := C 1 U ... U C/. If x =I y they are not both in a set PA , hence not in a block of B. If x = y the points (x, m), (x, m') are in a point class of each TA with x E Ao n G i , and hence in no block of a B A . Now consider two points (i, p), (i, q) with p, q E S. If p and q are in distinct point classes of T, they are joined by a unique A, and they are in a point class of T A.
§3. A construction of Wilson
697
As point classes in the point set V*, define the subsets (3. 1.d)
G1 := (Gi
x M) U ({i} x S)
for i E 1, ... , k,
and put G* := {Gi, .. ·, Gk}. Next we construct blocks in V* from any given block A EA in V. By (3.l.b) there is a TD[k; m + UA], say T A , with point set PA := (Ao x M) U (Nt x A'), point classes [(Ao n Gi) x M] U ({i} x A'), i = 1, ... , k, and block
setB A . Since T A has
UA
disjoint blocks, we may assume that, w.l.o.g.,
N1 x {Z},
ZEA',
are blocks of B A. (The case A' = 0 is not excluded.) Let B~ denote the set of the remaining (m + UA)2 - UA blocks of B A , and put (3.1.e)
B:=
U B~. AeA
Finally, we use (3.l.a) to construct further blocks. For each j E N~ there is a TD[k; h j ], say T j , with
:=N1
Pj x (SnHj ), {i} x (SnHj ),iE N
point set point classes
1,
and block set
Cj
•
Put (3.1.f)
A* :=BUC1 U ... UC/.
Note that the case S n Hj = 0, Cj = 0 is not excluded. We claim that (V*, A *, G*) is the desired TD[k; mg + s].
(b) The verification First we show that two points of a point class Gi are not joined by a block. Let x, y E G i and m, m' E M. The points (x, m) and (y, m') are not in a Pj , and hence not in a block of C := C 1 U ... U C/. If x =I y they are not both in a set PA , hence not in a block of B. If x = y the points (x, m), (x, m') are in a point class of each TA with x E Ao n G i , and hence in no block of a B A . Now consider two points (i, p), (i, q) with p, q E S. If p and q are in distinct point classes of T, they are joined by a unique A, and they are in a point class of T A.
699
§3. A construction of Wilson
Next we shall discuss some special cases of Theorem 3.1. We begin with the casesl = 1 and 1 = 2. In the case 1 = 1 we have UA E to, I}, and hence (3.1.b) reduces to m, m + 1 E TD(k). Thus we obtain 3.3 Theorem. If 0::: u::: g, then for mEN (3.3.a)
N(mg
+ u) ~ min{N(m), N(m + 1), N(g) -
1, N(u)}.
Proof. Put k := 2 + rnin{N(m), N(m + 1), N(g) - 1, N(u)}. Then let T be a TD[k + 1; g] and note that both (3.1.a) and (3.1.b) hold, where S is any usubset of HI' By Theorem 3.1, the assertion follows. (Recall the convention N(O) = N(1) = 00). • 3.4 Lemma. Let g ~ 3. In every TD[k; g] with g blocks.
~
k there are three disjoint
Proof. We first show the existence of two disjoint blocks. Let A be a block, say A = {al .... , ad. For i E there are g -1 blocks X # A with AnX = (ad. Hence the total number of blocks X with X n A # ¢ is 1 + keg - 1) ::: 1 + g2 - g < g2. Since the total number of blocks is g2, the assertion follows. Now let A and B be any two disjoint blocks. Then the total number of blocks Y with AnY # ¢or BnY # ¢isk(k-l)+2(g-k)+2:::g(g-3)+2g+2 < g2, implying the result. •
N1
3.5 Theorem. Ifu, (3.5.a)
N(mg
W E
N5, then
+ u + w) 2: min{N(m), N(m + 1), N(m + 2), N(g) -
2, N(u), N(w»).
Proof. Let k - 2 be the minimum on the right-hand side of (3.5.a) and start with T, a TD[k + 2; g]. For UA E to, 1, 2} there are TD[k; m + UA]'S. Since m + 1 ~ k, m + 2 > k, Lemma 3.4 implies that both (3. La) and (3.l.b) hold (with hi = U, h2 = w). Now Theorem 3.1 yields the assertion. • 3.6 Examples. (a) Let u::: g be an odd number and g (mod 18). Then, by Theorem 3.3, (3.6.a)
N(3g
+ u) ~2.
Thus it is easily seen that (3.6.b)
N(4n
+ 2) ~2
forn EN4'.
== 1,5,7,9,11,13,17
699
§3. A construction of Wilson
Next we shall discuss some special cases of Theorem 3.1. We begin with the casesl = 1 and 1 = 2. In the case 1 = 1 we have UA E to, I}, and hence (3.1.b) reduces to m, m + 1 E TD(k). Thus we obtain 3.3 Theorem. If 0::: u::: g, then for mEN (3.3.a)
N(mg
+ u) ~ min{N(m), N(m + 1), N(g) -
1, N(u)}.
Proof. Put k := 2 + rnin{N(m), N(m + 1), N(g) - 1, N(u)}. Then let T be a TD[k + 1; g] and note that both (3.1.a) and (3.1.b) hold, where S is any usubset of HI' By Theorem 3.1, the assertion follows. (Recall the convention N(O) = N(1) = 00). • 3.4 Lemma. Let g ~ 3. In every TD[k; g] with g blocks.
~
k there are three disjoint
Proof. We first show the existence of two disjoint blocks. Let A be a block, say A = {al .... , ad. For i E there are g -1 blocks X # A with AnX = (ad. Hence the total number of blocks X with X n A # ¢ is 1 + keg - 1) ::: 1 + g2 - g < g2. Since the total number of blocks is g2, the assertion follows. Now let A and B be any two disjoint blocks. Then the total number of blocks Y with AnY # ¢or BnY # ¢isk(k-l)+2(g-k)+2:::g(g-3)+2g+2 < g2, implying the result. •
N1
3.5 Theorem. Ifu, (3.5.a)
N(mg
W E
N5, then
+ u + w) 2: min{N(m), N(m + 1), N(m + 2), N(g) -
2, N(u), N(w»).
Proof. Let k - 2 be the minimum on the right-hand side of (3.5.a) and start with T, a TD[k + 2; g]. For UA E to, 1, 2} there are TD[k; m + UA]'S. Since m + 1 ~ k, m + 2 > k, Lemma 3.4 implies that both (3. La) and (3.l.b) hold (with hi = U, h2 = w). Now Theorem 3.1 yields the assertion. • 3.6 Examples. (a) Let u::: g be an odd number and g (mod 18). Then, by Theorem 3.3, (3.6.a)
N(3g
+ u) ~2.
Thus it is easily seen that (3.6.b)
N(4n
+ 2) ~2
forn EN4'.
== 1,5,7,9,11,13,17
701
§3. A construction of Wilson
assumption on U, V, W. To this end choose a first point Xo E Wand a v-subset V S;; H2 arbitrarily. Denote by X the (g - v)-subset of HI determined by the intersection points HI n B arising from the g - v blocks B joining Xo and the points of H2 \ V. Now choose any u-subset U C HI satisfying X S;; HI \ U, which is possible as u:S v; hence g - v:s g - u. Then the (g - u)(g - v) blocks joining some point in HI \ U to some point in H2 \ V determine at most (g - u)(g - v) - (g - v) + 1 distinct points of H3 (as Xo occurs g - v times in this way) and, as (g - u - 1)(g - v) < w we may include all these points in W. Thus any block A which misses both U and V will meet W, which proves our assumption. •
3.9 Corollary. (3.9.a)
N(90):::: 6.
Proof. Take m = 6, g = 11 and u = v = w
= 8 in Theorem 3.8. •
Wilson (1974a) proved N(n):::: 6 for all n:::: 91. It is remarkable that it was not realised that the critical result N (90) :::: 6 also followed from his Theorem 3.1 until some four years later. This shows that the task of finding suitable specialisations of a general result like Theorem 3.1 is not at all easy. Corollary 3.9 and (VIDA.30.c) allow us to prove N(n):::: 6 for n:::: 77, which will be done in the next section. A further interesting specialisation of Theorem 3.1 is the following result which is a little weaker than one obtained by Wojtas (1978).
3.10 Theorem. Assume 0 < s < g. Then (3.1O.a)
N(mg
+ s) 2:
min{N(m), N(m
+ 1), N*(m + s), N(g) -
s}.
Proof. Let k - 2 be the minimum occurring in (3.1O.a), take I = s in Theorem 3.1, and let S consist of s points of a block A. Then UA E {O, 1, s}, and (3.1.b) holds for UA = s, as there is a TD[k; m + s] with a parallel class, by Lemma VIDA.l3 . • 3.11 Corollary. (3.11.a)
N(91) 2: 7.
Proof. Take m = 8, g = 11, s = 3 in Theorem 3.10. •
701
§3. A construction of Wilson
assumption on U, V, W. To this end choose a first point Xo E Wand a v-subset V S;; H2 arbitrarily. Denote by X the (g - v)-subset of HI determined by the intersection points HI n B arising from the g - v blocks B joining Xo and the points of H2 \ V. Now choose any u-subset U C HI satisfying X S;; HI \ U, which is possible as u:S v; hence g - v:s g - u. Then the (g - u)(g - v) blocks joining some point in HI \ U to some point in H2 \ V determine at most (g - u)(g - v) - (g - v) + 1 distinct points of H3 (as Xo occurs g - v times in this way) and, as (g - u - 1)(g - v) < w we may include all these points in W. Thus any block A which misses both U and V will meet W, which proves our assumption. •
3.9 Corollary. (3.9.a)
N(90):::: 6.
Proof. Take m = 6, g = 11 and u = v = w
= 8 in Theorem 3.8. •
Wilson (1974a) proved N(n):::: 6 for all n:::: 91. It is remarkable that it was not realised that the critical result N (90) :::: 6 also followed from his Theorem 3.1 until some four years later. This shows that the task of finding suitable specialisations of a general result like Theorem 3.1 is not at all easy. Corollary 3.9 and (VIDA.30.c) allow us to prove N(n):::: 6 for n:::: 77, which will be done in the next section. A further interesting specialisation of Theorem 3.1 is the following result which is a little weaker than one obtained by Wojtas (1978).
3.10 Theorem. Assume 0 < s < g. Then (3.1O.a)
N(mg
+ s) 2:
min{N(m), N(m
+ 1), N*(m + s), N(g) -
s}.
Proof. Let k - 2 be the minimum occurring in (3.1O.a), take I = s in Theorem 3.1, and let S consist of s points of a block A. Then UA E {O, 1, s}, and (3.1.b) holds for UA = s, as there is a TD[k; m + s] with a parallel class, by Lemma VIDA.l3 . • 3.11 Corollary. (3.11.a)
N(91) 2: 7.
Proof. Take m = 8, g = 11, s = 3 in Theorem 3.10. •
§4. Six and more mutually orthogonal Latin squares
703
§4. Six and More Mutually Orthogonal Latin Squares The main result of this section is the following theorem on the existence of six MOLS which is essentially due to Wilson (1974a). Only the cases n = 76, 82 and 90 were settled later; they are due to Colboum, Yin and Zhu (1995), Mullin, Schellenberg et al. (1980a) and Wojtas (1978), respectively.
4.1 Theorem. One has (4. La)
N(n) 2: 6
for n 2: 76.
Proof. For the case n = 76, we refer to Colboum, Yin and Zhu (1995). The main part of the proof is an application of Theorem 3.5 with m = 7 which gives (4.l.b)
N (7 g
+ u + w) 2: 6
whenever 0 :s u, w and
:s g,
N (g) 2: 8
N(u), N(w) 2: 6.
Compare this result with the similar-looking formula (IX.4.3.c). Using (4.l.b) and (IX.4.3.d), we obtain (4.l.c)
N~~~g,7g+80) S; TD(8)
whenever N(g) 2: 8.
Using (4.l.c) for g = 79, 81, 83, ... and observing IX.4.3.(f), we obtain
(4. 1. d)
N560 S; TD(8).
To cover values n < 560 we will also require Theorem 3.7 for I E ~ (note that here g 2: 13 > (;)) and m = 7: (4.l.e)
N(7g
+ l) 2: 6
for I E N~ provided that N(g) 2: 12.
This allows an improvement in the lower bound in (4.l.c) as follows.
(4.1.f)
g g N;:{8 ,7 +80)
S; TD(8)
whenever N(g) 2: 12.
This now yields (4. 1. g)
by using (4.1.c) and (4.1.0 for g = 13, 16, 17, 19,23,25,27,29,32,37,41, 43,49,53,59,67 and 71 and by observing that for g = 13, 19,31 one even obtains N~P, Nl~~, N~g~ S; TD(8), which is seen from (4.1.b) and the table in
§4. Six and more mutually orthogonal Latin squares
703
§4. Six and More Mutually Orthogonal Latin Squares The main result of this section is the following theorem on the existence of six MOLS which is essentially due to Wilson (1974a). Only the cases n = 76, 82 and 90 were settled later; they are due to Colboum, Yin and Zhu (1995), Mullin, Schellenberg et al. (1980a) and Wojtas (1978), respectively.
4.1 Theorem. One has (4. La)
N(n) 2: 6
for n 2: 76.
Proof. For the case n = 76, we refer to Colboum, Yin and Zhu (1995). The main part of the proof is an application of Theorem 3.5 with m = 7 which gives (4.l.b)
N (7 g
+ u + w) 2: 6
whenever 0 :s u, w and
:s g,
N (g) 2: 8
N(u), N(w) 2: 6.
Compare this result with the similar-looking formula (IX.4.3.c). Using (4.l.b) and (IX.4.3.d), we obtain (4.l.c)
N~~~g,7g+80) S; TD(8)
whenever N(g) 2: 8.
Using (4.l.c) for g = 79, 81, 83, ... and observing IX.4.3.(f), we obtain
(4. 1. d)
N560 S; TD(8).
To cover values n < 560 we will also require Theorem 3.7 for I E ~ (note that here g 2: 13 > (;)) and m = 7: (4.l.e)
N(7g
+ l) 2: 6
for I E N~ provided that N(g) 2: 12.
This allows an improvement in the lower bound in (4.l.c) as follows.
(4.1.f)
g g N;:{8 ,7 +80)
S; TD(8)
whenever N(g) 2: 12.
This now yields (4. 1. g)
by using (4.1.c) and (4.1.0 for g = 13, 16, 17, 19,23,25,27,29,32,37,41, 43,49,53,59,67 and 71 and by observing that for g = 13, 19,31 one even obtains N~P, Nl~~, N~g~ S; TD(8), which is seen from (4.1.b) and the table in
§4. Six and more mutually orthogonal Latin squares
705
Put g := (n - u)/8; then N(g), N(u)?: 10 by MacNeish's theorem. If g?: u, i.e. n?: 9u, then (3.3.a) with n = 16 works. But n?: u ·239 = 2151 implies n?: 9u. Using Lemma IX.2.1, it is easily seen that n E B(P8') . •
4.4 Proposition. (4.4. a)
N~728 ~ TD(9).
Proof. Let n EN be even. We want to apply (3.3.a) with m uo, u 1, U as follows.
== 31 + n (mod 62),
(4.4.b)
Uo
(4.4.c)
u 1 = Uo
such that ul, n (4.4.d)
u=
ul
Ul
+ 62x,
x
E
= 31. Determine
0 < Uo < 62,
N~
¥= 0 (mod 3);
+ 186y,
Y ENci,
such that u, n - u ¥= 0 (mod 5, 7). Put g := n:;t. Then N(g), N(u)?: 10 and (3.3.a) works provided g ?: u, i.e. n ?: 32 . 929. •
4.5 Remark. A very important construction method not covered in this text is the use of "incomplete" transversal designs; loosely speaking, these are transversal designs with "holes", that is, sets of points which are not required to be joined by a line. Equivalently, one may talk about sets of mutually orthogonal Latin squares with holes. Such objects were first introduced by Dinitz and Stinson (1983); a nice discussion is contained in the survey by Heinrich (1991). For a recent paper with many relevant references, see Xu and Zhu (1995). Though MOLS with holes are interesting objects in their own right, their main use lies in constructing ordinary sets of M 0 LS; for a description of this method as well as the known construction methods for MOLS with holes, see Colboum and Dinitz (1996b). Another recent method which has resulted in many improved bounds on the number of M 0 LS of "small" orders is the uses of "thwarts" which constitutes a way to exploit Wilson's fundamental construction, see Colboum, Dinitz and Wojtas (1995) and Colboum, Dinitz and Stinson (1996). For recent discussions of construction techniques for MOLS, see Colbourn (1995a) and Colboum and Dinitz (1996b); the latter of these two papers explains how the extensive table of bounds (for orders:::: 10 000) given in Colboum and Dinitz (1996a) was prepared.
§4. Six and more mutually orthogonal Latin squares
705
Put g := (n - u)/8; then N(g), N(u)?: 10 by MacNeish's theorem. If g?: u, i.e. n?: 9u, then (3.3.a) with n = 16 works. But n?: u ·239 = 2151 implies n?: 9u. Using Lemma IX.2.1, it is easily seen that n E B(P8') . •
4.4 Proposition. (4.4. a)
N~728 ~ TD(9).
Proof. Let n EN be even. We want to apply (3.3.a) with m uo, u 1, U as follows.
== 31 + n (mod 62),
(4.4.b)
Uo
(4.4.c)
u 1 = Uo
such that ul, n (4.4.d)
u=
ul
Ul
+ 62x,
x
E
= 31. Determine
0 < Uo < 62,
N~
¥= 0 (mod 3);
+ 186y,
Y ENci,
such that u, n - u ¥= 0 (mod 5, 7). Put g := n:;t. Then N(g), N(u)?: 10 and (3.3.a) works provided g ?: u, i.e. n ?: 32 . 929. •
4.5 Remark. A very important construction method not covered in this text is the use of "incomplete" transversal designs; loosely speaking, these are transversal designs with "holes", that is, sets of points which are not required to be joined by a line. Equivalently, one may talk about sets of mutually orthogonal Latin squares with holes. Such objects were first introduced by Dinitz and Stinson (1983); a nice discussion is contained in the survey by Heinrich (1991). For a recent paper with many relevant references, see Xu and Zhu (1995). Though MOLS with holes are interesting objects in their own right, their main use lies in constructing ordinary sets of M 0 LS; for a description of this method as well as the known construction methods for MOLS with holes, see Colboum and Dinitz (1996b). Another recent method which has resulted in many improved bounds on the number of M 0 LS of "small" orders is the uses of "thwarts" which constitutes a way to exploit Wilson's fundamental construction, see Colboum, Dinitz and Wojtas (1995) and Colboum, Dinitz and Stinson (1996). For recent discussions of construction techniques for MOLS, see Colbourn (1995a) and Colboum and Dinitz (1996b); the latter of these two papers explains how the extensive table of bounds (for orders:::: 10 000) given in Colboum and Dinitz (1996a) was prepared.
§5. The theorem of Chowla, Erdos and Straus with 2h-
(5.3.b)
1
:::: x
+ 1 < 2h if n is odd, and h =
t;j=O,
707
0 if n is even, respectively, and
u;j=O(modq)
for all primes q :::: x.
Proof Since gcd (m, x!) = 1, the congruences (5.3.c)
t ;j= 0,
n - 2hmt ;j= 0 (mod q)
have a solution t for each prime q :::: x. By the Chinese Remainder Theorem, they have a simultaneous solution t. Now put
If t, u E Z satisfy (5.3.a) and (5.3.b), then t too. Hence we may assume
± Jrx
and u
+ 2hmJrx satisfy them,
(5.3.d) 5.4 Theorem. (SA.a)
lim N(n)
n-+co
= 00.
Proof. For given odd x EN, choose m as in Lemma 5.2. If the numbers u, t of Lemma 5.3 satisfy the inequality (5A.b)
then by MacNeish's theorem N(2 ht), N(u) > x, since x Condition (5A.b) certainly holds if (5.4. c)
+ 1 is not a prime.
n 2: m(m + 1)2h . Jrx =: no.
Now apply Theorem 3.3 with g = 2ht. Thus N(n) 2: x for all n 2: no . • As an application we have already introduced the numbers nr (r E N) in Remark 4.2. 5.5 Remark. Note that Theorem SA and the results quoted in 5.1 are of an asymptotic nature and do not provide us with an explicit function f such that
§5. The theorem of Chowla, Erdos and Straus with 2h-
(5.3.b)
1
:::: x
+ 1 < 2h if n is odd, and h =
t;j=O,
707
0 if n is even, respectively, and
u;j=O(modq)
for all primes q :::: x.
Proof Since gcd (m, x!) = 1, the congruences (5.3.c)
t ;j= 0,
n - 2hmt ;j= 0 (mod q)
have a solution t for each prime q :::: x. By the Chinese Remainder Theorem, they have a simultaneous solution t. Now put
If t, u E Z satisfy (5.3.a) and (5.3.b), then t too. Hence we may assume
± Jrx
and u
+ 2hmJrx satisfy them,
(5.3.d) 5.4 Theorem. (SA.a)
lim N(n)
n-+co
= 00.
Proof. For given odd x EN, choose m as in Lemma 5.2. If the numbers u, t of Lemma 5.3 satisfy the inequality (5A.b)
then by MacNeish's theorem N(2 ht), N(u) > x, since x Condition (5A.b) certainly holds if (5.4. c)
+ 1 is not a prime.
n 2: m(m + 1)2h . Jrx =: no.
Now apply Theorem 3.3 with g = 2ht. Thus N(n) 2: x for all n 2: no . • As an application we have already introduced the numbers nr (r E N) in Remark 4.2. 5.5 Remark. Note that Theorem SA and the results quoted in 5.1 are of an asymptotic nature and do not provide us with an explicit function f such that
§6. Transversal designs and orthogonal arrays
709
as in Theorem 11.8.8. Let us write ni for the number of points x =f:. p with Ax = i. Then we have (6.2.a) and rewriting equations (IT.8.8.b, c) yields (6.2.b)
:L>ni = LAx = r(sp, - 1), xip
(6.2.c)
L i (i -
l)ni
=L
We shall consider the function (6.2.d)
Ax(A x
-
1)
= r(r -
l)(fL - 1).
xip
i
f : Z -7 Z defined by
fey) = L(i - y)(i - 1 - y)ni' i
which is easily seen to be non-negative for all Y E Z. From (6.2.a, b, c, d) we obtain
fey)
= LiCi -1)ni i
- 2y Lini i
= p,[r(r - 1) - 2rsy + y(y
+ y(y + I) Ln; i
+ 1)s2)
- [r(r - 1) - 2ry + y(y + 1)]; and
f
(y) ?:: 0 implies
(6.2.e)
r(r - I) - 2ry + y(y + 1) --------=-..:......;--..,- r(r - 1) - 2rsy + y(y + l)s2
fL>
With (6.2.f)
C:= r -1- ys
one obtains (6.2.g)
D := r(r -1) -2rsy+ y(y+ l)s2 = res -1) - (C + 1)(s -1- C)
and then from (6.2.e) after some computation (6.2.h)
C))
S2 fL - 1 ( C(s --->r 1+---
s-1 -
D'
§6. Transversal designs and orthogonal arrays
709
as in Theorem 11.8.8. Let us write ni for the number of points x =f:. p with Ax = i. Then we have (6.2.a) and rewriting equations (IT.8.8.b, c) yields (6.2.b)
:L>ni = LAx = r(sp, - 1), xip
(6.2.c)
L i (i -
l)ni
=L
We shall consider the function (6.2.d)
Ax(A x
-
1)
= r(r -
l)(fL - 1).
xip
i
f : Z -7 Z defined by
fey) = L(i - y)(i - 1 - y)ni' i
which is easily seen to be non-negative for all Y E Z. From (6.2.a, b, c, d) we obtain
fey)
= LiCi -1)ni i
- 2y Lini i
= p,[r(r - 1) - 2rsy + y(y
+ y(y + I) Ln; i
+ 1)s2)
- [r(r - 1) - 2ry + y(y + 1)]; and
f
(y) ?:: 0 implies
(6.2.e)
r(r - I) - 2ry + y(y + 1) --------=-..:......;--..,- r(r - 1) - 2rsy + y(y + l)s2
fL>
With (6.2.f)
C:= r -1- ys
one obtains (6.2.g)
D := r(r -1) -2rsy+ y(y+ l)s2 = res -1) - (C + 1)(s -1- C)
and then from (6.2.e) after some computation (6.2.h)
C))
S2 fL - 1 ( C(s --->r 1+---
s-1 -
D'
711
§6. Transversal designs and orthogonal arrays
This is the result of Bose and Bush. As (b - p)(b + 1 - p)::: -~, we obtain 0= (b - p)(b
(6.4.d)
b
+ 1-
p) - s(b - 2p)::::
1
-4 -
s(b - 2p),
1
P -<-+-
and (6.4.c) simplifies to (6.4.e)
n>p.
This shows that Assumption 6.3 was reasonable. We have proved the following theorem of Bose and Bush (1952).
6.5 Theorem. a(s - 1)
(6.S.a)
If
a TDJL[r; s] with f.L > 0, s > 0 exists and 1, then
if
f.L - 1
+ b with a E No, 0 < b < s r<sf.L+f.L+a-p,
where p = -(s - b -
~) + JS(S
- 1 - b)
+~ .•
6.6 Remark. If ~ is small, then the order of magnitude of p may be seen from the Taylor expansion (6.6.a)
1 b (1 +b) 2 p >b+ - -s+Js(s -1- b) = - - -'----
2
2
8s
(1 +b)3
16s 2
Of course it is desirable to exhibit cases where (6.S.a) is best possible. To this end we introduce a construction using symmetric TD's. As these are simultaneously nets we may use the language of nets which is more convenient here. The basic idea goes back to Shrikhande (1964). It has been used by Hine and Mavron (1980) as well as by Jungnickel (1979) to construct TD's or nets with large values of r.
6.7 Lemma. Let:E be a symmetric (s, f.L)-net (i.e. an STDJL(s») with s I f.L, (cf I.7.17) and assume the existence of an (s, t; t;)-net. Then :E may be extended to an (s, Sf.L + t; f.L)-net by adding t new parallel classes of blocks. Proof. Let D be an (s, t; ;)-net and choose a bijection a mapping the Sf.L points of D onto the s f.L point classes of 'E. For each block B of D, write B" for
711
§6. Transversal designs and orthogonal arrays
This is the result of Bose and Bush. As (b - p)(b + 1 - p)::: -~, we obtain 0= (b - p)(b
(6.4.d)
b
+ 1-
p) - s(b - 2p)::::
1
-4 -
s(b - 2p),
1
P -<-+-
and (6.4.c) simplifies to (6.4.e)
n>p.
This shows that Assumption 6.3 was reasonable. We have proved the following theorem of Bose and Bush (1952).
6.5 Theorem. a(s - 1)
(6.S.a)
If
a TDJL[r; s] with f.L > 0, s > 0 exists and 1, then
if
f.L - 1
+ b with a E No, 0 < b < s r<sf.L+f.L+a-p,
where p = -(s - b -
~) + JS(S
- 1 - b)
+~ .•
6.6 Remark. If ~ is small, then the order of magnitude of p may be seen from the Taylor expansion (6.6.a)
1 b (1 +b) 2 p >b+ - -s+Js(s -1- b) = - - -'----
2
2
8s
(1 +b)3
16s 2
Of course it is desirable to exhibit cases where (6.S.a) is best possible. To this end we introduce a construction using symmetric TD's. As these are simultaneously nets we may use the language of nets which is more convenient here. The basic idea goes back to Shrikhande (1964). It has been used by Hine and Mavron (1980) as well as by Jungnickel (1979) to construct TD's or nets with large values of r.
6.7 Lemma. Let:E be a symmetric (s, f.L)-net (i.e. an STDJL(s») with s I f.L, (cf I.7.17) and assume the existence of an (s, t; t;)-net. Then :E may be extended to an (s, Sf.L + t; f.L)-net by adding t new parallel classes of blocks. Proof. Let D be an (s, t; ;)-net and choose a bijection a mapping the Sf.L points of D onto the s f.L point classes of 'E. For each block B of D, write B" for
713
§7. Completion theorems for Bruck nets
6.10 Exercise (Bose and Bush 1952). Let p be a prime, i a positive, and j a non-negative integer. Define a, b by j = ai + b with 0 -:s. b < i. Show that
(6. 10. a) Compare this to (Vill.3.12.a).
6.11 Remarks. (a) Some infinite series of transversal designs which nearly meet the Bose-Bush bound (Theorem 6.5) were constructed by de Launey (1987b). (b) In view of Lemma VIDA A, the Bose-Bush bound may be viewed as a bound on orthogonal arrays of strength 2. In this context, we mention the following bound for orthogonal arrays of strength t which is due to Bierbrauer (1995a). For large values of t, this result improves the bounds of Rao (1947) which are also derived in Bierbrauer's paper. It also covers the cases s of the bound of Bush (1952) which concerns the special case J.. = 1.
t:::
6.12 Theorem. Assume the existence of an OA). (t, k, s). Then (6.I2.a)
b=
ASt
>Sk(I- -I)k) .• + -
(s s(t
1)
§7. Completion Theorems for Bruck Nets In this section we consider nets with f.t = 1, i.e. dual structures of transversal designs with A = 1. Bruck (1963) has studied possible embeddings of such nets into affine planes (by adjoining new parallel classes of lines). His results have been improved significantly by Metsch (199Ia). Together with the Bruck-Ryser theorem their results imply the non-existence of nets for certain parameters.
7.1 Definition. Let D be an (s, r; I)-net, that is an affine Sr(1, s; S2) or the dual structure of a TD[r, s]. Then the deficiency d of D is defined by (7. La)
d:= s
+ 1- r;
thus d measures how far r falls short from its conceivable maximum s + 1, cf. Corollary II.2.13. If D can be embedded to an affine plane, say E, by adjoining new parallel classes of lines, then E is called a completion of D, and D is said to be completable (or embeddable).
713
§7. Completion theorems for Bruck nets
6.10 Exercise (Bose and Bush 1952). Let p be a prime, i a positive, and j a non-negative integer. Define a, b by j = ai + b with 0 -:s. b < i. Show that
(6. 10. a) Compare this to (Vill.3.12.a).
6.11 Remarks. (a) Some infinite series of transversal designs which nearly meet the Bose-Bush bound (Theorem 6.5) were constructed by de Launey (1987b). (b) In view of Lemma VIDA A, the Bose-Bush bound may be viewed as a bound on orthogonal arrays of strength 2. In this context, we mention the following bound for orthogonal arrays of strength t which is due to Bierbrauer (1995a). For large values of t, this result improves the bounds of Rao (1947) which are also derived in Bierbrauer's paper. It also covers the cases s of the bound of Bush (1952) which concerns the special case J.. = 1.
t:::
6.12 Theorem. Assume the existence of an OA). (t, k, s). Then (6.I2.a)
b=
ASt
>Sk(I- -I)k) .• + -
(s s(t
1)
§7. Completion Theorems for Bruck Nets In this section we consider nets with f.t = 1, i.e. dual structures of transversal designs with A = 1. Bruck (1963) has studied possible embeddings of such nets into affine planes (by adjoining new parallel classes of lines). His results have been improved significantly by Metsch (199Ia). Together with the Bruck-Ryser theorem their results imply the non-existence of nets for certain parameters.
7.1 Definition. Let D be an (s, r; I)-net, that is an affine Sr(1, s; S2) or the dual structure of a TD[r, s]. Then the deficiency d of D is defined by (7. La)
d:= s
+ 1- r;
thus d measures how far r falls short from its conceivable maximum s + 1, cf. Corollary II.2.13. If D can be embedded to an affine plane, say E, by adjoining new parallel classes of lines, then E is called a completion of D, and D is said to be completable (or embeddable).
§7. Completion theorems for Bruck nets
7IS
7.5 Lemma. Let D be an (s, r; I)-net of deficiency d, and let p and q be distinct points, L a line, and T a partial transversal ofD. Then: (7.S.a)
ITI ::: s with equality if and only ifT meets each line.
(7.S.b)
p is joined to n! := res - 1) points and not joined to n2 := des - 1) points.
(7.S.c)
Ifp is not on L, then p is collinear with r - 1 points of L and not joined to the remaining s - r + 1 = d points of L.
(7.S.d)
If T is a transversal not containing p, then p is collinear with r points ofT and not joined to the remaining s - r = d - 1 points ofT.
(7.S.e)
If p and q are collinear; then there are precisely P~1 = s - 2 + (r - 1)(r - 2)
points x
p12 =
=I p, q which are collinear with both p and q;
d(r - 1)
points x collinear with p but not with q; and
P~2 = d(d - 1) points x collinear with neither p nor q. (7.S.£)
Ifp and q are not collinear, then there are precisely pi!
= r(r -
points x pi2
1)
=I p, q which are collinear with both p and q;
= red -
1)
points x collinear with p but not with q; and
P~2
= s - 2 + Cd -
I)(d - 2)
points x collinear with neither p nor q. • Remark: The notation P~k (i, j, k E (I, 2}) is chosen as follows. The upper index denotes whether or not p and q (the given two points) are collinear; the lower ones whether or not the points x to be considered are collinear with p or q. Here "being collinear" is denoted by 1, and "not being collinear" by 2.
7.6 Lemma. Let D be an (s, r; 1)-net of deficiency d > 0, and let T be a transversal and S a partial transversal of D such that S % T but IS n TI ::: 2. Then one has (7.6.a)
IS n TI ::: d - 1 and lSI::: Cd -
1)2;
§7. Completion theorems for Bruck nets
7IS
7.5 Lemma. Let D be an (s, r; I)-net of deficiency d, and let p and q be distinct points, L a line, and T a partial transversal ofD. Then: (7.S.a)
ITI ::: s with equality if and only ifT meets each line.
(7.S.b)
p is joined to n! := res - 1) points and not joined to n2 := des - 1) points.
(7.S.c)
Ifp is not on L, then p is collinear with r - 1 points of L and not joined to the remaining s - r + 1 = d points of L.
(7.S.d)
If T is a transversal not containing p, then p is collinear with r points ofT and not joined to the remaining s - r = d - 1 points ofT.
(7.S.e)
If p and q are collinear; then there are precisely P~1 = s - 2 + (r - 1)(r - 2)
points x
p12 =
=I p, q which are collinear with both p and q;
d(r - 1)
points x collinear with p but not with q; and
P~2 = d(d - 1) points x collinear with neither p nor q. (7.S.£)
Ifp and q are not collinear, then there are precisely pi!
= r(r -
points x pi2
1)
=I p, q which are collinear with both p and q;
= red -
1)
points x collinear with p but not with q; and
P~2
= s - 2 + Cd -
I)(d - 2)
points x collinear with neither p nor q. • Remark: The notation P~k (i, j, k E (I, 2}) is chosen as follows. The upper index denotes whether or not p and q (the given two points) are collinear; the lower ones whether or not the points x to be considered are collinear with p or q. Here "being collinear" is denoted by 1, and "not being collinear" by 2.
7.6 Lemma. Let D be an (s, r; 1)-net of deficiency d > 0, and let T be a transversal and S a partial transversal of D such that S % T but IS n TI ::: 2. Then one has (7.6.a)
IS n TI ::: d - 1 and lSI::: Cd -
1)2;
717
§7. Completion theorems for Bruck nets
Thus the new lines also satisfy the parallel axiom (I.2.4.b) and thus E is indeed an affine plane of order s. Assume now that F is any completion of D; then the lines of F \ D are transversals of D, and there are exactly sd such transversals. By (7.8.a), these are all the transversals of D, proving E = F . • Note that Proposition 7.8 provides a prooffor the first part of Theorem 7.2. We stress that one may well have t < sd even if (7.6.b) is satisfied; in §9 we will obtain examples of such nets which are even transversal-free, that is, they have t = O. In order to prove the second part of Theorem 7.2 as well as Theorem 7.3, we will associate two strongly regular graphs with each net. The following result is an immediate consequence of Lemma 7.5.
7.9 Lemma. Let D be an (s, r; I)-net, and define a graph G on the s2 points ofD asfollows:
(7.9.a)
{p, q} is an edge of G if and only
if p and q are collinear.
Then G is a strongly regular graph with parameters
(7.9.b)
n = S2, k = res - 1), J-L = r(r -1) . •
A = s - 2 + (r - l)(r - 2)
and
7.10 Definition. In view of Lemma 7.9, we call a strongly regular graph G with parameters (7.9.b), where 0 < r :::: s, apseudonet graph with parameters s and r. If G is actually the graph of an (s, r; 1)-net as in Lemma 7.9, it is called a net graph (or a geometric pseudonet graph). With these definitions, we obtain the following stronger version of Lemma 7.9; again, the proof is immediate from Lemma 7.5.
7.11 Proposition. LetD be an (s, r; I)-net, and define a graph G as in Lemma 7.9. Then G is a net graph with parameters sand r, and its complementary graph G is a pseudonet graph with parameters sand d := s + I - r. • In what follows, the second assertion of Theorem 7.2 and its improvement in Theorem 7.3 will be established by proving that any pseudonet graph with parameters sand d satisfying (7.2.b) or (7.3.a) is actually geometric; thus the graph G of Proposition 7.11 arises from an (s, d; I)-net in these cases, and adjoining the lines of this net to D yields the required completion of D. We will require two special types of subgraphs in this approach.
717
§7. Completion theorems for Bruck nets
Thus the new lines also satisfy the parallel axiom (I.2.4.b) and thus E is indeed an affine plane of order s. Assume now that F is any completion of D; then the lines of F \ D are transversals of D, and there are exactly sd such transversals. By (7.8.a), these are all the transversals of D, proving E = F . • Note that Proposition 7.8 provides a prooffor the first part of Theorem 7.2. We stress that one may well have t < sd even if (7.6.b) is satisfied; in §9 we will obtain examples of such nets which are even transversal-free, that is, they have t = O. In order to prove the second part of Theorem 7.2 as well as Theorem 7.3, we will associate two strongly regular graphs with each net. The following result is an immediate consequence of Lemma 7.5.
7.9 Lemma. Let D be an (s, r; I)-net, and define a graph G on the s2 points ofD asfollows:
(7.9.a)
{p, q} is an edge of G if and only
if p and q are collinear.
Then G is a strongly regular graph with parameters
(7.9.b)
n = S2, k = res - 1), J-L = r(r -1) . •
A = s - 2 + (r - l)(r - 2)
and
7.10 Definition. In view of Lemma 7.9, we call a strongly regular graph G with parameters (7.9.b), where 0 < r :::: s, apseudonet graph with parameters s and r. If G is actually the graph of an (s, r; 1)-net as in Lemma 7.9, it is called a net graph (or a geometric pseudonet graph). With these definitions, we obtain the following stronger version of Lemma 7.9; again, the proof is immediate from Lemma 7.5.
7.11 Proposition. LetD be an (s, r; I)-net, and define a graph G as in Lemma 7.9. Then G is a net graph with parameters sand r, and its complementary graph G is a pseudonet graph with parameters sand d := s + I - r. • In what follows, the second assertion of Theorem 7.2 and its improvement in Theorem 7.3 will be established by proving that any pseudonet graph with parameters sand d satisfying (7.2.b) or (7.3.a) is actually geometric; thus the graph G of Proposition 7.11 arises from an (s, d; I)-net in these cases, and adjoining the lines of this net to D yields the required completion of D. We will require two special types of subgraphs in this approach.
719
§7. Completion theorems for Bruck nets
since each ofthes(s-l) pairs (v, w) E C X C is adjacenttos-2+(d -l)(d -2) vertices, s - 2 of which belong to C. From (7.13.e, f, g) we obtain (7.13.h) Henceai =
ofor i i=d -1 and ad-l =
s2-s, proving (7. 13.b), sinced -1 < s.
It remains to consider two cliques C and C'. If C U C' is not a clique, then C U C' contains two non-adjacent vertices, and thus (7.9.b) implies (7.13.c). Similarly, if C n C' contains two vertices, then (7.9.b) implies (7.13.d) . • Note that the lines of a net graph yield cliques of order s, that is the maximum feasible value according to (7 .13.a). The following two results are due to Metsch (1991a), improving the work of Bruck (1963).
7.14 Lemma. Let G be a strongly regular graph with parameters n, k, J... and fh, and let a be a positive integer. Then G contains no claw of order a provided that
(7.14.a)
k < (a
+ 1)(J... + 1) -
1
l(a
+ 1)a(fh -
1).
Proof. Assume the existence of a claw (v, A) of order a {vo, ... , va}, and denote by
hypothesis, IVi I =
J...
+1
+ 1. Write A
=
Vi the set of common neighbours of v and Vi' By
for all i. Note
(7.14.b)
since every vertex that is counted x times on the left-hand side is counted 1 + 1x(x - 1) times on the right-hand side and since x :s 1 + 1x(x - 1) for all positive integers x. For distinct indices i and j, we have IVi n Vj I :s fh - 1, since every vertex in V; n Vj is a common neighbour of Vi and vj which is different from v. Since the set U~=O Vi consists of some of the k vertices adjacent to v which are not in A, it follows from (7.14.b) that
1
(a + 1)J... :::: k - (a + 1) + l(a + l)a(fh - 1),
contradicting (7. 14.a). •
719
§7. Completion theorems for Bruck nets
since each ofthes(s-l) pairs (v, w) E C X C is adjacenttos-2+(d -l)(d -2) vertices, s - 2 of which belong to C. From (7.13.e, f, g) we obtain (7.13.h) Henceai =
ofor i i=d -1 and ad-l =
s2-s, proving (7. 13.b), sinced -1 < s.
It remains to consider two cliques C and C'. If C U C' is not a clique, then C U C' contains two non-adjacent vertices, and thus (7.9.b) implies (7.13.c). Similarly, if C n C' contains two vertices, then (7.9.b) implies (7.13.d) . • Note that the lines of a net graph yield cliques of order s, that is the maximum feasible value according to (7 .13.a). The following two results are due to Metsch (1991a), improving the work of Bruck (1963).
7.14 Lemma. Let G be a strongly regular graph with parameters n, k, J... and fh, and let a be a positive integer. Then G contains no claw of order a provided that
(7.14.a)
k < (a
+ 1)(J... + 1) -
1
l(a
+ 1)a(fh -
1).
Proof. Assume the existence of a claw (v, A) of order a {vo, ... , va}, and denote by
hypothesis, IVi I =
J...
+1
+ 1. Write A
=
Vi the set of common neighbours of v and Vi' By
for all i. Note
(7.14.b)
since every vertex that is counted x times on the left-hand side is counted 1 + 1x(x - 1) times on the right-hand side and since x :s 1 + 1x(x - 1) for all positive integers x. For distinct indices i and j, we have IVi n Vj I :s fh - 1, since every vertex in V; n Vj is a common neighbour of Vi and vj which is different from v. Since the set U~=O Vi consists of some of the k vertices adjacent to v which are not in A, it follows from (7.14.b) that
1
(a + 1)J... :::: k - (a + 1) + l(a + l)a(fh - 1),
contradicting (7. 14.a). •
§7. Completion theorems for Bruck nets
721
Step 3. Let (v, A) be a claw with IAI = a v , and let W E A. Then there exists a unique grand clique C containing both v and w. The uniqueness of C is guaranteed by Step 1. Denote by C the set of vertices which are adjacent to both v and w, but to no vertex in A \ {w}. Since wand every other vertex of A have exactly fL - 1 common neighbours distinct from v, we obtain ICI :::).. - (a v - 1)(fL - 1). Any two distinct vertices u and u' of C are adjacent, since otherwise (v, (A \ {w}) U {u, u'}) would be a claw of size a v + 1. Hence C is a clique. As a v :::: a, it follows that C' := C U {v, w} is a clique with at least).. + 2 - (a - 1)(fL - 1) vertices. Therefore any maximal clique containing C' is a grand clique. Step 4. Any vertex v lies in exactly a v grand cliques. Let (v, A) be a claw with IAI = avo In particular, (v, A) is a maximal claw. Step 3 shows that every vertex of A occurs in a unique grand clique containing v, and Step 2 shows that every grand clique through v contains a vertex of A, proving the assertion. Step 5. Any two adjacent vertices v and w occur together in a unique grand clique. Let (v, A) be any maximal claw containing the claw (v, {w}). Then IAI :::: avo By Step 4, there exist exactly au grand cliques containing v, and by Step 2 each of these grand cliques contains a vertex of A. Since distinct grand cliques through v cannot contain any other common vertex by Step 1, it follows that IAI = av and that every vertex of A occurs in a grand clique containing v. Step 6. Let C be a grand clique, and let v and w be two distinct vertices contained in C. Then ICI :::).. + 2 - (a v - 1)(aw - 1). By Step 5, every vertex u which is adjacent to both v and w gives rises to grand cliques C u and Du with u, v E C u and u, WE Du. If u tf. C, then C u, Du i= C and, by Step 1, Cu n Du = {u}. Hence distinct common neighbours u and u' of v and w which are not in C determine different pairs of cliques (C u , Du) and (Cu" D u')' Therefore Step 4 implies that v and w have at most (a v -1)(a w -1) common neighbours which are not in C. Hence).. :::: IC I - 2 + (a v - 1) (a w - 1). Step 7. Every grand clique has order s, and a v = d for every vertex v. Let v be some vertex of G, putt:= a v , and let C 1 , ••• , C t be the grand cliques containing v. By Step 5, each neighbour of v lies in exactly one of these t cliques. Hence t
(7.15.e)
L(IC;I - 1)
= k = des -
1).
;=1
By (7. 13.a), we have ICd :::: s for all i; hence t::: d, with equality if and only if all cliques containing v have order s. Now assume t ::: d + 1 and, without
§7. Completion theorems for Bruck nets
721
Step 3. Let (v, A) be a claw with IAI = a v , and let W E A. Then there exists a unique grand clique C containing both v and w. The uniqueness of C is guaranteed by Step 1. Denote by C the set of vertices which are adjacent to both v and w, but to no vertex in A \ {w}. Since wand every other vertex of A have exactly fL - 1 common neighbours distinct from v, we obtain ICI :::).. - (a v - 1)(fL - 1). Any two distinct vertices u and u' of C are adjacent, since otherwise (v, (A \ {w}) U {u, u'}) would be a claw of size a v + 1. Hence C is a clique. As a v :::: a, it follows that C' := C U {v, w} is a clique with at least).. + 2 - (a - 1)(fL - 1) vertices. Therefore any maximal clique containing C' is a grand clique. Step 4. Any vertex v lies in exactly a v grand cliques. Let (v, A) be a claw with IAI = avo In particular, (v, A) is a maximal claw. Step 3 shows that every vertex of A occurs in a unique grand clique containing v, and Step 2 shows that every grand clique through v contains a vertex of A, proving the assertion. Step 5. Any two adjacent vertices v and w occur together in a unique grand clique. Let (v, A) be any maximal claw containing the claw (v, {w}). Then IAI :::: avo By Step 4, there exist exactly au grand cliques containing v, and by Step 2 each of these grand cliques contains a vertex of A. Since distinct grand cliques through v cannot contain any other common vertex by Step 1, it follows that IAI = av and that every vertex of A occurs in a grand clique containing v. Step 6. Let C be a grand clique, and let v and w be two distinct vertices contained in C. Then ICI :::).. + 2 - (a v - 1)(aw - 1). By Step 5, every vertex u which is adjacent to both v and w gives rises to grand cliques C u and Du with u, v E C u and u, WE Du. If u tf. C, then C u, Du i= C and, by Step 1, Cu n Du = {u}. Hence distinct common neighbours u and u' of v and w which are not in C determine different pairs of cliques (C u , Du) and (Cu" D u')' Therefore Step 4 implies that v and w have at most (a v -1)(a w -1) common neighbours which are not in C. Hence).. :::: IC I - 2 + (a v - 1) (a w - 1). Step 7. Every grand clique has order s, and a v = d for every vertex v. Let v be some vertex of G, putt:= a v , and let C 1 , ••• , C t be the grand cliques containing v. By Step 5, each neighbour of v lies in exactly one of these t cliques. Hence t
(7.15.e)
L(IC;I - 1)
= k = des -
1).
;=1
By (7. 13.a), we have ICd :::: s for all i; hence t::: d, with equality if and only if all cliques containing v have order s. Now assume t ::: d + 1 and, without
X. Transversal designs and nets
722
loss of generality, a w :::: a v = t for every vertex w i= v. Then Step 6 shows IC;! :::}.. + 2 - (t - 1)2 for all i, and thus (7.I5.e) yields (7.I5.f)
teA
Using A + 1 equivalent to
=
+ 1) -
t(t - 1)2 :::: des - I).
s - 1 + (d - I)(d - 2), inequality (7.15.f) turns out to be
-
t(t - 1)2 - t(d - l)(d - 2) t-d
= t(t
+d -
s < 1+ ----------
In view of d+ 1 :::: t = a v
2)
::::
+d +
d(d - 1) t_ d .
a, this contradicts (7.I5.c) and proves the assertion.
Step 8. G is the graph of a uniquely determined (s, d; I)-net.
By Step 7, all grand cliques have order s and every vertex lies in exactly d grand cliques. Thus the only feasible definition for the desired netD is to take all grand cliques of G as lines. Then each line has order s and any two lines intersect in at most one point. It remains to check the parallel axiom. By (7.13.b), a vertex v outside a given grand clique C has precisely d - 1 neighbours in C. Therefore Step 5 and Step 1 show that v lies on precisely d - 1 grand cliques meeting C in a unique vertex, and thus there is a unique grand clique through v which is disjoint to C. Hence D indeed is an (s, d; I)-net, and in view of Step 5 G is the graph of this net. • We can now prove the following results of Bruck (1963) and Metsch (1991a) on pseudonet graphs.
7.16 Theorem. Let G be a pseudonet graph with parameters s::: 2 and assume (7.2.b) and s::: d > O. Then G is the graph of a uniquely determined (s, d; I)-net. Proof. We apply Theorem 7.15 with a = d. Thus we need to check the hypotheses (7.I5.a) and (7.I5.b) of Theorem 7.15. Using (7.15.d), this requires showing 3
s > 2d - 4d
2
+ 2d
and
14
s > -d - d 2
3
+ d 2+1-d 2
1.
The second inequality is just (7 .2.b). If d i= 3, then the second inequality implies the first, proving the theorem for d i= 3. Thus let d = 3.Then the two inequalities require s > 24 and s > 23, respectively. This proves the theorem for d = 3 and
§7. Completion theorems for Bruck nets
723
s > 24. For the single remaining case d = 3 and s = 24, we refer to the original paper of Bruck or the proof given in the first edition of this book. (It would be quite difficult to explain why the present approach does not cover this particular case.) •
7.17 Theorem. Let G be a pseudonet graph with parameters s::: 2 and assume (7.3.a) and s ::: d > O. Then G is the graph of a uniquely determined (s, d; I)-net. Proof. For d :s: 4, the statement follows from Theorem 7.16. Thus let d ::: 5. We apply Theorem 7.15 with a := (4d - 2 - R)/3. By the definition of R in Theorem 7.3, we see that a is an integer with a:::d + 1. Using (7.15.d), the three hypotheses of Theorem 7.15 can be written as follows:
+ (2a - l)(d 2 a(a + l)(d 2 s > 12 := 1 + s>
fl
:= I
d - 1) - (d - I)(d - 2),
d - 1) - 2(a + l)(d - l)(d - 2) 2(a + 1 _ d) ,
s > f3 :=a(a +d - 2) +d2. Since a = (4d - 2 - R) /3, it is tedious but not difficult to see that condition (7.3.a) is equivalent to s > fl if R Eto, I} and to s > 12 if R = 2. Moreover,
6(a
+1-
d)(fl - h) = (d 2 - d - 1)(2 - R)(4d - R - 5) - 6(d - 1)2.
Because of d::: 5, it follows that fI ::: 12 if R Eto, I} and 12::: fl if R = 2. Hence s > II and s > 12 for R E to, 1, 2}. Since 5 :s: d < a :s: 2d, we have
II - /3 =
(a - d)(2d 2
-
4d - a)
+ 2d3 -
7d 2 + 4d > O.
Hence also s > f3. •
7.18 Remarks. (a) Theorem 7.16 and Proposition 7.11 at once yield a proof of the second assertion of Theorem 7.2, and Theorem 7.17 and Proposition 7.11 at once yield a proof of Theorem 7.3.ln §9 we will give examples showing that the bounds in (7.2.b) and (7.3.a) are fairly good. We note that the Metsch bound (7.3.a) is better than the Bruck bound (7.2.b) for all values d::: 5. (b) Metsch (I991a) obtained Theorem 7.17 from a more general result for strongly regular graphs; we have here only treated the specialisation to pseudonet graphs. Metsch's more general theorem allows further applications to partial
X. Transversal designs and nets
724
geometries and to the embedding problem for linear spaces into projective planes, cf. Theorems IX.I1.14 and IX. I 1.15. Theorems 7.2 and 7.3 can be restated as follows: 7.19 Corollary. Assume the existence of an (s, r; I)-net of deficiency d. Then either there exists an affine plane (and hence also a projective plane) of order s, or d satisfies neither of the inequalities (7.2.b) and (7.3.a). • 7.20 Example. A net of order s and deficiency d is completable in each of the following cases: d = 1; d = 2, s ¥= 4, a result of Shrikhande (1961); d =3, s::: 24; d =4, s::: 82; d = 5, s::: 179 etc. Using the Bruck-Ryser Theorem II.4.8 and Remark II.4.12, we obtain
(7.20.a)
N(s) :::: s - 4
for s = 6,10,14,21,22;
(7.20.b)
N(s) :::: s - 5
for s = 30,33,38,42,46,54,57,62,66,69, 70,77,78;
(7.20.c)
N(s) :::: s - 6
for s = 86, 93, 94, etc.
We finally mention another consequence of Theorem 7.16 which is due to Shrikhande (1959b). 7.21 Corollary. Let G be a strongly regular graph with parameters
(7.21.a)
n
= s2,
k
= 2(s -
1),
A=s - 2
and
f.-L
= 2.
If s ¥= 4, then G is uniquely determined. Proof. This is easily checked for s = 1,2,3. Note that the hypothesis means that G is a pseudonet graph with parameters sand d = 2. By Theorem 7.16, G is geometric for s > 4. Thus G is constructed from an (s, 2; I)-net as in Lemma 7.9. But clearly any two (s, 2; I)-nets are isomorphic . • 7.22 Remark. The graphs occurring in Corollary 7.21 are usually called the L2-graphs. Let us remark that there is a unique exceptional graph G in the case s = 4, see Shrikhande (1959b). G is the Cayley graph based on Z4 x Z4 with S = {±(1, 0), ±(O, 1), ±(1, I)}, where we use the notation introduced in Remark III. 10.2.
§8. Maximal nets with large deficiency
725
§8. Maximal Nets with Large Deficiency In this section we are again only concerned with nets with intersection parameter I-t = 1. The completion theorems of the preceding section guarantee that such nets satisfying certain parametric restrictions (i.e. with small deficiency d = s + 1 - r in the order of magnitude of 0) may always be completed regardless of their specific geometric structure. This immediately poses some interesting questions. First of all, one would like to know whether the bounds (7.2.b) and (7.3.a) are best possible or at least reasonably good. Then the more general question arises: for which parameter pairs (s, d) does a net of deficiency d exist which cannot be enlarged to a net of smaller deficiency or which perhaps has no transversals at all? 8.1 Definition. An (s, r; 1)-netD is called maximal if it cannot be embedded into an (s, r + 1; I)-net (which means that D has no set of s pairwise disjoint transversals). If D is not maximal, it is called extendible;J more specifically, if D may be embedded into an (s, r + t; I)-net, it is said to be step-t-extendible.
In this section we will construct maximal nets with few parallel classes, i.e. with large deficiency. The more difficult case of maximal nets with small deficiency (which is relevant to the completion theorems) will be considered in the next section. Maximal nets for I-t > 1 will be studied in Section 11. For I-t = 1, it is sometimes helpful to consider Latin squares instead of nets. 8.2 Definition. Let L = (lij )i,j=J, ... ,s be a Latin square of order s with entries from the set S. A transversal of L is a set T of s cells of L (i.e. T c {I, ... , S}2) such that the cells in T contain each element of S exactly once and intersect each row and each column of L exactly once. 8.3 Observations. Let L be a Latin square of order sand D the corresponding (s, 3; I)-net; cf. Lemma VIII.4.4. Then the transversals of L correspond bijectively to those ofD. More generally, ifD is an (s, r; I)-net and {L J , ••• , L r - 2 } a corresponding set of mutually orthogonal Latin squares, then the transversals of D correspond to those sets of cells T which are transversals for each Li (i = 1, ... , r - 2). 8.4 Example. Bruck's completion theorem is best possible for s = 4. To prove this, we have to find a maximal (4, 3; 1)-net. Consider the Latin square L of order
1
This is not to be confused with the notion of extendibility treated in Chapter II, §7.
726
X. Transversal designs and nets
4 which is the group table of Z4. It is easy to check that L has no transversal; hence the corresponding net D has no transversal and thus is maximal. In fact, D is the net we displayed in Remark 7.22. Example 8.4 may be generalised. If L is the group table of a group G of even order with a cyclic Sylow 2-subgroup, then L has no transversal; see Corollary 12.4 below. This provides examples of maximal (s, 3; I)-nets for all even values of s. We will now give a construction for such nets if s == 1 (mod 4). The following two results and the subsequent exercises are all due to Mann (1944).
8.5 Lemma. Let L be a Latin square of order s = 4m + 1 with a Latin subsquare of order 2m. Then there is no Latin square which is orthogonal to L; that is, L has no orthogonal mate. Proof. We may assume L to have the form
where A is a Latin subsquare of order 2m, say on the symbol set R. As L is a Latin square, no symbol in R can occur in either B or C; thus all entries from R occur in A or D, and D contains exactly (2m + 1)2 - 2m (2m + 1) = 2m + 1 entries not in R. Now let T be any transversal of L; then T contains the 2m symbols in R and thus T has at least 2m entries in A or D (and, possibly, more entries not in R in a cell of D). Now there exists some Latin square L * orthogonal to L iff L has a set of 4m + 1 pairwise disjoint transversals. Assume this to be the case. Then at most 2m + 1 of these transversals contain an entry not in R in a cell of D; so there is a transversal T which has no entry r ERin a cell of D. Hence its entries in the union U of A and D are exactly the 2m entries in R. Thus the corresponding symbol of L * occurs exactly 2m times in the cells of U. But if this symbol occurs precisely x times in the cells of A, then it occurs 2m - x times in the cells of B, and (2m + 1) - (2m - x) times in the cells of D, and thus 2m - 2x + 1 times in the cells of U, a contradiction. •
8.6 Theorem. There exists a maximal (s, 3; I)-net whenever s == 1 (mod 4). Proof. Let s = 4m + 1; by Lemma 8.5, it suffices to construct a Latin square L or order 4m + I with a subsquare of order 2m. We start with a Latin square A of order 2m + 1 with an orthogonal mate. We thus may choose 2m pairwise
§8. Maximal nets with large deficiency
727
disjoint transversals T I , ... , T2m of A. Let A be defined on the symbol set S = {2m + I, ... , 4m + I}; we shall define L on {I, ... , 4m + I}. Put
where D is any Latin square of order 2m defined on {I, ... , 2m}. Furthermore A is obtained from A by replacing all entries in the cells of Tj by j (j = 1, ... , 2m); finally, the entry in cell (i, j) of B (i = 1, ... , 2m + 1; j = 1, ... , m) is the entry of the unique cell ofTj in row i of A, and similarly for C (using the columns of A). It is easily checked that L is indeed a Latin square .
•
8.7 Exercise. (a) Let L be a Latin square of order s with a subsquare of order k. Prove k ::: ~. (b) Let s, k be positive integers with k ::: ~. Show the existence of a Latin square of order s with a subsquare of order k if s - k ;;j::. 2, 6 and discuss the exceptional cases s - k = 2 or 6. The result is then still true; cf. Denes and Keedwell (1974), p. 43.
8.8 Exercise. Show that a Latin square of order 4m + 2 with a subsquare of order 2m + 1 has no transversal. Construct such Latin squares. Hint: let L = (~ ~) be as in the proof of Lemma 8.5, and let x be the number of cells of a transversal T of L in A. Then count the number of cells in T containing symbols different from those in A to obtain a contradiction. We will next give another construction yielding maximal nets with large deficiency, this is due to Drake (1977). 8.9 Theorem. Let D' = (V', B') be an (s, s + 1; I)-net (i.e. an affine plane of order s) and D" = (V", B") a (t, s + 1; I)-net, where s does not divide t. Then D := D' x D" = (V, B) has no transversal. Proof. Forthe definitionofD' xD", see I.7.21. Let T be a transversal ofD, say T = {(PI, ql), ... , (PSI, qSI)} with Pi E V', qi E V"(i = 1, ... , st), and define f : Vi -+ N U {OJ by (8.9.a)
f(p) := I{i E {I, 2, ... , Sf} : Pi = p}l;
X. Transversal designs and nets
728
thus f (p) is the number of points of T with first coordinate p. Let L be any line of D'; then L gives rise to t parallel lines of D (of the form L x M, M E B"). As T is a transversal of D, it intersects each of these t lines exactly once. Hence (8.9.b)
L
f(L) :=
f(p) = t
for each L EB',
pEL
as the union of the t lines under consideration is precisely the set of all points
(p,q)EVwithPEL. Now assume that s does not divide t. By (8.9.b), the average value of f(p) is = tis (using a parallel class of lines of D'). As sft there is a point p with f(p) > ;; say f(p) = ~ + x, x > O. If L is any line containing p, then f(L \{p}) = t - ~ - x = (s - l)~ - x. SinceD' is an affine plane, any point q i= p is on exactly one line L through p. Hence we get
stls 2
st =
L
f(q)
= f(p) + L
qEV'
=
G+x) +
f(q)
qip
(s
+ l)((S -1)~
-x)
<st,
a contradiction. •
8.10 Corollary. Let s = ql ... qn be the prime power factorisation of the integer s, where ql < q2 < ... < qn. Then there exists a maximal (s, ql + 1; 1)net. • Note that Corollary 8.10 which is due to Bruck (1951) is a strengthened version of MacNeish's Corollary 1.7.8. Corollary 8.10 shows that the MacNeish bound cannot be improved by trying to enlarge a net constructed by MacNeish's method, where one affine plane has been used. It may be remarked that Drake's proof is much simpler than the original one by Bruck. Actually, Drake's results are even more general, since he considers a generalisation of the direct product construction, where several distinct nets of order t may be used as in Theorem 8.9. Also, Theorem 8.9 has further interesting consequences.
8.11 Corollary. (a) There exists a maximal (n, 3; I)-net whenever n =2 (mod 4).
(b) There exists a maximal (n, 4; I)-net whenever n == 3 or 6 (mod 9), n i= 6. (c) There exists a maximal (n, 5; l)-netwhenevern =4,8 or12 (mod 16) and n >40.
§8. Maximal nets with large deficiency
729
(d) Let s be a prime power and K := {t EN: sit and s2 t t}. Then there exists a maximal (t, s + 1; I)-net for all butfinitely many t E K. Proof. We shall prove part (c); the other parts are similar. We may use Theorem 8.9 with s = 4 and t = n / s; then the existence of D" is guaranted by Theorem 3.14 as t = n/s > 10. For part (d), use the Chowla-Erdos-Straus Theorem 5.4 .
•
We conclude this section with a few remarks. 8.12 Remark. In Theorem 8.9 we gave a construction method for nets without transversals and so the examples of Corollaries 8.10 and 8.11 are not only maximal, but indeed have no transversals. In Exercise 8.8, Latin squares (and thus nets with three parallel classes) without transversals were constructed, and in Section 12 we will show the existence of such a Latin square for every even order s. In contrast to this result, it has been shown that every Latin square has a large partial transversal, the best known bound being at present s - c(log s)2 for order s, see Shor (1982). No comparable result is known for nets in general. 8.13 Remark. By Theorem 8.6 and Corollary 12.5, we know the existence of a Latin square without an orthogonal mate for all orders s i= 3 (mod 4). Regarding the case s = 7, the complete classification of all Latin squares of order 7 is known; this is essentially due to Norton (1939) who had missed one species which was discovered by Sade (1951). It turns out that only six out of the 147 species of Latin squares of order 7 possess orthogonal mates, see van Rees (1990). It is still not known whether there exists any Latin square of order s == 3 (mod 4), where s?:: 11, which has no orthogonal mate. In this context, the following general result of van Rees (1990) is quite interesting; the proof uses ideas similar to those used in proving the classical results Lemma 8.5 and Theorem 8.6. In fact, van Rees also proves these older results and gives a nice general account of the topic of subsquares and transversals in Latin squares. We also note that van Rees conjectures that asymptotically almost no Latin square of order s has an orthogonal mate. 8.14 Theorem. For every positive integer m, there exists a Latin square L of order 4m + 3 for which no two orthogonal mates can be orthogonal to one another. In other words, L is contained in a maximal set of MOLS of size::: 2.
•
In some cases, a more precise result is possible, as the following theorem of Drake, van Rees and Wallis (1999) shows.
730
x. Transversal designs and nets
8.15 Theorem. Let v =1= 1,19 be a positive integer that satisfies one 0/ the congruences v '= 1 or 7 (mod 9), v'= 11 (mod 18). Then there exists a pair 0/ orthogonal Latin squares 0/ order v which do not have a common orthogonal mate. • Evans (1991) proved the following quite general theorem on the existence of maximal nets which complements the results by Bruck and Drake, respectively, given in Corollaries 8.1 0 and 8.11. The idea of proof of his result will be sketched in §12, see Remark 12.21.
8.16 Theorem. There exists a maximal (s, r)-net in each cases:
0/ the following
+ 1;
(8.16.a)
s is a power o/someprime p, andr = p
(8.I6.b)
s = mpG and r = p + 1, where p is a prime not dividing m provided that there exists an (m, r)-net. •
=
8.17 Corollary. Let s mpG and r = p + 1, where p is a prime not dividing m. Then there exists a maximal (s, r )-net whenever m is sufficiently large. • What "sufficiently large" means for some values of p can be seen from the table given in Remark 4.2. For instance, if p = 3, we may take m 2: 7; and for p = 5, we may take m 2: 23. Drake, van Rees and Wallis (1999) provided two further constructions for small maximal sets of MOLS. The first of these uses Singer cycles for the classical symmetric designs, whereas the second one makes use of vectors V(k, s) as discussed in Remark VIIIA.31.
8.18 Theorem. Let nand n + 1 be prime powers and d a positive integer. Suppose either that d is even or that n + I is prime. Then there exists a maximal set o/n MOLS %rder n d + ... + n + 1. • 8.19 Theorem. There exists a maximal set o/three MOLS %rder 8t every positive integer t =1= 3,5 such that 6t + 1 is a prime power. •
+ l/or
8.20 Remark. We finally give a table of the known results concerning the existence of maximal sets of mutually orthogonal Latin squares of small order, namely for orders s :::: 13. A larger table of this type may be found in Section IY.27 of Colboum and Dinitz (1996a); the results of Drake, van Rees and Wallis
1
§9. Translation nets and maximal nets
731
1999 and of Bedford and Whitaker (1998) allow some improvements in this table.
In the following table, non-existence always is a consequence of Bruck's completion theorem, see Example 7.20, with the following two exceptions. The non-existence of a pair of orthogonal Latin squares of order 6 will be proved in § 13, and the non-existence of a maximal set of three MOLS of order 7 was established by Drake (1977) using the complete classification of all Latin squares of order 7 already discussed in Remark 8.13. Except for the trivial case of complete sets of MOLS corresponding to affine planes, a reason for the existing cases is given in the column "references", where we refer to a result in the present book whenever possible. Order
Exists
Does not exist
Undecided
2 3 4 5 6 7 8 9
1 2 1,3 1,4 1 1,2,6 1,2,3,7 1,2,3,5,8
1 2 2,3 2,3,4,5 3,4,5 5,6 6, 7
4 4
10 11 12
1,2 2,3,4,10 1,2,3,4,5
7,8 8,9 9,10
3,4,5,6 1,5,6,7 6,7,8,11
l3
1,2,3,4,6,12
10,11
5,7,8,9
References
8.13, 12.23 12.5, 12.18 8.6,8.16, 12.18,9.28 8.8, VIII.4.9 12.23, Evans (1987) 12.5.8.11, Chang, Hsiang and Tai (1965), 12.18, Bedford and Whitaker (1998) 8.6,12.26, 8.18, 12.23
§9. Translation Nets and Maximal Nets with Small Deficiency The aim of this section is the construction of maximal nets (with J-i = 1) with smaIl deficiency for square orders q2. All the examples which we will present admit a nice type of point regular automorphism group and are constructed with the aid of projective geometry. This requires some effort; we begin with a basic study of "translation nets", which applies to the case J-i > 1 as well. 9.1 Definition. LetD = (V, B) be a net, and let G be an automorphism group of D. Then D is called a translation net with translation group G if G acts regularly on the point set V and fixes each parallel class of D.
732
X. Transversal designs and nets
9.2 Remark. In the special case of affine planes, Definition 9.1 agrees with the usual definition of translation planes; for detailed studies of these objects, see the books by Liineburg (1980) and Kallaher (1981) as well as the survey by Kallaher (1995) However, the general case exhibits features which are quite different from what holds for translation planes. For instance, Sprague (1982) gave examples showing that a Bruck net may be a translation net with respect to different, even non-isomorphic translation groups. Therefore a general translation net should always be considered as a pair (D, G). Indeed, the study of individual "translations" of a net poses certain problems; the reader is referred to Hachenberger and Jungnickel (1990) for a detailed discussion of this topic. We now give a group theoretic description of translation nets which generalises the representation of translation planes in terms of "congruence partitions" as given by Andre (1954) and Bruck and Bose (1964).
9.3 Definition. Let G be an additively written group of order S2p." and let U I , ... , Ur (withr::: 3) be subgroups ofordersp., ofG. ThenU:= {UI, ... , Ur } is called a partial congruence partition (in short, an (s, r; p.,)-PCP) in G if one has (9.3.a) Note that this condition is equivalent to (9.3.b)
Ui
+ Uj
= G
for i, j = 1, ... , r with i ;6 j.
The subgroups in U are called the components of U. If U cannot be extended to a larger PCP by adjoining a further component, U is called maximal. If U is any set of subgroups of some group G, we define an incidence structureD(U) as follows: (9.3.c)
D(U) := (G, {U
+ g: U E U, g E G}, E).
An incidence structure D is said to be group constructible if it is isomorphic to some incidence structure of the form D(U). 9.4 Lemma. The group constructible nets are precisely the translation nets. Moreover, D(U) is an (s, r; p.,)-translation net with translation group G if and only ifU is an (s, r; p.,)-PCP in G. Proof. Let D be a group constructible net, say D = D(U), where U is a set of subgroups of G. Then the points of D are just the elements of G, and the blocks
733
§9. Translation nets and maximal nets
of D through the point 0 are just the components U of U, by (9.3.c). For U E U, all cosets of U are likewise blocks of D(U). Since parallelism in a net is defined by disjointness, the parallel class of U has to be the set of cosets of U. Thus G fixes each parallel class, and D(U) is a translation net. Conversely, let D be a translation net with translation group G. As G acts regularly on the point set V of D, we may identify V with G after choosing a "base point" p; thus q E V is identified with g E G if and only if p8 q. Let B be any block of D through 0 (= p); as each g E G maps B onto a parallel block, we see that the trace of B is identified with the stabiliser G B of B in G, that is, with a subgroup of G. Thus the blocks of D through 0 define a set U of r subgroups of G; as G fixes parallel classes, the parallel class of U E U is {U + g: g E G}, and thereforeD ';::'; D(U).
=
Finally, D(U) will be an (s, r; tL)-net if and only if G has order s2 tL, U has r components of order s tL each, and any two distinct subgroups in U intersect in tL common points; that is, if and only if U is an (s, r; tL)-PCP in G . •
9.S Corollary. Let U be an (s, r; tL)-PCP in G. Then
(9.S.a)
S2tL - 1 r<--- s -1 '
with equality ifand only if each non-zero element olG occurs in precisely s~-.:-;l components.
Proof. This follows by applying Theorem 11.8.8 to the translation net D( U). • 9.6 Example. Clearly, the (complete) nets AGd+l (d +2, q) are translation nets, and thus PCP's meeting the bound (9.S.a) exist whenever s = q is a prime power and tL = qd for some non-negative integer d; here G is elementary abelian. In fact, all examples of complete PCP's have these parameters and belong to an elementary abelian translation group, though they are not necessarily isomorphic to AGd+l (d + 2, q); see Jungnickel (198Ib) and Hine and Mavron (1983) for a more elementary proof. Andre (1954) proved Lemma 9.4 (and the result just mentioned in 9.6) in the special case of affine planes. Translation nets with tL = 1 in vector spaces were first considered by Ostrom (1968) as a tool for constructing translation planes; PCP's in general were used by Drake and Jungnickel (1978) (in somewhat different
X. Transversal designs and nets
734
tenninology) to construct certain "Klingenberg structures". Later Drake and Freeman (1979) considered group constructible nets, and the general existence problem for translation nets was first studied by Jungnickel (1981 b) and Sprague (1982). Ostrom (1968) and subsequent authors in fact considered a special type of PCP's which we will now introduce.
=
9.7 Definition. A partial t-spread is a collection U {UI, ... , Ur } of (t + 1)dimensional subspaces of the (d + 1)-dimensional vector space V (d + 1, q) over GF(q), where q is a prime power and d some positive integer with d ::: t + 1, satisfying (9.7.a)
Vi
n Vj
= {O}
for i, j
= 1, ... , r with i # j.
If V cannot be extended to a larger partial t-spread by adjoining a further component, V is called maximal. Trivially, this holds if each non-zero vector is contained in (precisely) one component of V; in this case U, is called at-spread. For t = l, the prefix t will be omitted. 9.8 Remarks. (a) A partial t-spread in V(2t + 2, q) may be considered as a PCP with J1, = 1 in the elementary abelian group EA(q2t+2); but - unless q is a prime - such a PCP is of a special type, as its components are subspaces of V (2t + 2, q) and not just subgroups of E A (q2t+2). Hence the translation net D corresponding to a partial t -spread has an interesting geometric interpretation, since it clearly can be considered as a substructure of the affine space AG(2t + 2, q). We also note thatD admits a cyclic group H of order q - 1 as an automorphism group, since the mapping x 1-+ XA (A E GF(q)*) fixes every subspace of V(2t + 2, q) and hence every component of U. The elements of H have a unique fixed point, namely 0, and also fix all parallel classes of D; therefore, they are called dilatations. In general, only the cyclic group of order p - 1 will act as a dilatation group of an arbitrary PCP (with f.L = 1) in an elementary abelian p-group. (b) A partial t -spread may also be considered from a projective geometry point of view, namely as a collection of pairwise disjoint t-dimensional subspaces of the projective space PG(2t + 1, q). In particular, partial spreads may be viewed as collections of pairwise skew lines of PG(3, q). In what follows, we will switch between the original definition of a partial t-spread given above and its interpretation in the corresponding projective space without any comment. This is, of course, an abuse of language, but a very convenient one, and should not cause any confusion.
§9. Translation nets and maximal nets
735
For results on (partial) t -spreads, we refer the reader to the books of Dembowski (1968), Hirschfeld (1985) and Hirschfeld and Thas (1991). We just mention that a t-spread in V(d + 1, q) exists if and only if t + 1 divides d + 1. Some constructions of maximal partial t-spreads were given by Beutelspacher (1975, 1976, 1980), and Heden (1993, 1995). A brief survey on bounds and existence results for maximal partial t-spreads may be found in §4 of Jungnickel (1996). We now state a result of Drake and Freeman (1979) which shows that partial t-spreads always give rise to PCP's (not only in the case d = 2t + 1); Drake and Freeman used this result together with the Bose-Bush bound (Theorem 6.5) to obtain new bounds for the size of a partial t -spread in V (d + 1, q).
= {VI, ... , Ur } be a partial t -spread in V (d + I, q). Then there exists a (qt+l, r; qd-I-2t)_PCp in EA(qd+l) which consists of sub-
9.9 Proposition. Let V spaces of V (d
+ 1, q). The converse holds provided that q is a prime.
Proof. Apply a polarity 8 to PG(d, q); for instance, let W" be the orthogonal complement of W for each subspace W of V (with respect to the standard inner product). Then dim = d - t and dim n UJ) = 2(d - t) - (d + 1) = d - 1 - 2t whenever i =1= j. The converse follows similarly. •
vl
(Ul
We now return to our goal of constructing maximal Bruck nets with small deficiency, where we take "small" to mean that (7.2.a) is satisfied; that is s> (d - 1)2, d <
v's + 1,
or equivalently r > 2 + v's.
Apart from the more or less trivial cases s = 4 (see Example 8.4) and s = 5 (see Theorem 8.6) and a series due to Dow (1983), all known examples of such nets are constructed from partial spreads in V (4, q), where q is a prime power; they are due to Bruen (1972, 1975) and Jungnickel (l984a, 1993b). The construction of these examples consists basically of two steps; (i) constructing a maximal partial spread in V(4, q), which is not a spread; (ii) proving that the corresponding translation net is maximal. Even assuming (i), Bruen's proof of part (ii) for his examples (which only concern prime orders q) was rather involved; the systematic use of translation nets will allow us to give simpler proofs for stronger results. We will therefore consider transversals and possible extensions of translation nets of small deficiency following Jungnickel (1993b). By a result of lungnickel (1984a), s has to be a prime power and G an elementary abelian group in this case, unless s = 2 or 4. We begin with a simple lemma which is due to Ostrom (1966).
738
X. Transversal designs and nets
9.15 Definition. A regulus in PG(3, q) is a set R of q + 1 skew lines such that any line of PG(3, q) which intersects three of the lines in R in fact intersects all of them; such a line is called a transversal of R. 9.16 Remark. The points of a regulus in PG(3, q) form the set of absolute points of a "ruled quadric", and any such quadric gives rise to exactly two such reguli, as we shall see in the following lemma. In normal form, a ruled quadric has the matrix
(! ~~ ~) For more on ruled quadrics, we refer to Hirschfeld (1985). 9.17 Lemma. Any three mutually skew lines A, B, C ofPG(3, q) determine a unique regulus R = R(A, B, C). Moreover, the transversals ofRform another regulus, the opposite regulus R' of R. Proof. Choose any point a of A. We first show that there is a unique line L = L(a, A, B, C) intersecting A in a and also meeting both Band C. For a and B span a plane of PG(3, q), and thus C meets this plane in a unique point; in other words, exactly one of the lines ax, x E B, also meets C. It is now easily seen that the lines L(a, A, B, C), where a varies over the points of A, form a regulus R' = R' (A, B, C). Repeating our argument for any three lines of R' in the role of our given three skew lines, we see that the transversals of R' include A, B, C and form a regulus R = R(A, B, C) whose transversals are the lines ofR' . •
9.18 Definition. A spread Sin PG(3, q) is called regular if, for any three of its components, all lines in the corresponding regulus are in S. We now give an explicit construction for a regular spread in PG(3, q), following Bruck and Bose (1964). This strengthens Lemma VI.12.7. 9.19 Lemma. Let K = GF(q2) and F = GF(q), and let {I, e} be a basis ofK over F. Then the one-dimensional subspaces xK of K2 form a regular spread of K2, considered as the vector space F4 over F. Proof. We consider the two-dimensional subspaces of F4 as lines in P = PG(3, q), and putS:= {aK: a E K2 \ {O}}. Note that any two distinct elements
§9. Translation nets and maximal nets
739
of S intersect in 0 only. HenceS consists of (q4 _l)/(q2 -1) = q2+ 1 mutually skew lines of P := PG(3, q), as any spread does. Now let A, B, C be any three lines of S; we have to show that the regulus R determined by these lines is contained in S. By Lemma 9.17, there is a line in P which intersects A, Band C, say in aF, bF and (a + b)F. Considering the vectors a, b E F4 as vectors in K2, we obtain A = aK, B = bK and C = (a + b)K. We now claim that the subset (9.l9.a)
R a .b := {aK} U {(Aa
+ b)K: A E F}
of S is just the regulus R. Note first that R a •b consists of q + 1 mutually skew lines of P and that each of the q. + 1 mutually skew lines of P in (9. 19.b)
R~.b := {aF
+ bF} U {(IL + s)aF + (IL + s)bF : IL E F}
intersects each line in Ra,b' Hence Lemma 9.17 implies that these two sets of lines form a pair of opposite reguli of P. • We digress for a moment to note the following interesting consequence of Lemma 9.19 which is due to Andre (1954). 9.20 Corollary. For each prime power q > 2, there exists a non-desarguesian affine translation plane of order q2. Proof. Let A = AG(2, q2) be the desarguesian affine plane of order q and S the regular spread determined by A (as in Lemma 9.19). Now choose any regulus R in S and replace the lines in S by those of its opposite regulus R'; by Lemma 9.17, this yields a spread S' and thus a translation plane A' = D(S') of order q2. By Exercise 9.22, S contains a regulus R" which has exactly one line in common with R, say L. Because of q > 2, we may choose three lines A, B, C ER" \ {L}. Since L ER(A, B, C) \ {R'}, the spreadS' is not regular. It is not difficult to see that the spread belonging to a translation net is essentially unique, cf. Liineburg (1980), Theorem 1.11. Hence A' ;t A, by Lemma 9.19 . • 9.21 Remarks. (a) Bruck and Bose (1964) also proved the converse of Lemma 9.19: if S is a regular spread of PG(3, q), then D(S) is isomorphic to AG(2, q2). (b) The planes constructed in the proof of Corollary 9.20 are precisely the socalled Hall planes; see Hall (1943) and Albert (1961) as well as Dembowski (1968), Hughes and Piper (1982), Liineburg (1980) or Kallaher (1981).
X. Transversal designs and nets
740
9.22 Exercise. Show that each regular spread S contains two reguli which have exactly one line in common. Hint: This follows from counting the reguli containing one or two lines of S. We now give the following general construction principle for maximal partial spreads over GF(qS) which stay maximal when considered as partial (2s + 1)spreads over the subfield GF(q). This result of Jungnickel (1993b) makes use of ideas going back to Bruen (1971) and Beutelspacher (1980). 9.23 Theorem. Let S be a regular spread in P = PG(3, qS), where qS :::: 4, and let Ro and RI be two reguli in S which intersect in exactly two lines, say L and M. Let U denote the partial spread of deficiency 2q s obtained by omitting the 2q s lines in Ro URI, and let W be a set of lines in P satisfying the following conditions: (9.23.a)
W is contained in the union R~ U R~ of the opposite reguli afRo and R I·
(9.23.b)
Each point in L U M is on at most one line of W.
(9.23.c)
W contains at least one line from each of R~ and R~ .
(9.23.d)
For each line G E (R~ U R~) \ W, at least one of the two points of intersection of G with L U M is on a line of W.
Then the set F (9.23.e)
= U U W is a maximal partial spread in P with deficiency
d = 2q s -IWI.
Moreover, any line not in F either lies entirely in P(R o) U P(RI), where P(X) denotes the set of points covered by the lines in a set of lines X, or contains at most four points not in P(F). Now denote by V = V(4, qS+I) the underlying vector space ofP, consider V as a (4s + 4)-dimensional vector space over the subfield GF(q) of GF(qs+I), and denote the corresponding projective space PG(4s + 3, q) by Q. (Thus the points and lines ofP are particular s -dimensional and (2s + I)-dimensional subspaces ofQ.) Then the maximal partial spread F constructed above remains maximal when considered as a partial (2s + 1)spread ofQ. Proof. Since the point set P (R) covered by the lines in a regulus R is the same as the point set covered by the opposite regulus R', conditions (9.23.a, b) imply that F is a partial spread of P. Now let K be any line not in F and consider the set K' := K \ P(F). As every point not in P(R o URI) is on a line in U, the set K' is contained in P(R o URI). By Remark 9.16, the point set peR) covered
§9. Translation nets and maximal nets
741
by the lines of a regulus R is a ruled quadric in P, and thus the only lines of P which intersect P(R o URI) = P(R o) U P(R I ) in more than four points are the lines which are entirely contained in P(Ro) or in P(RI)' that is, the lines in the reguli Ro, R 1, R~ and R~ . In particular, since q ::::: 4, any line X which is skew to every line of F has to belong to one of these four reguli. Because of (9.23.c), X cannot belong toRo or R I , and because of (9.23.d), X cannot be in R~ or R~ either, and thus F is indeed a maximal partial spread of P.
It remains to show that F remains maximal when considered over Q. Note first that the points of P form a partition T of Q into s-dimensional subspaces;2 thus each point v of Q is contained in a unique s-dimensional subspace Sv in T. Now assume the existence of a (2s + I)-dimensional subspace U of Q which is skew to every subspace in F. Then the point Sx of P cannot belong to P (F) for any x E U. Now choose a fixed point U E U, and let v be any point in U \ SUo The line Luv of P determined by Su and Sv is an (2s + I)-dimensional subspace of Q; since F is a maximal partial spread of P, we must have U =1= Luv. Therefore U is the union of the proper subspaces U n Luv, where v runs over the points in U \ Su. Obviously, this requires that each point u of U lies in at least three distinct lines Luv of P. On the other hand, the q + 1 points w of U on the line uv of Q (for any v E U \ Su) give rise to q + 1 ::::: 5 distinct points Sw of Luv none of which lies in P(F). Hence Luv is entirely contained in P(Ro U R j ) and therefore belongs to one of the reguli Ro, R I, R~ and R~. Since we may assume that Su lies in only one ofthe two ruled quadrics P(Ro) and P(Rl), say in P(Ro), we conclude that Luv has to be either the line in Ro or the line in R~ through Su, contradicting the fact that there are at least three distinct choices for Luv. This contradiction finishes the proof of the maximality of F as a partial (2s + I)-spread of Q . •
9.24 Construction. We now use Theorem 9.23 to construct the desired examples of maximal partial t-spreads. This requires the construction of suitable line sets W; we will achieve this by direct computation, using the representation of a regular spread given in Lemma 9.19. Thus let V = GF(q4s); then V is a two-dimensional vector space over its subfield K = GF(q2s), say with base {I, d}, which in tum is a two-dimensional vector space over its subfield F = GF(qS). If {I, e} is a basis for Kover F, then {l, e, d, ed} is a basis of the four-dimensional vector space V over F. As in Lemma 9.19, (9.24.a)
2
S = {aK : a E V \ {O}}
T is what is (among geometers) usually called a "geometric s-spread" of PG(4s
will not really require this tenninology and thus prefer to avoid it.
+ 3, q), but we
X Transversal designs and nets
742
is a regular spread of the projective space P = PG(3, qS). In the following examples, we use (in the notation of (9.19.a» the reguli Ro = R[,d and R[ = Rl,ed. Note that these two reguli intersect in the two lines L := lK and M := d K. It will be convenient for the subsequent explicit examples to list the lines in these reguli and their opposites explicitly. Because of (9.19 .a, b), we have:
(9.24.c)
= (A+d)K: AEF*}; R[ = (L} U (T). = (A + 8d)K: A E F*};
(9.24.d)
R~ = (L' = F
(9.24.e)
R~ = (L" =
(9.24.b)
Ro={L}U{S).
+ dF} U {S~ = (fL + 8)F + (fL + 8)dF: fL E F*}; F +sdF} U {T: = (fL+8)F + (fL+8)8dF: fL E F*}.
As (9.24.e) indicates, the computations depend upon the choice of the base element 8 of K: since the term 8 2 appears, the minimal polynomial of 8 is of importance in determining the intersections of lines in R~ and R~ .
9.25 Example. As a first application of Theorem 9.23 and Construction 9.24, we give a simple computational proof for the construction of maximal partial spreads with deficiency qS -1 in PG(3, qS) due to Bruen (1971). This requires finding two lines each in R~ and R~ which cover the same four points of L U M, provided that qS :::: 5 is odd, and turns out to be very easy if we choose the base element 8 such that v = S2 is contained in F. Thus let v be any non-square in F, and let s be a root of v in K. Then the lines L' and S~ of R~ contain the points F, sF, dF and sdF of L U M; and the lines L" and T~ of R~ contain the points F, sdF, sF and s2dF = dF of L U M. Thus the set
W:= {L', is a set of qS
S~} U
(R; \ {L", T~})
+ I lines satisfying the conditions in Theorem 9.23.
9.26 Example. We now show how one finds sets of qS, respectively of qS - 1, lines satisfying the conditions of Theorem 9.23, provided that qS :::: 5 is odd. Since the details of the computation are a little more involved, they will be omitted and we just indicate the respective definitions of W. One again begins with a suitable choice of the base element s which is adjoined to F in order to construct the quadratic extension K. Consider the equation (9.26.a)
x 2 - X = Y over F = GF(qS).
This equation has a unique solution for y = -1/4 and two solutions each for (qS -1)/2 values of y. Thus there are (qS -1)/2 choices of y for which (9.26.a) has no solution in F. We choose such an element y (so that the polynomial
743
§9. Translation nets and maximal nets
x 2 - x - y is irreducible over F) and also require y ¢ {I, 2}. In view of the preceding remarks, this is clearly possible for qS 2: 7; and for qS = 5, we may take y = 3. Thus we may assume our base element to satisfy
(9.26.b)
8
2
= 8 + y,
where
y
E
F \ {I, 2}.
We now define a set W of qS lines by putting (9.26.c)
W:=
{s~} U (R~
\ {T~, L"l).
It can be verified that W indeed satisfies conditions (9.23.a, b). Similarly, a suitable set W' of qS - 1 lines is obtained by putting
(9.26.d)
L'
= {s~, S;} U (R; \ {T~, L", T{, T;-d).
Applying Theorem 9.23 to Examples 9.25 and 9.26, we obtain the following series of maximal partial t-spreads due to Jungnickel (1993b).
9.27 Theorem. A maximal partial t-spread in PG(2t of exactly r components exists in the following cases: (9.27.a)
t
= 2s + 1, with
q 2: 5
odd,
r
= l+ I
-
+ 1, q) which consists
qS+ I
+0
OE{O, I,2} . •
We can now apply Theorem 9.27 together with Corollary 9.13 to obtain some series of maximal nets with small deficiency; this is due to Jungnickel (1993b) and considerably strengthens previous results of Bruen (1972, 1975). In fact, we shall include some further series of examples which may be derived by similar methods in the following result. These examples show that the bounds of §7, in particular the Metsch bound (7.3.a), for the completion of Bruck nets indeed are quite good.
9.28 Theorem. Let q be a power of the prime p. Then there exist transversalfree translation nets of order q2 and deficiency d in each of the following cases: (9.28.a)
d = q - I,
(9.28.b)
d=q,
P 2: 5;
p2:5;
(9.28.c)
d=q+l,
p2:5;
(9.28.d)
d = q - 1,
p= 2
(9.28.e)
d=q = 3.
and q = 2r for some prime r;
744
X. Transversal designs and nets
Proof. The results presented in this book suffice to prove (9.28.a, b) as follows. By Theorem 9.27, there exist maximal partial (2s + I)-spreads with deficiency ps+1 + 8 in PG(4s + 3, p), where 8 E {-I, 0, +1}. By Corollary 9.13, the corresponding translation nets are transversal-free if 8 =1= 1. Similarly, the case 8 = 1 leads to series (9.28.c) if one uses Corollary 9.30 below instead of Corollary 9.13. The example in (9.28.e) is obtained by applying Corollary 9.13 to a maximal partial spread with deficiency 3 in PG(3, 3); the reader may try and construct such a maximal partial spread as an exercise or consult Bruen (1971). Finally, series (9.28.d) was constructed by Jungnickel (I993b) using ad hoc arguments. • As mentioned above, series (9 .28.c) rests on the following analogues of Theorem 9.11 and Corollary 9.13; these results are again due to Jungnickel (1993b).
9.29 Theorem. Let D be a translation net of order s > 4 with translation group G, and assume that D has critical deficiency, say s = m 2 and d = m + 1. Then any net E extending D is either a translation net with the same translation group G, or E is a transversal-free net with deficiency m. • 9.30 Corollary. Any maximal partial t-spread of critical deficiency over a prime field GF(p) gives rise to a maximal (and hence transversal-free) net. • 9.31 Remarks. (a) Dow (1983) gave a construction of maximal nets of order q2 and deficiency q for each prime power q =1= 2, which are no longer translation nets but extend a translation net D of deficiency q + I; hence the second case in Theorem 9.29 can occur. In fact, D may be described as the translation net in common to the Desarguesian and the Hall planes of order q2; using the notation of the proof of Corollary 9.20, D is the net associated with the partial spreadS\R.
(b) Jungnickel (1984a) proved that any maximal partial spread U of deficiency q in PG(3, q) gives rise to a netD(U) which is not completable. In contrast to this result, he also constructed a maximal partial spread U of deficiency q - 1 in PG(3, q) such that D(U) is completable for each prime power q which is not a prime. On the other hand, if d = q - 1 and q == 2 (mod 3), one may also construct a maximal partial spread U for which D(U) cannot be completed to an affine plane; this is due to Johnson (1989). It would be very interesting to know if any of the non-embeddable examples actually give rise to extendible nets, since this would produce considerably larger maximal nets than we know up to now: the smallest deficiency known to occur is still q - 1.
§9. Translation nets and maximal nets
745
We conclude this section by discussing some related results. First we describe the present state of the existence theory for translation nets; as the results below will show, an extremely difficult problem such as the existence problem for nets can become much more tractable if one assumes the existence of a reasonably nice group. We first introduce some notation. Let G be a group of square order s2. Then we denote by T(G) the maximum cardinality of an (s, r; I)-PCP in G; we also write T (s) for the maximum value of T (G) as G runs over all groups of order S2. The following results due to lungnickel (198Ib, I989c) which are based on previous work of Frohardt (1987) determine the numbers T(s) and thus show the validity of the MacNeish conjecture for translation nets.
9.32 Proposition. Assume the existence ofan (s, r; I)-PCP in G, and let P bea Sylow p-subgroup o/G, say IPI = p2a. Then there also exists a (pa, r; I)-PCP in P. Hence (9.32.a)
T(G) ::: min {T(P): P E Syl(G)}.
IfG is nilpotent, one has equality in (9.32.a) . •
9.33 Corollary. Let s (9.33.a)
= ql ... qk be the prime power factorisation ofs. Then:
T(s) = min {qi
+ I : i = 1, ... , k} . •
The equality in (9.33.a) may be realised by taking G as the direct product of elementary abelian groups; however, note that these are in general not the only groups of order s2 satisfying T(G) = T(s). It is also known that one does not in general have equality in (9.32.a); the simplest counterexample is provided by taking G as the direct product of the alternating group of degree 5 with itself. Proposition 9.32 reduces the existence problem for translation nets with a nilpotent translation group to the same problem for p-groups. Bailey and lungnickel (1990) completely settled the abelian case by proving the following result.
9.34 Theorem. Let G be an abelian p-group containing an (s, 3; I)-PCP. Then G is isomorphic to a group of the form U ED U. If one has (9.34.a) where the qi are pairwise distinct powers ofp and where ai then (9.34.b)
T(U)
= pa + 1
with
a = min {ai: i
=1= 0 for i
= I, ... , n,
= 1, ... , n} . •
746
X. Transversal designs and nets
More generally, Bailey and Jungnickel (1990) considered splitting translation nets, that is, translation nets belonging to a partial congruence partition U in a group G which is the direct product of two components of U. The problem of determining T(G) for an arbitrary p-group G has not yet been settled and seems to be very difficult. Of course, one has T(EA(p2n» = pn + 1. Every other group G of order p2n has a considerably smaller value of T(G), as was first shown by Jungnickel (1981b). The best results known at present are due to Hachenberger (1992, 1993).
9.35 Theorem. Let G be a p-group of order p2n which is not elementary abelian, and assume the existence of a (pn, r; I)-PCP in G. Then: (9.35.a)
forn = 2;
(9.35.b)
forn = 3;
(9.35.c)
r ::: 4
(9.35.d)
r ::: pn-2
for p
+ pn-3 + ... + P + 1
= 2 and n =
3;
for n :::: 4 . •
The groups achieving equality in (9.35.a, b) have all been determined by Hachenberger (1992). As an example, we mention his results for the case n = 2.
9.36 Theorem. Let G be a p-group, and assume the existence of a (p2, 3; 1)PCP in G. Then one has one of the following cases:
+ 1.
(9.36.a)
G is elementary abelian and T (G) = p2
(9.36.b)
G
(9.36.c)
G ~ 71.,2 EB D4, where D4 denotes the dihedral group of order 8, and T(G) = 3.
(9.36.d)
p is odd and G is metacyc[ic, that is
~
Zp2 EB Zp2 andT(G)
= p + 1.
G = (x, y: Xp2 = yp2 = 1, y-1xy = x p+1 )
andT(G) = p (9.36.e)
+ 1.
p is odd and G is the direct product ofZp with the extraspecial group E(p3) of order p3 and exponent p, and T(G) = p + 1. Here E(p3) = (x, y, z: x p = yp = zp = [x, z] = [y, z] = 1, [x, y] = z) . •
9.37 Remark. The only 2-group achieving equality in (9.35.c) is the one already given in (VI.9.9.b); this result is due to Sprague (1982) and Gluck (1989). Regarding odd primes p, Hachenberger (1992) proved that, given p, there is
§9. Translation nets and maximal nets
747
precisely one group for which equality holds in (9 .35.b) which will be described in Theorem 9.38, a result generalising Theorem VI.9.9. Hachenberger (1993) obtained the even stronger result that, given p, there are precisely four groups of order p6 satisfying T(G) 2: P + 2. It is not known whether bound (9.35.d) is sharp for any pair (p, n). However, the results of Hachenberger (1991) show that this bound is not too bad. For instance, he constructs groups G of order p8 with T(G) 2: p2 + I and groups H of order pl2 with T(H) 2: p4 + 1 for all odd primes p.
9.38 Theorem. Assume the existence of a (pu, u; I)-PCP in some group G, where p is the smallest prime dividing IGI./f p = 2, then G is one ofthe groups described in Theorem VI.9.9./fp is odd, then G is a p-group; moreover, Gis either elementary abelian, or one ofthe groups oforder p4 described in Theorem 9.36, or the group of order p6 with exponent p and generators a, b, c, x, y, Z for which one has Z(G) = (x, y, z) and the relations [a, b] = x, [a, c) = y and [b, c) = z. In the last case, one actually has T(G) = p2 + 1. • 9.39 Remark. Recently, there has also been considerable interest in dual nets with a translation group, i.e. in translation transversal designs, which are defined in the obvious way. We refer the interested reader to Schulz (1984, 1985a, b, 1987a, b, 1988), Biliotti and Micelli (1985), Herzer and Schulz (1989) and Hachenberger (1994). As we have seen in this section, maximal partial spreads playa crucial role in the construction of maximal nets with small deficiency. For this reason, we mention a few results concerning these objects.
9.40 Theorem. Let F be a maximal partial spread in PG(3, q), and assume that F consists of r lines. Then one has the following: (9.40.a)
2q
:s r :s q2 + 1 - -li4.
for q > 2.
If q is not a square, then (9.40.b)
and even (9.40.c)
r
:s q2 + 1 -
(q
+ 3)/2
ifq is a prime.
i
X. Transversal designs and nets
748
If q is a square and if q = ph, where h > 2 is even and where p 2:: 5 is a prime, and if r 2:: q2 - q2/3 + 1, then the deficiency d = q2 + 1 - r ofF satisfies (9.40.d)
d
= s(.,jq + 1)
for some integer s 2:: 2;
similarly, if q = p2, where p is an odd prime, and ifr 2:: q2 + 1 - (q then d likewise satisfies (9.40.e)
+ 1)/2,
d = s(.,jq + 1) for some integer s 2:: 2.
Moreover, in either case the set of points of PG(3,q) not covered by F is the disjoint union of s Baer subgeometries. • The lower bound in (9.40.a) is due to Glynn (1982), and the upper bound to Blokhuis and Metsch (1993). For q = 5, we have a small improvement, namely r 2:: 11, given by Heden (1990). The improved upper bound (9.40.b) for non-squares q is due to Metsch (1991a); it is a consequence of Theorem 7.3. In particular, (9.40.a) shows that the case of deficiency d = .[ii + 1 can only occur for q = 4, disproving a conjecture stated by Hirschfeld (1985); the case q = 4 actually occurs, and all such maximal partial spreads have been classified by van Dam (1993). The bound (9.40.c) is due to Blokhuis (1994), and the strong restrictions for the case of square orders in (9.40.d, e) are recent results of Metsch and Storme (1999). All these results give some evidence to the long-standing conjecture that the smallest possible deficiency of a maximal partial spread in PG(3, q) is actually q - 1. We also mention the following very general construction for maximal partial spreads due to Heden (1993).
9.41 Theorem. A maximal partial spread in PG(3, q) consisting of exactly r lines exists for all odd q 2:: 7 and for all r in the range (9.41.a)
5q2
+ 4q 8
1 < r < q2 _ q
--
+ 2.
•
There are some similar results due to Heden (1995); a summary of the present state of knowledge regarding maximal partial spreads and maximal partial tspreads is given by Jungnickel (1996).
9.42 Remark. We finally mention the following interesting "q-analogue" of a t-design: A t-(n, k, A)-design over GF(q) is a collection B of k-dimensional subspaces of Yen, q) (called blocks) such that any t-dimensional subspace is
§10. Completion results for Ik > I
749
contained in exactly A blocks in B. In this terminology, a spread in Yen, q), n even, is the same as a I-(n, 2, I)-design over GF(q); a similar remark holds for (A-fold) t-spreads. A 2-(n, k, A)-design over GF(q) yields an ordinary 2-design S).(2, qk-l + ... + q + 1; qn-l + ... + q + 1) on the set of one-dimensional subspaces of V (n, q) by identifying every block with the set of one-dimensional subspaces which it contains. The first examples of t-designs over GF(q) with t:::: 2 were given by Thomas (1987) who constructed a 2-(n, 3, 7) design over GF(2) for all n with (n, 6) = 1. Ray-Chaudhuri and Singhi (1989) obtained an asymptotic existence result for t-designs over GF(q). Some other relevant references are Suzuki (1990a, b, 1992), Ray-Chaudhuri and Schram (1992), Abe and Yoshiara (1993), Ray-Chaudhuri and Schram (1994), Miyakawa, Munemasa and Yoshiara (1995) and !toh (1998).
§10. Completion Results for IL> 1 In this section we will consider the question of "completing" an (s, r; Ik)net with IL > 1 to an affine design A 2(1k); cf. Theorems II.8.7 and 11.8.8. As in Section 7, we will define a notion of "deficiency" and try to complete (s, r; Ik)nets of small deficiency. Unfortunately, this is much harder than in the case Ik = 1, and only the cases d :<:: 3 are more or less settled. We will only deal with the cases d = 1 and d = 2. We begin with a definition generalising 7.1. Note first that the problem just described makes sense only if s - 1 divides Ik - I, for otherwise no As (IL) can exist. Thus the case to be studied here is the one not covered by the Bose-Bush theorem of Section 6. In this section we always assume J-L > 1.
10.1 Definition. LetD be an (s, r; Ik)-net (i.e. an affine Sr(1, Slk; S21k» where s - 1 divides Ik - 1. Then the deficiency of D is the integer d defined by (10. 1. a) i.e. d measures how far r falls short of its conceivable maximum S:,~l which is reached precisely when D is an (affine) 2-design; cf. Theorem II.8.8. Assume thatD may be embedded in an affine design As (11) with parameters as described in Theorem II.8.7, say A; then A is called a completion of D, andD is said to be completable. The notions "extendible" and "step-t-extendible" are defined as in the case Ik = 1 (see Definition 8.1). We shall prove thatD is always completable for d = 1, and likewise for d = 2 provided s 1:4. It is helpful to introduce the notion of a transversal also for (s, r; J-L)-nets with J-L > 1. In view of (7.4.a) the
X. Transversal designs and nets
750
following definition is indeed a generalisation of the corresponding notion for fL=l.
10.2 Definition. Let D be an (s, r; fL)-net. Then a subset T of the point set of D is called a transversal of D if T intersects each block of D in precisely fL points. 10.3 Lemma. Any transversal of an (s, r; fL)-net D consists of SfL points. A further parallel class may be added to the blocks of D if and only if D has s pairwise disjoint transversals. • We will make essential use of Exercise n.8.20; therefore this result due to Jungnickel and Sane (1982) will be restated and proved.
lOA Theorem. LetD bean (s, SfL+t; fL)-netwhere t isanon-negative integer. Then (lO.4.a)
I(p, q)l::: t
moreover; I(p, q)1
for any two distinct points p, q of D;
= t implies that I(p, x)1 = I(q, x)lfor all points x =1= p, q.
Proof. Choose any two points p, q and assume I(p, q)1 = u. Removing the u parallel classes of D determined by the u blocks joining p and q leaves an (s, SfL + t - u; fL)-netD'. Now I(p, q)1 = 0 inD' and thus SfL + t - u :::: SfL by the dual of Theorem n.8.18; i.e. (lOA.a) holds. In the case t = u, the dual of n.8.18 also shows I(p, x) I = I(q, x) I in D' for all points x =1= p, q; but then clearly also I(p, x)1 = I(q, x)1 in D, as the blocks through p in D \D' arc precisely the t blocks joining p and q. • 10.5 Corollary. Let D be an (s, SfL + t; fL)-net where t is a non-negative integer. Then the point set ofD is partitioned into disjoint classes called cosets; we write p ~ q if p and q are in the same coset, where (l0.5.a)
p
~
q
~
I(p, q)1 = tor p = q
and
(lO.5.b)
I(p, q) I >
t
for p "" q .
Moreover; the joining number of two points in distinct cosets only depends on their cosets.
§1O. Completion results for J-t > 1
751
Proof. Exercise. • 10.6 Examples. (a) If D is a symmetric (s, J-t)-net (so t = 0), all cosets have size s. First examples were obtained in PropositionI.7.IS; also, each (s, sJ-t; J-t)difference matrix yields an example by Theorem VIII.3 .6, Lemma VIII.3.3 and Theorem II.S.21. Hence more examples are provided by Corollary VIII.3.12 and Theorem VIII.3.l4. (b) If D is a complete (s, J-t)-net (i.e. an affine design AI"(s», then any two points are joined by s:~ I = r - s J-t points and so there is just one coset of s2 J-t points. Removing one or two parallel classes from D leaves a net with coset sizes s J-t or J-t, respectively.
(c) Any symmetric (s, J-t)-netD is step-I-extendible. Take the SJ-t cosets (of size s each) and form a new parallel class Tby joining them to s blocks of J-t cosets each (in any way). This is dual to I.7.l6.(d); cf. the proof of Corollary VIII.3.S. By adjoining T to D we obtain an (s, SJ-t + 1; J-t)-net D' (still with cosets of size s). Now remove a parallel class distinct from T from D' to obtain again an (s, s J-t; J-t )-net D I ; then DI has only cosets of size l. We will now turn our attention to the completion problem again. For J-t > 1, there arises a further difficulty compared to the case J-t = l. If an (s, r; J-t)-net D of deficiency d is completable, then any completion E is an affine design AI" (s) and therefore any two points of E are joined by exactly A = ':_--/ blocks (see Theorem II.S.7). Thus a completion of D clearly is impossible ifthere exist points x, y ED with l(x,y)I
or
l(x,y)I>A.
Now the first of these cases is impossible by Theorem lOA, but the second one was an open problem for a long time. For a given d, it is now settled for all but finitely many values of s by the following result due to Shrikhande and Singhi (I979a).
10.7 Theorem. Let D be an (s, r; J-t )-net of deficiency d with (1O.7.a)
s?:.. 2d(d - 1)
and strict inequality for d = 2. Then (1O.7.b)
A - d ~ I(x, y)1 ~ A for any two distinct points x, y ofD,
X. Transversal designs and nets
752 where
(1O.7.c)
S{L - 1 A=--. s-1
Proof. The lower bound in (10.7.b) is clearly valid if d 2: A. Thus let d < A and note that r = s:-~l d = S{L + A-d. Then I(x, y)l2: A - d by (lOA.a).3 It remains to prove the upper bound in (10.7.b). To this end, writebz = I(x, z)1 and Cz = I(y, z)1 for points z with z i= x and z i= y, respectively. Standard counting arguments yield the equations (using r = S {L + A - d)
-
(1O.7.d)
L
b z = (S{L
+A-
d)(s{L - 1) =
z#x
L
Cz
z#y
and (1O.7.e)
L bz(bz -
1) = (S{L
+A-
d)(s{L
+A-
d - 1)({L - 1)
z#x
We want to evaluate Lz#x,y bzcz and thus count pairs (B, C) with x, z E B and y, z E C. Abbreviating by = Cx by h, we obtain h(s{L - 2) such pairs with B = C; and h(h - 1)({L - 2) pairs where B i= C, X E C and y E B; and 2h(s{L + A - d - h)({L - 1) pairs where B i= C and either x E Cory E B; and finally (S{L + A - d - h)(s{L + A - d - h - 1){L pairs with B i= C, x rf. C and y rf. B. Altogether we have
L
(10.7.f)
bzcz = h[(s{L - 2)
+ (h -
1)({L - 2)]
z#x,y
+ (S{L + A -
d - h)[2h({L - 1) + (S{L
+ A-
d - h - 1){L].
We want to evaluate (henceforth, all summations are over all z i= x, y) CY
= 1;(bz + Cz - 2A)(bz + Cz + 1 - 2A)
= 1;b; + 1;c; + 21;bzcz + (1 + (s2 {L - 2)2A(2A - 1). 3
Here (IO.7.a) is not necessary.
4A)(1;bz + 1;cz)
§10. Completion results for f.L >
1
753
Using (1O.7.d) and (1O.7.e), note first (1O.7.g)
}:bz
= }:cz = (Sf.L + '). -
d)(sf.L - 1) - h
and (1O.7.h)
Also let us rewrite (l0.7.f) as (10.7.i)
}:bzcz
= -h(sf.L + 2').. -
2d)
+ f.L(SfJ, + '). -
d)(sf.L
+ '). -
d - 1).
From (1O.7.g, h, i) we obtain after some computation (J'
"2
= (sf.L+')..-d)[(sf.L - h 2 - h(1 - 2')..
= -h(h
+').. -d -1)(2f.L -1) + (sf.L -1)(2-4')..)]
+ SfJ, -
+ 1- 2').. +Sf.L -
2d)
+ (s2f.L -
2 2d) +A (2f.L
2)(2')..2 - ')..)
-1- 4sf.L + 2S2f.L)
+A[1 +Sf.L - 2d - f.L(s -1)(4d - 4sf.L -s +2)] + (Sf.L - d)(sf.L - 1)[(2f.L
+ 1) -
d(2f.L - I)].
Using A = s:~/ we rewrite this as (J'
- = -(h - A)(h - A + 1 +Sf.L - 2d) +2(sf.L -1)2f.L 2 +f.L(sf.L-1)(-4sf.L -s +2) 2
+ (d - d)(2f.L - 1)
= (d
2
- d)(2f.L -
+ (Sf.L -
1)(2sf.L
+ SfJ,)
I) - (h - A)(h - A + 1)
- (h - A)(Sf.L - 2d).
Since the bz , Cz are integers, we have (J' 2: 0 and therefore writing y = h - A, (l0.7.j)
y(y
+ 1) + Y(Sf.L -
2
2d).::s (d - d)(2f.L - I).
Note that the assertion is just y .::s 0; thus assume the contrary. Then (1O.7.j) implies
X. Transversal designs and nets
754 or, equivalently,
d2
-
3d + 2.::: JL(2d(d -1) - s),
which is clearly impossible because of (10.7.a). Thus indeed y .::: 0 and our assertion is proved. • The following result of Shrikhande and Bhagwandas (1969) is an easy consequence of Theorem 10.7.
10.8 Theorem. Any (s, r; JL)-net D of deficiency 1 is completable. Proof. By Theorem 10.7 with d = 1, we have (10.8.a)
A-I.::: I(x,
y)l.::: A
for any two distinct points x, y of D.
The cosets of D (cf. 10.5) are here defined by the joining number A. - 1 and in view of Example iO.6.(b) we have to show that each coset has SJL points. Thus choose a point x and let the number of points y with I(x, y) I = A-I (resp. A.) be denoted by ex (resp. f3). Then clearly ex + f3 = S 2 JL - 1 and (10.7.d) yields the further equation (A. - l)ex + Af3 = (s2JL + A - l)(sJL - 1). Substituting A = s:_~1 one obtains ex = s JL - 1 and thus the coset of x has indeed s f.h points. Then adjoining all cosets as new blocks yields an S)" (2, s JL; s2 JL) which clearly is resolvable; by Bose's Theorem 11.8.7 it is affine. Of course it is also possible to check that each coset is a transversal by another counting argument and then to use Lemma 10.3. • The case d = 2 is much more involved. Shrikhande and Bhagwandas (1969) proved thatD can be completed in this case if s =/= 4 and if no two points of Dare joined by A. + 1 blocks; but they could verify this condition only for s = 2 and 3. The problem remained open for 12 years till Shrikhande and Singhi (1979a) proved Theorem 10.7 and thus the condition needed for all s > 4. Let us state this as a lemma:
10.9 Lemma. Let D be an (s, r; JL)-net of deficiency 2, with s =/=4. Then (iO.9.a)
A. - 2 .::: I(x, y) I .::: A.
for any two distinct points x, y of D.
Proof. For s > 4 this is immediate from Theorem 10.7. The cases s .::: 3 follow from a more general bound for certain divisible designs due to Agraval (1964). We refer the interested reader to this paper as this result only covers the cases d = 1 and d = 2, s = 2 or 3. •
§lO. Completion resultsjor f-t > 1
755
We will prove the completability of D for d = 2, s =1= 4 using cosets, an approach due to Jungnickel and Sane (1982). The original proof of Shrikhande and Bhagwandas (1969) uses eigenvalue techniques and is rather involved. In view of Example 1O.6.(b), the cosets should have size f-t; let us begin by proving this.
10.10 Lemma. Let D be an (s, r; f-t)-net oj deficiency 2, with s =1= 4. Then all cosets oj D have size f-t.
Proof. Choose a point x of D and let a, (3, y denote the number of points y with I(x, y)1 = ).. - 2,).. - 1,).., respectively. Clearly (1O.1O.a)
a
+ (3 + Y =
S2f-t - 1;
and (10.7.d, e) yield (1O.1O.b)
().. - 2)a
+ ().. -
1)(3 +)..y = (Sf-t
+).. -
2)(sf-t - I)
and (1O.1O.c)
().. - 2)().. - 3)a = (Sf-t
+A -
+ ().. -
2)(sf-t
1)().. - 2)(3
+A -
+ )..(A -
1)y
3)(f-t -1).
Multiplying (lO.1O.a) by ).. and subtracting (lO.1O.b) from the result yields (lO.lO.d)
2a
+ (3 =
where we used sf-t equation (lO.1O.e)
2(sf-t - 1),
+A=
2(A - 2)a
S:-~l. Similarly we obtain from (lO.1O.b, c) the
+ (A -
1)(3 = 2(sf-t + A - 2)(f-t -1).
Then (lO.1O.d, e) yield a = f-t - I, i.e. the coset of x has size f-t . •
10.11 Theorem. Let D be an (s, r; f-t)-net oj deficiency 2 and assume s =1=4. Then D is completable.
Proof. In this case all cosets have size f-t by Lemma 10.10; we will denote cosets by capital letters P, Q, R. In view of Corollary 10.5 we may define (lO.l1.a)
[P,
Q] := I(x, y)l,
where x E P and Y E Q are arbitrary. We now define a graph G on the S2 cosets as vertices as follows: (P, Q) is an edge iff [P, Q] = ).. - 1. We write P ~ Q
X. Transversal designs and nets
756
iff (P, Q) is an edge of G. Our aim is to show that this yields a pseudonet graph with parameters sand 2 and then to apply Corollary 7.21. Now from (W.W.d) we see that any point x is joined to 2(s - 1)p, points by exactly A - 1 blocks and therefore each vertex of G has degree 2(s - 1). We next want to show that there are precisely s - 2 vertices joined to both P and Q, whenever P ~ Q. Put (lO.l1.b)
N(P):= {R: R ~ P, R,# Q},
N(Q):= {R: R ~ Q, R,# Pl.
Then clearly IN(P)I = IN(Q)I = 2(s - 1) - 1; we want to evaluate IN(P) n N(Q)I. Now choose x E P, Y E Q, let z ,#x, y be any point of D and define bz and Cz as in the proof of Theorem 10.7. Then we obtain from (lO.7.g, h, i) with d = 2 and h = A - I the following equation (1O.I1.c)
L (bz -
c z )2 = 2(sp,
+ 1..- 2)(2 -
A) + 2(1.. - 1)(sp, + 1..- 3)
z ;6x,y
= 2(sp, - 1).
Now if Z E P, then bz = 1..- 2 and Cz = A - 1; i.e. (b z - cz )2 = 1. A similar argument holds if Z E Q. Therefore (lO.I1.c) yields (10.I1.d)
L
(b z - c z )2 = 2(sp, -1) - 2(p, -1) = 2(s -1)p,.
z¥PUQ
For Z !if P U Q, we have bz, Cz E {A - 1, A} and thus (b z - cz? E {O, I}. As this value is independent of z as long as z E R for some fixed R and as each R contains p, choices for z, we obtain from (10.11.d) (1O.I1.e)
L
(b R
- CR)2
= 2(s -1),
R;6P,Q
where b R := [P, R) and CR := [Q, R). As noted, each term (b R -CR)2 is either Oar 1; and in fact (b R
- CR)2
= I{:=:::? [P, R) '# [Q, RJ {:=:::? R E (N(P)\N(Q)) U (N(Q)\N(P)).
Hence (1O.11.e) yields 2(s - 1) = IN(P)\N(Q)I
as IN(P)I
+ IN(Q)\N(P)I =
2IN(P)\N(Q)1
= IN(Q)I; but then clearly IN(P) n N(Q)I = s -
2, as asserted.
757
§10. Completion results for f-L > I
Next, let us show that there are precisely two vertices joined to both P and Q whenever P "" Q. With notation as before, this means (1O.1l.f)
IN(P)
n N(Q)I =
2
whenever p"" Q.
The proof of (1 0.11.f) is in complete analogy to that of the case P ~ Q; we will therefore just give the formulas corresponding to (10.II.c, d, e). These are: (1O.11.c')
L (bz - cz)2 = 4(sf-L - 2); L (bz - c z )2 = 4f-L(s - 2);
z ";'x,y
(10. I 1.d')
z\/,PUQ
(10.1 I.e')
L
(bR - CR)2
= 4(s -
2).
R";'P,Q
One then has IN(P)\N(Q)I = 2(s - 2) which yields (1O.1l.f). We have now verified that G satisfies the assumptions of Corollary 7.21; thus G is the graph of an (s, 2; I)-net. Hence we may label the S2 cosets as Pij (i, j = 1, ... , s) such that
.
(lO.n.g)
.,
OfJ=J.
We now define
T:= {
UPij: J=1
i = I, ... ,
s}
and
T':=
{U
Pi{ j= 1, ... ,
,=1
Adjoining T U T' to D gives a resolvable S)" (2, s f-L; II.8.7, this design is affine. •
S2 f-L).
s}.
By Bose's Theorem
10.12 Remarks. (a) The proof of Theorem 10.11 is basically that ofJungnickel and Sane (1982), but we have simplified the last part of the proof by using Bose's theorem; also, we have inserted a discussion of condition (10.11.f) omitted in their paper.
t=
(b) The use of Corollary 7.21 made it necessary to assume s 4 in Theorem 10.11. In fact we will see in the next section that the case s = 4 is truly exceptional; cf. (11.13.a).
(c) There are two further general completion results for (s, r; f-L)-nets with f-L> 1. Completion is possible for s = 2 and d :s 7 (Verheiden (1978), using the language of Hadamard matrices) and for d = 3, s :::: 104 (Shrlkhande and Singhi 1979b). The case d = 3 already requires the use of strongly regular multigraphs
758
X. Transversal designs and nets
and is much more involved than the cases d ::::: 2 or JL = 1. There is one partial result for d:::: 4: whenever s is large enough (e.g. s 2:: 2(d -1)2(2d 2 - 2d + 1», an (s, r; JL)-net D of deficiency d may be embedded into an 5)..(2, SJL; S2JL), say E. As it is not known whether E is resolvable or not, an application of Bose's theorem is not possible and the existence of E says nothing about the completion problem. Nevertheless, this result (due to Shrikhande and Singhi 1979b) supports the conjecture that D is completable if s is sufficiently large. Nothing better than what is true for JL = 1 can be proved, however, as we will see in the next section; thus it seems to be unimportant how large JL is.
§11. Extending Symmetric Nets In this section we will consider extensions of symmetric (s, JL)-nets with JL > 1. This will allow us to construct examples of maximal nets with small deficiency for JL > 1 as well (using the results of §9 with JL = 1). Our account will follow Jungnickel and Sane (1982). We will always assume JL > 1. We shall require a characterisation of the transversals of a symmetric net; we will derive this as a corollary to a more general result of Hine and Mavron (1983) characterising the subnets of a net as its sub-I-designs.
11.1 Lemma. Let e be a set of blocks of an (s, r; JL )-net D and assume that each point of D is on t blocks of e for some positive integer t. Then e consists of t parallel classes ofD. Proof. As e induces a I-design on the point of D, we have lei = st. Now let BEe and assume that C contains x blocks parallel to B. Counting pairs (p, X) where pI B, X and B =1= X E C, yields sJL(t - 1) = (st - x)JL; i.e. x = s . • 11.2 Corollary. Let T be a set of points of a symmetric (s, JL)-net D. Then T meets every block in t points if and only if T is a union of t point classes. In particular, T is a transversal ofD if and only if it is a union of JL point classes. Proof. As D is symmetric, it is simultaneously a TD. Hence the dual of Lemma 11.1 yields the assertion. • Corollary 11.2 implies the result of Exercise 1.7.16.(d) which we state here again for completeness.
11.3 Corollary. Any symmetric (s, JL)-net is step-I-extendible. •
759
§11. Extending symmetric nets
11.4 Theorem. Let D be a symmetric (s, J.1)-net and t 2:: 2 an integer. Then every step-t-extension E ofD induces an (s, t; fLls)-net N(EID) on the set of cosets of D, where the blocks of D' are the transversals of D used in the extension. Moreover, each (s, t; fLls)-net may be obtained in this way. Proof. Let E be any step-t-extension of D and let T be the set of blocks in
E \ D. Then T consists of transversals of D and thus each T E T is a union of J-L cosets of D, by Corollary 11.2. If x is a point of D, denote its coset by x. Define (l1.4.a)
X1T
{:=:}
x~
T
for any point x and any T
E
T.
The resulting incidence structure N (E ID) has s 2 m points with blocks of sm points each, where m = fL Is. Clearly each parallel class of E \ D induces a parallel class of N(EID). If T, T' are non-parallel blocks in T, we have IT n Til = fL inE and therefore IT n T'l = m in N(E ID). Thus N(E ID)is an (s, t; m)-net. The second assertion has already been proved in Lemma 6.7 . •
11.5 Corollary. Let D be a symmetric (s, fL)-net where s does not divide fL. Then the only extensions ofD are (s, sfL + 1; fL)-nets, and all such extensions are maximal. • 11.6 Corollary. A symmetric (s, fL )-net D is step-t -extendible for t > 2 if and only if s divides fL and if there exists an (s, t; fLls)-net. Hence D is step-7extendible if s I fL and s =j:. fL· Proof. The first assertion is immediate from Theorem 11.4; the second assertion follows from the existence of an (s, 7; m)-netfor all m 2:: 2 (the dual of Han ani's Theorem 2.1) . • In particular, we have the following result of Hine and Mavron (1980).
11.7 Corollary. A symmetric (s, fL)-net is completable if and only if there exists an (s, (SfL - 1)/(s -1); fLls)-net, i.e. an affine 2-design A/L/s(s) . • We will require the following strengthening of Theorem 11.4.
11.8 Theorem. Let E be a step-t-extension of a symmetric (s, fL)-net D with t:::: 2. Then E is step-n-extendible if and only if N(D IE) is step-n-extendible. In particular, E is maximal if and only if N (E ID) is maximal.
X. Transversal designs and nets
760 Proof. Exercise. •
11.9 Corollary. Let D be a symmetric (s, f.L)-net with s I f.L. Then D can be extended to a maximal (s, Sf.L + t; f.L)-net if and only if there exists a maximal (s, t; f.L/s)-net. • These results give us two general constructions for maximal nets with f.L > 1, due to Jungnicke1 and Sane (1982). 11.10 Theorem. Lets and c be positive integers, where c does not divide s, and assume the existence ofa symmetric (s, f.L )-net with f.L = cs n for all non-negative integers n. Then there also exists a maximal (s, r; f.L)-net with (1 LlO.a)
r
= cs n+ 1 + cs n + ... + cs + 1
for all n.
Proof. As that of Corollary 6.8, using Corollaries 11.5 and 11.9. • 11.11 Examples. In Theorem 11.10, one may choose e.g. (cf. Example 6.9): (ll.11.a)
s = pi, C = pj whenever p is a prime and i, j are positive integers with i > j (by Corollary Vrn.3.12);
(ll.11.b)
s an odd prime power, c
= 2 (by Theorem vrn.3.14).
The second construction is of even greater interest as here the corresponding nets might be completable (i.e. s - 1 divides f.L - 1). Thus we may speak: of the deficiency of the nets constructed in this case. 11.12 Theorem. Let s be a prime power and assume the existence ofa maximal (s, r; I)-net of deficiency d. Then there also exists a maximal net of order sand deficiency d for all f.L = sn with n E N. Proof. Symmetric (s, sn)-nets exist for all positive integers n by Proposition 1.7.18. Now use Corollary 11.9 and induction. • 11.13 Examples. Let n be any non-negative integer. Then there exists a maximal (s, r; sn)-net of deficiency d in each of the following cases: (l1.13.a)
d = 2, s = 4 (by Example 8.4);
(11.13.b)
d = 3, s = 5 or 9 (by Theorem 8.6 and Theorem 9.28);
§12. Complete mappings, difference matrices and maximal nets (11.13.c)
d = 4, s = 7, 8, or 25 (by Remark 8.20 and Theorem 9.28);
(11.13.d)
d = 5, s = 7, 8 or 25 (Remark 8.20 and Theorem 9.28).
761
More generally, we have the following series: (l1.l3.e)
s a prime power with s even or s == 1 (mod 4), d (Theorem 8.6 and Corollary 12.5 below);
(ll.l3.£)
s = q2, where q is a power of a prime p?::. 5, and dE {q - I, q, q + I}, by Theorem 9.28.
=s-
2
We remark that (11.13.a) shows that the condition s # 4 in Theorem 10.11 is necessary. Theorem 11.12. also shows that nothing better can be expected regarding the completion of (s, r; J.L)-nets with 11- > 1 than what is true for 11-=1.
§12. Complete Mappings, Difference Matrices and Maximal Nets In this section we will consider some more advanced results on difference matrices. We will deal with three interrelated topics, namely non-existence results for difference matrices and, in particular, generalised Hadamard matrices; connections to the notion of "complete mappings" which is one of the major themes in the study of Latin squares; and the use of difference matrices in the construction of maximal nets. Our first goal is the proof of a major non-existence result for difference matrices due to Drake (1979). For this purpose, we require the following auxiliary result from group theory which is a special case of Satz IV.2.8 in Huppert (1967); for the convenience of the reader, we will include an ad hoc proof.
12.1 Lemma. Let G be a finite group of even order, and let T be a Sylow 2subgroup of G. Assume that T is cyclic. Then G has a unique maximal normal subgroup O(G) of odd order; moreover, T is a complement of O(G), that is G = T· O(G). Proof. Since the product of any two normal subgroups of odd order is again a normal subgroup of odd order, it is clear that 0 (G) exists and is in fact a characteristic subgroup of G. Let a be a generator of T and note that G acts on itself by the right regular representation, see Exercise V.I.I. The permutation induced by a consists of 10 (G) I cycles of length IT I each and is therefore odd. Thus the subgroup of even permutations in G is a proper subgroup G I of index 2,
762
X. Transversal designs and nets
and O(G J ) = O(G), as each element of O(G) induces an even permutation. By induction, we may assume G 1 = O(G) . (T n G 1 ), as T n G 1 is either a Sylow 2-subgroup of GJ or of order 1. The assertion is now immediate. •
12.2 Theorem. Let G be a finite group of even order s, let T be a Sylow 2subgroup of G, and let A be an odd positive integer. Assume that T is cyclic. Then there is no (s, k; A)-difference matrix over G with k?: 3. Proof. It suffices to consider the case k = 3; thus assume the existence of an (s, 3; A)-difference matrix over G. By Lemma 12.1, T ;:::: G/O(G); thus Lemma VIII.3.l1 implies the existence of a (t, 3; Au)-difference matrix over T, where t := ITI and u := IO(G)I. As AU is odd, it suffices to prove the assertion for the special case T = G. Thus assume the existence of an (s, 3; A)difference matrix Dover Zs, where A is odd and s is a power of 2. We may assume that the first row of Dis (0, ... ,0). For i,j = 0, ... , s - 1, let f(i,j) denote the number of columns (O,i ,jl of D. As D is a difference matrix, we obtain the three conditions s-J
(12.2.a)
s-J
Lf(x, j)
s-J
= Lf(i,x) =
Lf(x,x +k)
= A,
where i, j, k are arbitrary elements of Zs and where all arguments are to be considered modulo s. Also note
(l2.2.b)
~
~n
=
s(s - 1)
2
n=O
s
=- (mods). 2
We now consider the f(a, f3) modulo s; then (12.2.a, b) imply
s
s-1
+ L(s -k) Lf(x,x +k) k=J
s-J
-
x=O s-J
Lf(i,j)(j-i+s-(j-i»=s. Lf(i,j)=O, i,j=O
i,j=O
a contradiction. • The special case A = I yields a famous result on Latin squares which is due to Hall and Paige (1955) and which yields an interesting series of transversal-free nets. We need a definition first.
§12. Complete mappings, difference matrices and maximal nets
763
12.3 Definition. Let G be a quasigroup, and let IX : G -+ G be a bijection. Then IX is called a complete mapping of G provided that f3 : x ~ xx a is a bijection as well. In other words, IX is a complete mapping if and only if the set of cells {(x, x a ) : x E G} is a transversal of the Latin square given by the multiplication table of G. Any permutation T) of G for which the mapping x ~ X-I x~ is a bijection is called an orthomorphism of G; in other words, T) is an orthomorphism if and only if x ~ x-I x~ is a complete mapping. 12.4 Corollary (The Hall-Paige Theorem). Let L be the Latin square given by the multiplication table of a group G of order s.lf s is even and if the Sylow 2-subgroup of G is cyclic, then L has no transversal. Proof. Assume otherwise; then G admits a complete mapping IX. Writing G additively, both IX and f3: x ~ x + x a are bijections of G. But then, with G = {gl, ... , gs},
g~) -g~
is an (s, 3; I)-difference matrix over G, contradicting Theorem 12.2. •
12.5 Corollary. Let s be an even positive integer. Then there exists a transversal-free (s, 3; I)-net. • The case s = 4 of Corollary 12.5 gives the net presented in Remark 7.22. For odd values of s, group tables cannot yield maximal nets as the following result of Sade (1963) shows.
12.6 Lemma. Let G be a group or a commutative quasigroup of odd order. Then G admits a complete mapping. Proof. It suffices to show thateach element of G has a (unique) square root, since then the mapping x ~ x 2 is a bijection which implies that the identity is a complete mapping for G. Quasigroups with this property are called diagonal. If G is a group and if g E G has odd order 2n -1, then g2n = g and hence gn is the square root of g. Now let G be a commutative quasigroup and note that each g E G occurs exactly IGI times in the multiplication table of G. Since there is an even number of ordered pairs (x, y) of elements of G satisfying x =1= y and xy = y x = g, there exists at least one element Z E G with Z2 = g. •
_I
\-...-
§12. Complete mappings, difference matrices and maximal nets
765
12.10 Theorem. Let s and A be odd integers, G an abelian group of order s, P a prime dividing s, and m some divisor ofthe square-free part of n := SA which is not a multiple ofp.lfm has even order modulo p, then there is no GH(s, Je) over G. Proof. Since G is abelian, there exists an epimorphism from G onto a (multiplicatively written) cyclic group C of order p. By Lemma VIII.3.II, the existence of some GH(s, A) over G implies that of a GH(p, UA) over C, where u := s/ p. Thus let H be a GH(p, UA) over C. Let m =PI ... Pt be the prime factorisation of m, and denote the order of m modulo Pi by ei, for i = I, ... , t. Then me == I (mod p), where e = el ... e t • Now assume that m has even order modulo p, that is, assume that ei is even for some i. Then me;/2 == - I (mod p), which will allow us to apply Lemma VI.3.10 to the element A := det H ofZC. As n is odd, the p-part of nn / p is a square, say q2. Hence Lemma 12.9 shows AA(-l) == 0 (mod pq2), and Lemma VI.3.10 implies A == 0 (mod pq). But then AA(-l) == 0 (mod p2q2), a contradiction. • 12.11 Examples. (a) Assume the existence of an abelian GH(s, 1) with s ::: 100. Then s is a prime power or s E {39, 55, 63}. This is a consequence of Corollary 12.8 and Theorem 12.10, as noted by de Launey (1986); the reader should check this assertion as an exercise. (b) There is no abelian GH(s, 1) withs == 15 (mod 18). To see this, lets = 18a+ 15 = 3(6a + 5) and choose p = 3. Clearly some prime m == 2 (mod 3) divides the square-free part of 6a + 5 (hence of s), and thus Theorem 12.10 implies the assertion. This application of Theorem 12.10 was noted by Jungnickel (1990a); the result in question was also obtained by Woodcock (1986) in the special case of cyclic groups. The following non-existence result is due to Brock (1988); its proof uses theorems on the hermitian congruence of matrices and is considerably more involved than that of Theorem 12.10. 12.12 Theorem. Assume the existence of some GH(s, A) over a (not necessarily abelian) group G, where n := SA is odd. Then the equation
has a non-trivial solution in integers x, y, zfor every t of a homomorphic image of G. •
# 1 which is the order
§12. Complete mappings, difference matrices and maximal nets
765
12.10 Theorem. Let s and A be odd integers, G an abelian group of order s, P a prime dividing s, and m some divisor ofthe square-free part of n := SA which is not a multiple ofp.lfm has even order modulo p, then there is no GH(s, Je) over G. Proof. Since G is abelian, there exists an epimorphism from G onto a (multiplicatively written) cyclic group C of order p. By Lemma VIII.3.II, the existence of some GH(s, A) over G implies that of a GH(p, UA) over C, where u := s/ p. Thus let H be a GH(p, UA) over C. Let m =PI ... Pt be the prime factorisation of m, and denote the order of m modulo Pi by ei, for i = I, ... , t. Then me == I (mod p), where e = el ... e t • Now assume that m has even order modulo p, that is, assume that ei is even for some i. Then me;/2 == - I (mod p), which will allow us to apply Lemma VI.3.10 to the element A := det H ofZC. As n is odd, the p-part of nn / p is a square, say q2. Hence Lemma 12.9 shows AA(-l) == 0 (mod pq2), and Lemma VI.3.10 implies A == 0 (mod pq). But then AA(-l) == 0 (mod p2q2), a contradiction. • 12.11 Examples. (a) Assume the existence of an abelian GH(s, 1) with s ::: 100. Then s is a prime power or s E {39, 55, 63}. This is a consequence of Corollary 12.8 and Theorem 12.10, as noted by de Launey (1986); the reader should check this assertion as an exercise. (b) There is no abelian GH(s, 1) withs == 15 (mod 18). To see this, lets = 18a+ 15 = 3(6a + 5) and choose p = 3. Clearly some prime m == 2 (mod 3) divides the square-free part of 6a + 5 (hence of s), and thus Theorem 12.10 implies the assertion. This application of Theorem 12.10 was noted by Jungnickel (1990a); the result in question was also obtained by Woodcock (1986) in the special case of cyclic groups. The following non-existence result is due to Brock (1988); its proof uses theorems on the hermitian congruence of matrices and is considerably more involved than that of Theorem 12.10. 12.12 Theorem. Assume the existence of some GH(s, A) over a (not necessarily abelian) group G, where n := SA is odd. Then the equation
has a non-trivial solution in integers x, y, zfor every t of a homomorphic image of G. •
# 1 which is the order
766
X. Transversal designs and nets
Let us once more consider Corollary 12.4. From its proof, it is clear that an (s,2; I)-difference matrix over G is "maximal" if and only if the corresponding (s, 3; I)-net (or Latin square) has no transversal. In order to generalise this observation, we give the following definition.
12.13 Definition. Let G be an additively written (but not necessarily abelian) group of order s and let D (dij)i=l, ... ,r;j=l, ... ,s). be a matrix with entries from G. One calls D a left (s, r; ).,,)-difference matrix over G, if the following condition is satisfied:
=
(12.13.a)
{-dij
+ dhj
:j
= I, ... , s} = )"'G
for any two distinct indices h, i
E
{I, ... , r }.
In other words, the "left" differences of corresponding entries of any two rows in D contain each element of G exactly)." times. D is called maximal if no further row may be adjoined to obtain a larger left difference matrix.
12.14 Remark. Similarly, one may define right difference matrices. Of course, it makes no essential difference which one of these two species of difference matrices is considered: D is a left difference matrix if and only if - D is a right difference matrix. In the abelian case, both conditions are trivially equivalent, anyway. Usually, the term "difference matrix" in the literature means "right difference matrix", as in Definition VIll.3.4. However, in the present context, where we want to consider the associated nets, the use ofleft difference matrices is more appropriate. We know that any right difference matrix D yields a TD[r + 1; s], by Corollary VIII.3.8, and thus an (s, r + 1; I)-net D; this net can be described explicitly by using the corresponding left difference matrix - D. The following result provides such a description and also shows thatD is transversal-free if and only if D is maximal.
12.15 Theorem. The existence ofa left (s, r; 1)-difference matrix D over some group Goforder s implies that ofan (s, r+ I)-net N(D). Moreover; D is maximal if and only ifN(D) is transversal-free. Proof. Let D be a left (s, r; I)-difference matrix over G. We construct N := N(D) as follows. As point set, we choose V={(i,x): i=I, ... ,S;xEG}.
§12. Complete mappings, difference matrices and maximal nets
767
We now define sets of lines
Po:={Lj:j=l, ... ,s}
and
Pi:={Li,x:XEG}
for i = I, ... , r, where
Lj:={(j,X):XEG}, Li,x := {(j, dij
+ x) : j
= 1, ... , s}.
It is obvious that all the Pi are parallel classes of lines. It remains to check that any two non-parallel lines intersect in a unique point. This is clear if one of the lines belongs to Po; thus consider any two lines L i •x and Lh,y where h i= i. Note that a point (j, z) is on both of these lines if and only if z = dij + x = d"j + y. This condition is equivalent to the equations
z
= dij + x
and
-dhj
+ dij
= Y - x;
the second of these equations has a unique solution for j by the defining property of a difference matrix, and then z is determined from the first equation. Thus N is indeed a net. Now consider any s-set T of points of N which intersects each line of the parallel class Po uniquely (and thus is a candidate for a transversal of N). Hence T has the form
T:= {(j,tj):j = 1, ... ,s}. By our definition of the lines Li,x, the intersection points of T with Li,x correspond to the solutions of the equation (l2.IS.a)
-tj
+ dij = -x.
Hence T is a transversal for N if and only if, for any given pair x and i, equation (12.lS.a) has a unique solution for j; but this just means that the row (t\, ... , ts ) extends D to a left (s, r + 1; I)-difference matrix. •
12.16 Remark. The equivalence between certain types of nets (or mutually orthogonal Latin squares) and difference matrices was first established by Jungnickel (1979, 1980b); see Hachenberger and Jungnickel (1990) for a considerably more detailed study. Sitnilar but weaker results also hold in the case of an arbitrary A, see Jungnickel (1979). Note that the net N constructed in the proof of Theorem 12.15 is a very special type of net (an observation similar
X. Transversal designs and nets
768
to that made for transversal designs in Theorem VIII.3.6): G acts on N as a group of "central translations" if we let the element g E G map the point (j, x) to (j, x + g) and use the induced action on the set of lines, namely, if we let g fix each line in Po and map the line Li,x to Li,x+g' Moreover, N is a so-called "Po-transitive" net. It can be shown that nets with such a "transitive direction" are equivalent to difference matrices. For a detailed study of the geometry of these nets, we refer the reader to Hachenberger and Jungnickel (1990). Given this geometric interpretation of a difference matrix as a net with a transitive direction, the maximality result in Theorem 12.15 can be recognised as a consequence of a theorem of Ostrom (1966) even though it was first explicitly stated in the first edition of the present book (with a minor mistake, because no attention was paid to the distinction between left and right difference matrices, and thus the proof given then only worked for the abelian case). 12.17 Exercises. (a) Show that Theorem 12.15 does not hold for (s, r; p,)difference matrices with p, > 1. Hint: consider Example 6.9.
=
(b) In the situation of Theorem 12.15, assume that N N(D) has small deficiency. Use Lemma 7.6 to prove that every extension of N corresponds to a left difference matrix extending D. This observation is due to Hachenberger and Jungnickel (1990). (c) Use Theorems 12.15 and IT.8.21 to prove the result already mentioned in Remark VIII.3.9: an (s, r; p,)-difference matrix D satisfies r = sp, if and only if - DT is also a difference matrix. 12.18 Examples. We will now use maximal difference matrices to construct some transversal-free nets of small orders. The examples we present are due to Jungni.ckel and Grams (1986) and were found by computer search, but their correctness can be checked directly, even though this is a bit tedious. Jungnickel and Grams (1986) in fact determined maximal (s, r )-difference matrices of all possible sizes r in all groups of order s :::: 10. This was extended to s :::: 15 by Bedford and Whitaker (1998). (a) The following matrix is a maximal (8, 4)-difference matrix over the elementary abelian group of order 8:
(~
000 000 000
000 000 000 100 010 110 101 100 001 111 011 100
000 000 001 101 010 111 110 001
000 011 110 101
111 ~) 011 . 010
§I2. Complete mappings, difference matrices and maximal nets
769
(b) The following matrix is a maximal (8, 3)-difference matrix over the quaternion group of order 8, written as G = {±1, ±i, ±j, ±k}:
G
1 -1
k
-i j
j -j
1 -j
-k
1 k -1
-!) -I
(c) The following matrix is a maximal (9, 6)-difference matrix over the elementary abelian group of order 9: 00 00 00 00 00 00
00 10 20 21 01 22
00 20 10 12 02 11
00 00 00 00 01 11 21 02 22 12 02 11 10 01 22 20 12 10 11 21 02 21 10 01
00 12 01 11 22 20
00 22 21 02 20 12
(d) The (12, 6; I)-difference matrix given in Lemma VllI.3.13 is maximal and hence yields a transversal-free (12, 7; I)-net. The following result shows that (s, r; f.L )-difference matrices with f.L > 1 can also be useful in the construction of maximal Bruck nets. This somewhat surprising fact was discovered by Evans (1991).
12.19 Theorem. Let D be an (s, r; I)-difference matrix over G, and assume that mD is a maximal (s, r; m)-difference matrix, where mD denotes the matrix obtainedfrom D by concatenating m copies ofD. If there exists some (m, r + 1)net, then there exists a maximal (sm, r + I)-net. • The proof of Theorem 12.19 is rather more involved than that of Theorem 12.15, though similar in spirit. As Evans noted, the maximality of D is trivially necessary but in general not sufficient for that of mD. Theorem 12.19 may be applied to give difference matrix proofs for some of the older results which were discussed in §8. However, its real interest comes from the fact that it allowed Evans to produce many previously unknown series of maximal nets. To this end, he first gave the following construction for maximal difference matrices of prime power order.
12.20 Theorem. Let s = pa be a power of some prime p. Then there exists an (s, p; I)-difference matrix Dover Zs for which mD is maximal whenever m is not a multiple of p. More precisely, every matrix of the form (12.20.a)
D
= (dij)i=I ..... p;j=I, ... ,s
with
dij
= CPU -
l)j
X. Transversal designs and nets
770
for some mapping : tl p -+ tl s satisfying
(12.20.b)
(i):=i (mods) for all i
has the desired properties. •
12.21 Remark. In particular, one may choose the desired matrix D in Theorem 12.20 to consist of the first p rows of the multiplication table of 7l s • Combining Theorems 12.19 and 12.20 gives a proof of Evans' Theorem 8.15. One wonders if it is possible to extend his work to prove the existence of a maximal (mpa, pb + I)-net, where b :::: a, provided that there exists some (m, pb + I)-net. This would follow if one could generalise Evans' work to solve the following problem which should have a positive answer at least when a is a mUltiple of b. 12.22 Research Problem. Construct a (pa , pb; I)-difference matrix D, where b :::: a, such that mD is maximal whenever p does not divide m, or show that no such matrix exists. We conclude this section with another important construction due to Evans (1992a) and some variations discussed by Pott (1993).
12.23 Theorem. Let p > 5 be a prime. Then there exists a maximal (p, r; 1)difference matrix, where
(12.23.a)
r = {(p -1)/2 (p
ijp:=3 (mod 4)
+ 1)/2 if p:= 1 (mod 4).
Hence there exist maximal (p, r)-nets with
(12.23.b)
(P+1)/2 r= { (p+3)/2
ijp:=3(mod4) ijp-=1(mod4) . •
12.24 Remark. Evans' proof of Theorem 12.23 is given in the language of orthomorphisms and uses permutation polynomials, in particular Hermite's criterion; see Lidl and Niederreiter (1983) for background. Pott gave a somewhat simpler and definitely more transparent proof in the language of difference matrices. We shall sketch his approach which is based on Redei's theorem; see Rectei (1973) and Lovasz and Schrijver (1981) for a simpler proof. The idea
§12. Complete mappings, difference matrices and maximal nets
771
is to use some rows of the trivial (p, p; I)-difference matrix D* given by the multiplication table of Zp. Thus we take G = (Zp, +) and D* := (mx)m,xEG'
We now want to extend a selected set of rows of D* by adding a row of the form (12.24.a)
/= (f(X»XEG,
where f is a suitable non-linear function on G. Let us consider the set of slopes determined by the graph of f in the affine plane AG(2, p), that is (12.24.b)
Sf= {
f(x) - fey) } :X,yEG,X¥Y· x-y
Then it is easily checked that f is compatible with the m-th row d m of D* if and only if m ¥ Sf; let D consist of all such rows together with row f Of course, there is no reason why D should be maximal in general, even though all further possible extensions g are quite restricted, as they have to yield the same set of slopes as f. By Redei' s theorem, Sf contains at least (p + 3) /2 elements, since f is non-linear, and so D has at most (p -1)/2 rows. Evans' result comes from choosing f as follows:
(12.24.c)
f(x) := X(p+1)/2 =
I~ -x
if x = 0 if x is a square if x is a non-square.
It turns out that the corresponding matrix D is indeed maximal if and only if p == 3 (mod 4). In the case p == 1 (mod 4), a maximal difference matrix is obtained by adjoining to D the further row (12.24.d)
-f = (- f
(x»x
E
G·
Pott's proof then makes use of Redei's theorem and of one ofits variations, i.e. the complete description of all functions giving a minimal set of slopes obtained by Lovasz and Schrijver (1981). Pott also considers possible extensions of Evans' result. Regarding a generalisation to prime powers, Pott explained why the construction cannot work for q = 9, a fact already noted by Evans; computer checks show, however, that it does work for q = 25, 27 and 49. Thus we have the following problem.
X. Transversal designs and nets
772
12.25 Research Problem. Does the analogue of Theorem 12.23 hold for prime powers q > 9? 12.26 Remark. Pott (1993) also considered possible extensions of Evans' results obtained from replacing the function f defined in (I2.24.c) by using higher power residues instead of squares. He suggested taking
(12.26.a)
ax f(x) = { bx
if x if x
E
Co
Ii Co '
where Co is the set of e-th powers if GF(q) for some divisor e of q - 1. In this case, the set of slopes of f has cardinality q - 1 - g, where ge = q - 1. With this approach, Pott used a computer search to show the existence of the maximal (p, r; I)-difference matrices given in the following table. Unfortunately, there is no theoretical explanation for his results and not even a reasonable conjecture comparable to the one suggested in Research Problem 12.25. p
g r
13 3 3
13 4 5
17 3 5
17 4 4
19 3 4
19 4 7
§13. Tarry's Theorem In this section, we prove the result of Tarry (1900) concerning the nonexistence of two MOLS of order six. Following Betten (1983) we show:
13.1 Theorem. A TD [3; 6] cannot have six mutually disjoint parallel classes.
• The proof will need some preparation.
13.2 Notation and Definition. We denote the cell of a Latin square in the i-th row and j-thcolumn by (~) and, if the entry in this cell is k = ioj we write k). Two Latin squares are called equivalent if their TD[3; 6]' s are isomorphic. Recall that the points of the TD of a Latin square are Xi (x E {I, 2, 3,4,5, 6}, i E {I, 2, 3}) and that the lines are the point sets {Xl, X2, (X 0 Yh}. Permutations of rows, columns, or symbols 1, ... , 6 yield equivalent Latin squares. The same holds for mappings Xi 1-+ X"i with J( E S3, e.g. replacing the quasigroup operation 0 by * which is defined by X 0 Y = z ~ y =X * z. Note that the transversals of a Latin square correspond to the parallel classes of its TD[3; 6].
e
773
§13. Tarry's theorem
13.3 Definition and Observations. The permutation n (i, j) of the rows i and j is the permutation n(i,j): iox
1-+
jox
of S6. As n has no fixed points, only the following permutation types can occur. b c
G G
c d
d e
b c
c d
(~
b c
~) (~
~),
e f
more briefly (a b c d e f),
type 6,
~) (; ~),
more briefly (a b c d)(e f),
type 4,4
~),
more briefly (a b c)(d e f),
type 3,4
more briefly (a b)(c d)(e f),
type 2,4
e f
(~ ~) (~ ~) (; ~),
The permutations of type 4 and 3 are even, the others odd. The main idea of Betten's proof is to establish and use the fact that each equivalence class of Latin squares of order 6 contains a Latin square with n(1, 2) of type 3.
13.4 Observation. Given two rows of permutation type 6, 4, or 2, there are three columns containing all six symbols. Given two rows of permutation type 4 or 2, any pair of columns with four distinct entries can be extended to three columns with six distinct entries. 13.5 Lemma. Any colouring of the 15 edges of the complete graph K6 with two colours, say red and blue, contains at least one monochromatic triangle. Proof. Any vertex x is joined to five other vertices, hence x is on three edges of equal colour, say red. Then either x is a vertex of a red triangle or the other vertices of the three red edges form a blue triangle. • This is the simplest case of Ramsey's theorem; cf. Ryser (1963). All the following reasoning is to be understood up to equivalence.
13.6 Corollary. The rows i E {I, 2, 3, 4,5, 6} and the row pairs ii, j} form the vertices and edges of the complete graph K6. Let the edge ii, j} be red
4
Note that this would usually be called type (4,2), etc.
X. Transversal designs and nets
774
if n(i,
j) is even, and blue otherwise. Then there is a red triangle, as a blue triangle is clearly impossible. •
13.7 Lemma. In each equivalence class there is a Latin square with n(1, 2) of type 3. Proof. Otherwise there are three rows, w.l.o.g. the first three ones with n(1, 2), n(2, 3), and n(l, 3) of type 4 (Corollary 13.6). Wl.o.g. the first two columns are
12) 21 . ( 34
By Observation 13.4, they can be extended to
(125---) 21- 6 -- . 346521
Then
203 E {3,4}, say 203 = 3 (w.l.o.g.). As n(1, 2), n(2, 3) and n(l, 3) are of type 4, the entries in the first three rows must be either
1 2 5 3 4 6) 2 1 3 6 5 4 (3 4 6 5 2 1
1 2 5 or
4 6 3) .
213 6 5 4 ( 346 521
In the first case the columns 3 and 4 are of permutation type 3. In the second case define * by x 0 y = z ¢:=} x * z = y. Then the Latin square belonging to * has the last two columns of type 3. • 13.8 Corollary. Each Latin square is equivalent to one ofthefollowing shapes, where the shaded cells are occupied by 4, 5, or 6 and the other cells by 1, 2, or 3.
Shape A
Shape B
Shape C
Proof. Exercise. • 13.9 Lemma. A Latin square of shape A has no transversal. Proof. A transversal must contain two entries from {I, 2, 3} in the upper left 3 x 3-square and one in the lower right one, or vice versa. Then three entries
§13. Tarry's theorem
775
from {4, 5, 6} must lie in one column of the lower left square and in one row of the upper left square, which is impossible. •
13.10 Corollary. A Latin square of shape B has at most four mutually disjoint transversals. Proof. Any transversal T avoiding the four central cells would also be a transversal of shape A. Thus it cannot exist. • Henceforth we only consider Latin squares of shape C.
13.11 Lemma. A transversal T of a Latin square of shape C has exactly one entry both in the upper left and the upper right 2 x 3-rectangle. Proof. Suppose T has two entries in the upper left 2 x 3-rectangle. Then it cannot have a third entry in the upper left 3 x 3-square, as the number of entries from {4, 5, 6} in the lower right 3 x 3-square cannot be two. Hence the third entry from {I, 2, 3} is in the third row on the right-hand side. Two of the entries from {4, 5, 6} are in (!), (;), (~), and the third one is in the lower left 3 x 3-square. But then one of the three rows 4, 5, 6 would contain two identical entries, a contradiction. Note that the entries in (!), (D, (~) must be distinct. The case that two entries of T are in the upper right rectangle is treated analogously. • 13.12 Lemma. If a transversal T of a Latin square of shape C has an entry in one of the cell sets {(~), (;), (~)} or {(!), (;), (~)}, then it has exactly one entry in each of these two sets. Proof. First assume (w.l.o.g.) that T has two entries in the set [(~), (;). m}. By Lemma 13.11, it has one entry in the upper left 2 x 3-rectangle, hence it must have exactly one entry (from [4.5, 6}) in {(!), n), (~)} and two in the upper right rectangle, which contradicts Lemma 13.11. Now assume that T has exactly one entry in {(i), (D, (~)}. By Lemma 13.11, T contains exactly two entries from {l, 2, 3} in the first three columns, and hence exactly two entries from {4. 5, 6} in the last three columns. Again by Lemma 13.11, one of these entries must be in {(:). (D, (~)}. •
13.13 Observation. Using the previous lemma, each Latin square (of shape C) with six mutually disjoint transversals contains one of the following two partial
X. Transversal designs and nets
776
squares (of course, up to equivalence), where three of the transversals are denoted by a, band c and where a letter a next to an entry means that the corresponding cell belongs to the transversal a, etc.
3
2 3 la
1 -
-
-
2b
-
I 2 3c
5 6 -
6 4
4 5
-
-
4c
-
-
-
5a
-
-
-
6b
5 4 -
or 2 3 la -
-
1 -
I 2
6 5
4 6
-
-
-
-
4c
-
2b
-
-
-
3c
-
3
5a -
-
6b
Note that this arrangement of the partial transversals a, b, c may be obtained by suitable pennutations of the rows, columns, and symbols. Moreover both of these partial squares remain unchanged under the simultaneous pennutation (4 5 6) of rows, (1 23) (45 6) of columns, (1 23) (4 5 6) of the symbols, and (a b c) of the transversals. Let r be the group generated by this simultaneous pennutation. Now we try to extend the partial transversals a, b, c. 13.14 Lemma and Observation. In the left-hand partial square of Observation 13.13, the partial transversal a cannot be completed. Proof. A complete transversal a must contain either (~ 3) and (~6) or (; 2) and 4), by Lemma 13.11. In both cases one cannot find an entry in row 6. •
(!
Henceforth only the right-hand partial square of 13.13 needs further attention. This partial square is also preserved under the pennutation
(J:
(45) (15)(24)(36) (15)(24)(36) (bc)
I
ofrows of columns of symbols of transversals.
§13. Tarry's theorem
777
13.15 Observations. In the last three rows the transversals a, b, and e can take the following places.
Ia
4
5b
I
5a
2
4a 3
6
4
2b
3 2a
6
5
6b
3
5a
2
6e 2
5
4
Ia
4e
4
2b
3e 4e
3b 5
2
6b
3
6
5
3e
Ie 6
Note that the group G generated by rand (J is isomorphic to the symmetric group S3 and permutes the above six 3 x 6-rectangles regularly. r permutes the upper three rectangles cyclically, and the three lower ones too, and (J interchanges each upper rectangle with a lower one. Now we combine the above rectangles. Up to equivalence there are only the following two possibilities.
Ia 6e
5b 2b 4a
4e
Ia 5a
6e
6b
3e
5b
4e
3e
2a
2b
5a 6b
These two rectangles can be uniquely extended to the following two partial squares (using the preceding lemmas).
2
3b
s=
Q=
4
3 Ie
I
6a
4b
5e
2a
5 Ib
6
4
2e
3a
4e 3 2
3
2
5 6
6
4
5
2b 4a
Ia 6e
5b 3e
5a
I
1
6b 5e 4a
2
3a
1
6
4b
3b
Ie
2
5
6
2e
6
6a 5b
Ib
Ia 6e
4e
3
3 2a
5a
3 2 1
I
6b
2b
4
3e
The four empty cells in Q have to be occupied by one of the Latin subsquares
(~~)
or
(~~).
778
X. Transversal designs and nets
13.16 Lemma. The partial squares Sand Q cannot be completed to Latin squares with six mutually disjoint transversals. Proof. The entries of a hypothetic fourth transversal T of S through (~ 5) in rows 3, 1,2 must be (; 6), 3) and 4). Now the extension to row 4 is impossible.
G
a
(!
G
a
A hypothetic fourth transversal T of Q through 2) must contain 1), 3), and (~ 6). The remaining two entries must form a transversal of a Latin subsquare (~~) or (~~), which is impossible. This contradiction proves Lemma 13.16 and hence also Theorem l3.1. • The method ofthis section, i.e. of Betten (1983), may also be used to determine all "species" of Latin squares of order 6; cf. Betten (1984) and Fisher and Yates (1934). Another simplified proof of Tarry's theorem is due to Stinson (1984) and a further proof, similar to that of Stinson, was given by Dougherty (1994); these proofs use the codes of a corresponding TD or net, respectively.
§14. Codes of Bruck Nets fu this final section, we report on some very interesting results due to Moorhouse (1991a, b, 1993) regarding the p-rank of Bruck nets; we remind the reader that we have discussed the p-rank of designs in §II.II. The proof of the following lemma is a simple exercise.
14.1 Lemma. Let A be an affine plane of order n, let P be the projective plane extending A, and let p be a prime dividing n. Then rankpA = rankpP - 1. • Combining Lemma 14.1 with Theorem II.I1.6, we have the following result:
14.2 Theorem. Let A be an affine plane oforder n, and let p be a prime dividing· the square-free part of n. Then rankpA = n(n + 1)/2. • Moorhouse (1991a) offered the following interesting conjecture regarding the p-rank of a Bruck net.
14.3 Coujecture. Let N be any (n, r; I)-net, and let p be a prime dividing the squarejree part of n. Moreover, let M be any (n, r - 1; I)-net contained in N. Then (14.3.a)
rankpN::;: (n - r + 1) +rankpM.
§14. Codes of Bruck nets
779
In particular;
(14.3.b)
rankpN;::rn - r(r - 1)/2.
This conjecture holds trivially for r = 1 and r = 2; also, Theorem 14.2 shows that (14.3.b) is true for r = n + 1. Moorhouse established that his conjecture is also true for r = 3; in fact, he proved the following considerably stronger result.
14.4 Theorem. Let N be any (n, 3; l)-net, and let p be a prime such that pe II n. Then
(14.4. a)
3n - 2 - e::::: rankpN::::: 3n - 2. •
14.5 Remark. More precisely, rankpN can be determined as follows by using some results from the theory of loops. For background and undefined terms, we refer the reader to Jungnickel (1990a) and Moorhouse (1991a). Coordinatise N by a loop G. Then (14.5.a)
rankpN
= 3n -
2 - s,
where pS = [G: K] for the unique minimal normal subloop of G for which G/K is an elementary abelian p-group. Theorem 14.4 immediately implies the validity of Conjecture 14.3 for r = 3. Moorhouse (1991 a) also obtained the following characterisations of nets based on a cyclic Latin square.
14.6 Theorem. Let N be any (n, 3; I)-net and assume that n is square-free or that N can be coordinatised by a nilpotent group G. Then rankpN = 3n - 3 for every prime p dividing n if and only if G is cyclic, that is, N belongs to the cyclic Latin square of order n given by the group table ofZn . • Moorhouse (1991a) also verified Conjecture 14.3 for many examples of nets of small orders. Moreover, he proved the validity of his conjecture for nets which are part of a Desarguesian affine plane AG(2, p), where p is any prime; his proof additionally provides an explicit basis for the GF(p )-code of AG(2, p). Note that one has to have equality in Conjecture 14.3 then, as is easily seen using Theorem 14.2. That Moorehouse's conjecture is extremely interesting but, presumably, also extremely difficult becomes clear from his following major result.
X. Transversal designs and nets
780
14.7 Theorem. Let A be an affine plane ofsquare-free order n, and assume the validity of Conjecture 14.3 for n. Then n is a prime and A is the Desarguesian plane AG(2, n) . • Moorhouse (1991b) proved the validity of Conjecture 14.3 for certain nets admitting translations; combining this with Theorem 14.4 we have the following results.
14.8 Theorem. Let N be an (n, r; I)-net. Conjecture 14.3 is valid in each of the following cases:
= 3;
(14.8.a)
r
(14.8.b)
n = p is a prime, r = 4, and N belongs to a (p, 3)-difference matrix over Zp;
(14.8.c)
N is a translation net with abelian translation group. •
Note that (14.8.c) contains a case previously mentioned, namely that N consists of some parallel classes of AG(2, p).
14.9 Remark. Moorhouse (1993) considers the p-ranks of direct products of nets, cf.1.7.21, and shows that Conjecture 14.3 holds in this situation if it holds for the direct factors involved. Codes of nets were also studied by Dougherty (1993) who gave a reformulation of Conjecture 14.3. A related problem is the determination of the p-ranks of net graphs; for this and for more general results on the rank and structure of graphs, we refer to Peeters (1995a, b).
XI Asymptotic Existence Theory Spat kommt Ihr, doch Ihr kommt! Der weite Weg entschuldigt Euer Saumen.
(Schiller)
§ 1. Preliminaries
1.1 Introduction. The aim of this chapter is the proof of Wilson's main theorem: the necessary conditions (IX.2.9.c, d) for the existence of an SA. (2, K; u) are sufficient eventually, i.e., for all but finitely many v E N; Wilson (1975). The proof uses the following previous results. (a) Dirichlet's theorem on primes in arithmetic progressions; see, for instance, Serre (1973) or Ireland and Rosen (1990). (b) The infinity of the closed sets B(k, A), cf. Theorem VII.6.6 and Corollary Vll.6.7. (c) The sufficiency of the necessary existence conditions for large values of A, cf. Theorem Vrn.7.1 and Corollary Vrn.7.15. (d) The theorem of Chowla, Erdos and Straus (1960) on the existence of arbitrarily many mutually orthogonal Latin squares of order g for almost all values of g, cf. Theorem X.5.4. First we shall prove some special cases of the main theorem: (i) the existence of infinitely many v E B(k) in any residue class u+k(k-l)N, if u == 1 (mod k - 1) and u(u - 1) == 0 (mod k(k - 1)), using Theorem Vll.6.6, Corollary Vlll.7.15, and linear algebra; (ii) the eventual periodicity (see below) of closed subsets K = B(K) of N; (iii) the main theorem for A = 1.
1.2 Definition. Let M be a subset of No = N U {O} and mEN. An m-fibre of
781
782
Xl. Asymptotic existence theory
M is any non-empty subset (1.2.a)
Ma,m := {x EM: x =.a (modm)} = Mn(a+mNo)
An m-fibre is called complete if it contains almost all x E a (a + mN) \ Ma,/n is finite.
with a E No.
+ mN, i.e. if
A non-empty subset M of N is called eventually periodic with period m if all m-fibres of M are complete. Then M is infinite. If M is eventually periodic with period m then it is eventually periodic with period nm for each n E N.
1.3 Lemma. If M is eventually periodic with periods m and n, then it is eventually periodic with period d = gcd(m, n). Proof. There are positive integers a, f3 such that d = am-f3n.Foreacha EM, there is an element b E M with b =. a (mod mn) such that b + mx, b + nx E M for all x E No. If hEN, thenb+hd = b+ham -hf3n. Letha = rn+y with 0:::: y < n. Then b + hd = b+ ym + (rm - hf3)n = b + ym + (h~ - y~)n. If h > Y;;', then b + hd E M; i.e. the d-fibre Ma,d is complete. • 1.4 Corollary. The periods of an eventually periodic subset M consist of the multiples of a primitive period d, i.e. the gcd of all periods. •
1.5 Lemma. Let K s:::: N with K Lemma IX.2.9. Then (l.S.a)
a(B(K» = a(K),
(l.S.b)
f3(B(K» = f3(K).
Proof. Obviously a(B(K»
i= rtJ, {I},
and define a(K), f3(K) as in
Ia(K) and f3(B(K» I f3(K). By Lemma IX.2.9
v-I =. 0 (mod a(K», v(v - 1) =. 0 (mod f3(K»
for all v E B(K), hence a(K) I a(B(K» and f3(K) I f3(B(K» . • 1.6 Observations. (a) Obviously, f3(K) is always even, and a(K) divides f3(K). If we define (1.6.a)
y(K) := f3(K)/a(K),
then a(K) and y(K) are relatively prime.
§2. Steiner systems with v in given residue classes
783
Proof. Let d be a common divisor of a(K) and y(K). Then d . a(K) divides k(k - 1) for each k E K. Now a(K) divides each k - 1, hence both a(K) and d are relatively prime to each k E K. This is only possible if d . a(K) divides k - 1. But then d . a(K) I a(K), i.e. d = ±1. (b) Let a, f3 be positive integers such that 2 and a divide f3, and gcd(a, ~) = 1. Then there is a set K S; N such that a = a(K), f3 = f3(K), namely K {v EN: a I (v - 1) and f31 v(v - I)}.
=
Proof. Exercise. (c) Let
(1.6.b)
y(K) =
n
pe(p)
pEP
be the prime decomposition of y(K), and v sary condition (1.6.c)
v(v - 1)
== 1 (mod a(K». Then the neces-
== 0 (mod f3(K»
holds iff (1.6.d)
v
== 0 or 1 (mod pe(p»)
for all pEP.
Proof. (1.6.d) is equivalent to v(v - 1)=0 (mod y(K», i.e. to pe(p)lv or pe(p) I (v - 1) for all pEP . • 1.7 Lemma. Let K be finite, say K (1.7.a)
kJk2 .... kn
= {kJ, ... , kn }. Then
== 0 (mod y(K».
Proof. Let pEP be as in (1.6.b). Then pe(p) divides each ki (k i - 1) but not eachki -1 (i = 1, '" , n). Thus there is an i E {I, ... , n} with pe(p) I k i (k i -1) and Pf(ki - 1), as ki and ki - 1 are relatively prime. Hence pe(p) I ki, which implies the assertion 0.7.a) . •
§2. The Existence of Steiner Systems with v in Given Residue Classes 2.1 Introduction. Following Wilson (1975) and Brouwer (1979c), we shall prove that every admissible residue class mod k(k -1) contains some v E B(k).
784
XI. Asymptotic existence theory
The idea is to use Theorem vn.6.6 and Corollary Vrn.7.1S and thus to reduce the existence question for A = 1 to the question for large values of A which was answered in Corollary vrn.7.1S.
2.2 Theorem. If there exists an 8J,.(2, k; u), where A = q is a prime power, and also an 8(2, k; qd), and if, furthermore, q 2: u + 2 and d 2: (;), then there is a Steiner system S(2, k; uqd) . • The proof will take several steps.
2.3 Notation and Construction. Let U = Ni = {1, ... , u}, and let (U,A) be an 8q (2, k; u). Let V = GF(q)d be a q-vector-space with d 2: (;) and with coordinates
of each vector x E V. Thus there are (;) coordinates with double index ij, i < j, and d' = d - (;) remaining coordinates with single index i. For each pair {i, j} E (~) there are exactly q blocks A E A containing i and j. Let
(2.3.a)
N ij
:
{A : i, j I A E A} -+ GF(q)
be a bijection of the set of these q blocks onto GF(q). Now assume IVI = v = qd E B(k), as in the hypotheses of Theorem 2.2. In order to construct an 8(2, k; uv) on the point set U x V, we first construct an S(2, k; v) on each "level" {i} x V. Let ({i} x V, B i ) be this S(2, k; v). Let W be a generator of (GF(q)*, .), and let H C V be the hyperplane consisting of all vectors x with
(2.3.b)
Xl2
+ ... + Xu-l,u + Xl + ... + Xd' = O.
For each i E U let T; : x ~ Y be the linear mapping defined by
(2.3.c)
Yjl
={
X'l J i XjlW
if i = j or i = l, otherwise,
and
(2.3.d)
Yj
= XjW i
for j = 1, ... , d'.
§2. Steiner systems with v in given residue classes
785
Finally, for each A E A, let fA : A -+ V be a mapping which will be specified later. Now we construct blocks B(x, y, A) for all (x, y, A) E V x H x A by (2.3.e)
B(x, y, A) := {(i,
z) : i
E A and
z=
x
+ 1i(Y) + fA(i)}.
We have to show that for an appropriate choice of the mappings fA these blocks, together with the blocks B E Bi (i = 1, ... , u), form an S(2, k; uqd). First note that we have got the right number of blocks. For given i, j with i < j there are q2d pairs {(i, z), (j, z')}. On the other hand, there are exactly q blocks A E A throughi andj. Thus the number of blocks B(x, y, A) defined by (2.3.e) is also q . qd . qd-l = q2d. We have to choose the mappings fA such that any pair {(i, z), (j, z')} with i < j is covered by at least (hence exactly) one block (2.3.e). This is true if and only if each pair {( i, 0), (j, z' - z)} is covered, where 0 is the zero vector.
2.4 Problem. For every A E A, find a mapping fA : A -+ V such that, for E (~), the expression
each {i, j} (2.4.a)
takes all values z
E
V, where (y, A) runs through H x A.
2.5 Observations. In order to solve Problem 2.4, we study the system of equations (2.5.a) for a given vector z coordinatewise. Assume i < j. For a given find y.
Case 1. Let I
E
{I, ... , d'}. Put
(2.5.b) Then (2.5.a) yields (2.5.c) hence the Yl are uniquely determined.
z we have to
Xl. Asymptotic existence theory
786 Case 2. Let{i', j'} (2.S.d)
E
(~), i' < j', and (i, j}
fA(i)i'j' := 0
ifi' < j' and i
n Ii', j'} = ¢. Put
'I Ii', n·
Then (2.S.a) reads
and the Yi' j' are uniquely determined. Case 3. i = i' < j' =1= j. Now (2.S.a) reads Tj(Y)ij' - T;(Y)ij' + fA(j)ij' - fA(i)ij' = Zij', j Yij'(W - 1) = ZiP - fA(j)ij' + fA (i)ij"
Hence the Yij' are uniquely determined, regardless of the choice of fA. The same holds for i =1= i' < j' = j and the coordinates Yi'j' and the similar cases i < j = i' < j' and i' < j' = i < j. Case 4. i = i' < j' = j. (2.5.a) reads Tj(Y)ij - T;(Y)ij
(2.S.e)
+ fA(j)ij -
fA(i)ij = Zij, i.e.
fA (j)ij - fA (i)ij = Zij'
Now, using (2.3.a), define for i < j (2.S.f) and (2.S.g)
fA (i)ij := O.
Then (2.5.e) has a solution with a unique A E A and arbitrary Yij' Hence Yij can be chosen such that Y E H; see (2.3.b). This proves Theorem 2.2. • 2.6 Theorem. Let m, u, kEN and k > 2. Moreover, let u - 1 == 0 (mod k - 1),
(2.6.a) (2.6.b)
u(u - 1) == 0 (mod k(k -
1».
Then there are infinitely many W E B(k) with
(2.6.c)
W
== u (mod mk(k -
1».
Proof. Let A. == 1 (mod k(k - 1». If A. is sufficiently large, then there is an S),,(2, k; u), by Corollary VIII.7.IS. In particular, A. can be taken from an infinite set of primes q == 1 (mod mk(k In addition choose d:::: (~) and
1».
§3. The main theorem/or Steiner systems S(2, k; v)
787
q :::: u + 2 such that qd E B(k), which is possible by Wilson's Theorem VII.6.6. Then w := uqd == u (mod mk(k Now the assertion follows from Theorem2.2. •
1».
2.7 Example. There are infinitely many S(2, 10; v)'s with v 90), even though no explicit example seems to be known. 2.8 Corollary. Define Rk := {x EN: (k - l)x Select mEN arbitrarily, and assume (2.8.a)
r(r - 1)
+I
== 46 or 55 (mod
E B(k)}, as in IX.3.7.
== 0 (mod k).
Then there are infinitely many x E Rk with (2.8.b)
x
== r (mod mk).
Proof. Put u := (k - 1)r + 1. Then (2.8.a) is equivalent to (2.6.a, b). There are infinitely many W E B(k) satisfying (2.6.c). Set -1 E Rk • k-l
W
. x:= - Then (2.8.b) holds. •
§3. The Main Theorem for Steiner Systems S(2, k; v) 3.1 Introduction. In this section we shall prove that r E Rk for almost all r satisfying (2.8.a). This is equivalent to the main theorem for the case K = {k} and J... = 1. Recall that Rk is closed, i.e. (3. 1.a)
also, from Lemma IX.3.13, (3.l.b)
if r, s
E
Rk and r E TD(k), then rs E Rk.
3.2 Lemma. There is an arbitrarily large ro E N with ro, ro ro == 0 (mod k).
+I
E Rk and
Proof. By the Chowla-Erdos-Straus TheoremX.5.4, there is a TD[k; u] for all sufficiently large u. As B(k) is infinite (cf. Corollary VII.6.7), we may choose
Xl. Asymptotic existence theory
788
u from B(k). Starting with a TD[k; u - 1] and a TD[k; u], we obtain k(u - 1)
+ 1 E GD(k, u -
1) + 1 S; B({k, u}) = B(k),
ku E GD(k, u) S; B(k, u) = B(k).
Now set ro = k~':}) .•
3.3 Lemma. If r, r + t E Rk.
+ 1, s, t
E Rk with s :::: t and s E TD(r
+ I),
then
rs
Proof. From a TD[r + 1; s] remove s - t points of one point class and thus get an S(2, {s, t, r, r
+ I}; rs + t). Hence rs + t
E B(Rk) = Rk . •
In the following we shall use this lemma without explicit reference. 8 > 0 be given. Then Rk contains an infinite sequence r[, r2, ... such that ri+1 > ri and ri+1/ri < 1 + 8 for all i E N.
3.4 Lemma. Let
Proof. Let ro be as in Lemma 3.2. Take any tI, t2 E Rk with tl < t2 and another mE Rk withm :::: t2 andm E TD(r+ I). Nowputu := rOm+tI, v := rOm+t2. If m is large, then 1 <
~
< 1 + 8 and u, v E Rk n TD(k).
Now let n be the least integer such that (~)n :::: u. Every i EN can be uniquely written as i sn + t with s E No and 1 :5 t :5 n. We define
=
in particular rl = un-I, r2 = U n- 2V, ..• , rn = all ri are in Rk. If i = sn + t with t < n, then
V n,
rn+I = unv, ... By (3.l.b)
ri+I/ri = v/u < 1+8,
and if i
= sn + n, then
3.5 Lemma. There is a positive integer m and an increasing sequence SI, S2, S3, ... of elements of Rk such that 0 < Si+ I - Si < m for all i E N.
Proof. Let ro be as in Lemma 3.2. Select a sequence TI, T2, (3.5.a)
1 < ri+1/ri < (ro
+ l)/ro
for all i EN,
T3, ...
such that
§3. The main theoremfor Steiner systems S(2, k; v)
789
and, w.l.o.g., using Theorem X.5.4, (3.S.b)
ri
E
TD(ro
+ 1).
Hence, for all i E N and for all t E Rk U {O} with t ::; ri, we have (3.S.c) Now we take m := rorl and define the sequence SI, S2, S3, ... inductively as , follows. Put So := 0, SI := rOrl, S2 := rorl + rl' Assume that we have defined SI, ... ,Sn E Rk (n > 1) such that 0 < Si+l - Si < m for i = 1, ... , n - 1. Let I be the least integer such that (ro+ 1)rl > Sn (note I > 1). Then (ro+ 1)rl_l ::; Sn and, by (3.5.a), (3.S.d)
o<
Sn - rorl < Sn.
There is a unique j E {l, ... , n} such that Sj_1 ::; Sn - rorl < Sj. Hence 0 < Sj - (sn - rorl) < m. Now define
t := rnin{rl, Sj}
Sn+l:= rorl
and
Then 0 < Sn+1 - rorl < Sn+1 - Sn = rorl
+t -
+ t. Sn ::; rorl
+ Sj
- Sn ::; m . •
3.6 Lemma. Let hEN be a number such that y E TD(ro + 1) for all y > h, where ro is as in Lemma 3.2, and let x E R k . Then, for every n E No, there is an r(n) E Rk such that (3.6.a)
h < r(n)
+ iro + x
Rk for i = 0, ... , n.
E
Proof. For n = 0 select any S E Rk with S > max{x, h}. Then ros Define reO) := ros + x.
+x
E
Rk.
Now assume that r(n) exists such that (3.6.a) holds. By Lemma 3.2, there is an s' > r(n) + nro + x with s', s' + 1 E Rk. Then
ros' + r(n)
+ iro + x
E
Rk
for i
and
ro(s' + 1) + r(n) Define r(n
+ nro + x
+ 1) := ros' + r(n) . •
E
Rk.
= 0,1, ... , n
790
Xl. Asymptotic existence theory
3.7 Lemma. Let ro be as in Lemma 3.2 and x any element of Rk. Then Rk contains almost all y == x (mod ro), that is, all such y with finitely many exceptions. Proof. By Lemma 3.5, there is an mEN and an infinite sequence Sl, S2, S3, ... of elements of Rk such that 0 < Si+l - Si < m for all i E No Put n := rom and apply Lemma 3.6. In detail, put ti := r(n)
+ iro + x
(i = 0, ... ,n)
and choose j EN such that s := Sj 2:: r(n) agE N such that (3.7.a)
ros
+ ti
= rOsj
+ nro + x. By Lemma 3.3, there is
+ r(n) + iro + x
E Rk
for i E {O, 1, ... , n} and for all j ::: g. As rO(sj+l - Sj) < n, these numbers (3.7.a) cover almost all integers y == x (mod ro) . • 3.8 Theorem. The necessary conditions (3.8.a) (3.8.b)
v(v -
°
== (mod k -1), 1) == 0 (mod k(k - 1»
v-I
for the existence of an S(2, k; v) are sufficient for almost all v E N.
Proof. We have to show that Rk contains all sufficiently large numbers r satisfying the condition r(r - 1) == 0 mod k, cf. (2.8.a). Let ro be as in Lemma 3.2. Select any r EN such that r(r - 1) == 0 mod k. We have to show that r E Rk if r is sufficiently large. By Corollary 2.8 with m = ro/ k, there is an x E Rk with x == r (mod ro). By Lemma 3.7, the set Rk contains almost all numbers in the residue class of x mod roo But this is the required result. • §4. The Eventual Periodicity of Closed Sets In what follows, K always denotes a closed subset of N which contains at least one element k > 1. Note that all the following results are trivial if 2 E K, since then K = No
4.1 Theorem. Let K =1= (/), {I} be a closed subset ofN. Then K is eventually periodic with period f3 (K). • The proof needs several steps.
791
§4. The eventual periodicity of closed sets
4.2 Corollary. Let K I- 0, {1) be a closed subset ofN, furthermore k, u E K and u == 1 (mod k(k - 1». Then K is eventually periodic with period u - L Proof. This follows immediately from Definition 1.2, assuming Theorem 4.1.
4.3 Lemma. Corollary 4.2 implies Theorem 4. L Proof. Choose k E K arbitrarily. By Theorem 3.8, there are u, v E B(k) with u =mk(k - 1) + 1, v =nk(k - 1) + u and gcd(m, n) = 1, hence gcd(u - 1, v - 1) = k(k - 1). By hypothesis, Corollary 4.2 holds, i.e. K is eventually periodic with periods u-1 and v- L By Lemma 1.3, K is eventually periodic withperiodk(k-1) and, by Corollary 1.4, the primitive period of K divides f3(K). Another application of Lemma 1.3 shows that it suffices to prove Corollary 4.2. • Henceforth we fix k, u
E
K with u
==
1 (mod k(k - 1).
4.4 Lemma. Letk, f, u E K andu == 1 (modk(k-1». Then there are infinitely many v E K with v == f (mod u - 1).
Proof. By Theorem 3.8, t(u - 1)
+1E
B(k) S; K
for sufficiently large tEN.
If t is large, then there is a TD[f; t(u - 1)
v := f(t(u - 1) Hence v
== f
+ 1]; i.e.
+ 1) E GD(f, t(u -
1)
+ 1) S;
B(K) = K.
(mod u - 1). •
4.5 Lemma. Let u E K and u == 1 (mod k(k - 1». Furthermore, let f E K and f - 1, f E TD(u). Then there are GD[K, {f - 1, u - I}; v] 's with both v = uf - 1 and v = uf - u which in case u I- f contain exactly one point class of size f - L Proof. In the case v = uf -1, construct aTD[u; fl which may be considered as an S(2, {u, f}; uf), say D. Delete one point x and consider as new point classes the point sets B \ {x} where B is any block of D containing x. If v = uf - u construct a TD[u; f - 1] and adjoin a point CXl to its point classes. From the
XI. Asymptotic existence theory
792
resulting S(2, {u, f}; uf - u + 1), say D', delete one point x and take as new point classes the point sets B \ {x} as before . • We finish the proof of Theorem 4.1 by establishing the following result which is equivalent to Corollary 4.2.
4.6 Proposition. Let k, u, f E K be given with u n(u -1) + f E Kfor almost all n EN.
== 1 (mod k(k - 1». Then
Proof. (a) By Lemma 4.4, we may assume that f - 1, f E TD(u). Hence the GDD's of Lemma 4.5 exist. (b) We will show that for almost all mEN with m allt E to, 1, ... ,m} (4.6.a)
==
1 (mod (u - 1)) and for
m(f - l)u
+ f(U -
1) E GD(K, {f - 1, u - I});
m(f - l)u
+ f(U -
1)
hence (4.6.b)
+1E
B(K U {f, u}) = K.
First note that m == 1 (mod k(k - 1)). Hence m E B(k) s;:; K for large m, by Theorem 3.8. Also (4.6.c)
m(f - l)u
+ t(u -
1) + 1 ==
f
(mod u - 1).
We will now find a GD[K, {f -1, u -I}; m(f -1)u+t(u -1)] for sufficiently largem EN. (c) In order to achieve this we use the recursive construction of Theorem IX.3 .2. But we describe it independently in the language of auxiliary matrices. LetD and E be the GDD's of Lemma 4.5, with incidence matrices
Mo No and .........
Here Mo and No have rows each.
f - 1 rows, and the other matrices Mi, Ni have u - 1
§5. The main theoremjor}.. = 1
793
Now let m == 1 (mod u - 1) be sufficiently large. Then there is a TD[f + 1; m J, say T. Delete m - t points from the second point class of T and obtain a GD[{f, f + I}, {t,m}; fm + tJ, sayG. Each block B ofGwith block length f + 1 or f determines a column in the incidence matrix of G. Replace the entries I in this column by Mo, M" ... , Mf if B has f + I points (one of them in the second point class of G), and by No, N" ... , N f-I otherwise. Replace the zeros by appropriate zero matrices. The result is the desired GD[K, {f - 1, u - I}; v] with v = m(f - l)u + t(u - I). (d) If m is sufficiently large we may assume that m(f - l)u
+ m(u -
1) 2: (m
+u -
1)(f - l)u.
Thus the numbers m(f - l)u + t(u - 1) cover the residue class f - 1 mod (u - 1) with finitely many exceptions, as required. •
§S. The Main Theorem for ,\ = 1 It suffices to prove the following lemma, with notation as before. 5.1 Lemma. Let fEN with (5. 1. a)
(5.tb)
f - 1 == 0 (mod a(K», f(f - 1)
Then there is a v
==
== 0 (mod f3(K».
f(mod f3(K» with v E B(K) . •
5.2 Main Theorem. The necessary conditions (5.1.a) and (5.l.b)for the existence of an S(2, K; f) are sufficient for almost all fEN. Proof. This follows from Lemma 5.1 and Theorem 4.1, as B(K) is closed. •
5.3 Proof of Lemma 5.1. (a) There is a finite subsetL of K witha(L) = a(K) andf3(L) = f3(K). HencewemayassumethatK is finite, say K = {k" ... , kn}. As B(K) is eventually periodic with period f3 (K), there are elements Xi E B(K) with Xi == k i (mod f3(K» for i = 1, ... , n, and Xi+' E TD(x, ... Xi)
fori = 1, ... , n - 1.
§5. The main theoremfor A. = 1
795
sinceki == 1 (moda(K)) fork = 1, ... , n which shows thata(K) divides k-l. Hence, by Observation 1.6.(a),
(5.3.1)
v == f(mod (J(K)).
Now Theorem 3.8 ensures v E B(k) s;: K for almost all values of v with the above congruence properties. This proves Lemma 5.1 and hence the Main Theorem for A = 1. • As a consequence, we obtain the following result due to Wilson (1972c), cf. Remark IX.7.3.
5.4 Corollary. For every K
~
B(L) = B(K). Hence every K
N there is a finite subset L
s;:
K such that
s;: N has a unique finite basis.
Proof. Let H s;: K be a finite subset with a(H) = a(K) and (J(H) = (J(K). Then B(H) contains almost all v with v -1 == 0 (mod a(K)) and v(v -1) == 0 (mod (J(K)). Hence K \ B(H) s;: B(K) \ B(H) is finite. Put L := H U (K \ B(H))
s;:
K.
Then B(K) ;2 B(L) ;2 B(H) U (K \ B(H)) ;2 K. As B(L) is closed, B(L) B(K) . •
=
# 0, {I} is eventually periodic with period if and only if d == 0 (mod a(K)) and d is odd.
5.5 Corollary. A closed set K d := {J(K)/2
Proof. (a) If K has period d, then there are x, x x - 1, x
+ dE
K, hence
+ d - 1 == 0 (mod a(K))
which implies d == 0 (mod a(K)). Furthermore (J(K) divides x 2 - x and (x + d)(x + d - 1) which impli~s d 2 + 2dx - d == 0 (mod (J(K)); this holds iff d is odd. (b) Let d == 0 (mod a(K)) and d == 1 (mod 2). By definition, x == 1 (mod a(K)) and x(x - 1) == 0 (mod a(K)) for all x E K. If y == x (mod (J(K)) is sufficiently large, then y E B(K), by Theorem 4.1. If y:=x +d (mod (J(K)) is sufficiently large, then y := 1 (mod a(K)) and y(y - 1):= x(x - 1)
and, since d
+ 2dx + d(d -
= (J(K)/2, by Theorem 5.2 also y
1) == x(x
E B(K) . •
1) (mod (J(K)),
§5. The main theorem for A. = 1
795
sinceki == 1 (moda(K» fork = 1, ... , n which shows thata(K) divides k-l. Hence, by Observation 1.6.(a), (5.3.1)
v == f(mod fJ(K)).
Now Theorem 3.8 ensures v E B(k) s;: K for almost all values of v with the above congruence properties. This proves Lemma 5.1 and hence the Main Theorem for A = 1. • As a consequence, we obtain the following result due to Wilson (1972c), cf. Remark IX.7.3. 5.4 Corollary. For every K
s;: N there is a finite subset L s;: s;: N has a unique finite basis.
K such that
B(L) = B(K). Hence every K
Proof. Let H s;: K be a finite subset with a(H) = a(K) and fJ(H) = fJ(K). Then B(H) contains almost all v with v -1 == 0 (mod a(K» and v(v -1) == 0 (mod fJ(K». Hence K \ B(H) s;: B(K) \ B(H) is finite. Put L := H U (K \ B(H»
s;:
K.
Then B(K) ;2 B(L) ;2 B(H) U (K \ B(H» ;2 K. As B(L) is closed, B(L) = B(K) . •
# 0, {I} is eventually periodic with period if and only if d == 0 (mod a(K)) and d is odd.
5.5 Corollary. A closed set K d := fJ(K)/2
Proof. (a) If K has period d, then there are x, x
x-I, x
+ dE
K, hence
+ d - 1 == 0 (mod a(K))
==
0 (mod a(K». Furthermore fJ(K) divides x 2 - x and (x + d)(x + d - 1) which implies d 2 + 2dx - d == 0 (mod fJ(K)); this holds iff d is odd. which implies d
(b) Let d == 0 (mod a(K)) and d == 1 (mod 2). By definition, x == 1 (mod a(K» and x(x - 1) == 0 (mod a(K» for all x E K. If y == x (mod fJ(K» is sufficiently large, then y E B(K), by Theorem 4.1. If y ==x +d (mod fJ(K)) is sufficiently large, then y == 1 (mod a(K)) and y(y - 1) == x(x - 1)
and, since d
+ 2dx + d(d -
= fJ(K)/2, by Theorem 5.2 also y
1) == x(x
E B(K) . •
1) (mod fJ(K)),
Xl. Asymptotic existence theory
796
5.6 Remark. An interesting application of Theorem 5.2 to partition lattices was given by Ganter and Gronau (1991). §6. The Main Theorem for .A> 1 In this section, we will finish the proof of the main result due to Wilson (1975).
6.1 Main Theorem (Wilson 1975). The necessary existence conditions for a B[K, A; v], i.e. (6. La)
(6.1.b)
A(V - 1) == 0 (mod a(K», AV(V - 1) == 0 (mod f3(K»
are sufficient for almost all v E N. • The proof requires some preparation.
6.2 Remark. If A satisfies (6. La, b), then for A' = mA+n.B(K) withm, n E Z obviously (6. La) and (6.1.b) with A' instead of A hold. If, in particular, A' is the greatest common divisor of Aand f3 (K), then the necessary existence conditions for A and A' are equivalent. Hence, in order to prove the main theorem it suffices to consider the special case that A divides f3(K). Henceforth we always make this assumption.
6.3 Lemma. Define M := B(k, A), where A divides k(k - 1), and (6.3.a)
k(k - l)/A .Bo = .Bo(k) := { 2k(k - 1)/A
if this number is even, otherwise.
Then (6.3.b)
.B(M) = .Bo.
Proof. As AV(V - 1)
== 0 (mod k(k -
1» for v
V(V - 1)
== 0 (mod k(k -
l)/A).
As {3(M) is even, f30 I f3(M), say (6.3.c)
f3(M) = m.Bo.
E
M,
§6. The main theoremfor J... > 1
797
By VII.S.13, all sufficiently large prime powers q with (6.3.d)
q
== 1(mod k(k -
1)/J...)
are in M. In particular, let q E M be a large prime of the form
q = fJo(fJo
+ l)x -
fJo
+1
(x EN).
The existence of q is assured by Dirichlet's theorem. Then q(q - 1)
== 0 (mod fJ(M».
If q > fJ(M), thenq and fJ(M) are relatively prime, hence fJ(M) divides q -1 = fJo(fJo + l)x - fJo and m divides (fJo + l)x - 1. Therefore, (6.3.e)
gcd(fJo + 1, m) = gcd(fJo
+ 1, fJ(M»
= 1.
Again, by Corollary VII.8.13 and Dirichlet's theorem, there are large primes p=fJ(M)t+fJo+lEM
withtEN,
hence pep - 1)
== 0 (mod fJ(M».
If p > fJ(M), then p is relatively prime to fJ(M). This implies p - 1 = fJ(M)t
+ fJo == 0 (mod fJ(M».
Hence fJ(M) divides fJo and, by (a), fJ(M) = fJo . •
6.4 Lemma. Let J... divide fJ(K), and let M := B(K, J...). Put (6.4.a)
fJ, :=
fJ(K)/J... { 2fJ(K)/)"
if this number is even, otherwise.
Then
(6.4.b)
fJ(M) = fJ,·
Proof. If v E M, then v(v - 1) divides fJ(M).
==
0 (mod fJ(K)/J...). As fJ(M) is even, fJ,
Xl. Asymptotic existence theory
798
For each k E K obviously B(k, A) s:; M = B(K, A). Hence (3(M) divides (3(B(k, A». Thus, using Lemma 6.3, f3(M) divides gcd{f3o(k) : k E K}. Let
e(k)
={
I
if k(k - l)/A is even,
2
otherwise.
Then gcd {f3o(k) : k E K} = gcd { e(k)
Case 1. e(k) = 1 for all k hence f3(M) divides f31'
E
k(k - 1) } A :k E K .
K. Then gcd {f3o(k) : k
E
K} = f3<{) = f31,
Case 2. e(k) = 2 foratleastonek E K. Then(3(K)/Ais odd and f31 = 2f3(K)/A, hence gcd {f3o(k) : k E K}, divides gcd {2k(~-1) : k E K}=2§J[l =f31' In both cases f31
= f3(M) . •
6.5 Theorem. B(K, A) is eventually periodic with period f3(K)/A. Proof. If f3(K)/A is even, then f3(K)/A = f3(M) with M = B(K, A), and the assertion follows from Theorem 4.1. Now let f3(K)/A be odd. It will suffice to exhibit an element x E B(K, A) with
x
==
I +d (mod2d),
whered := f3<{) = f3(~). For then x == x+d == 1 (moda(M», asa(M) divides 2d, hence d == 0 (mod a(M». Then the assertion follows by Corollary 5.5. Now let us find x. As f3(K)/). is odd, there is a k odd. There are infinitely many n E N with (6.5.a)
2/l - I
== 0 (modk(k -
E
K such that k(k - 1)/)" is
1)/),,),
which, by Corollary VII.8.13, implies 2/l E B(k,)..) s:; M if n is sufficiently large. Obviously 2" - I == 0 (mod f3(K)/)") and 2" - 1 'f:. 0 (mod 2f3(K)/A). Thus we found x := 2" == d + 1 (mod 2d). • 6.6 Lemma. Theorem 5.2 (for).. = 1) implies Theorem 6.1 (for A> 1).
799
§6. The main theorem/or A> 1
Proof. Let /
E
N be given such that
(6.6.a)
A(f - 1) == 0 (mod a(K»,
(6.6.b)
J..f(f - 1) == 0 (mod {3(K».
We try to find a u (6.6.c)
N with
== / (mod {3(K)/A), u == 1 (mod a(K», 1) == / (mod {3(K». u
(6.6.d) (6.6.e)
E
u(u -
If we succeed, then there is a w == u (mod {3(K» with W E B(K) ~ B(K, A) = M, by Theorem 5.2. As also W == / (mod {3(K)/A), Theorem 6.5 implies that almost all v == / (mod {3(M» are in 8(K, A). Thus Theorem 6.1 will be proved if we can find u with properties (6.6.c, d, e). We proceed similarly to 5.3. Let (6.6.f) PEP
(6.6.g)
y(K) _ gcd(J.., y(K» -
n
pe(p)
pEQ
be the prime decompositions of y(K) and y(K)/gcd(A, y(K». Of course Q ~ P and e(p) :::: d(p) for p E Q. Furthermore (6.6.h)
/ == cp
(mad pe(p))
For PEP \ Q define cp
E
with cp E {O, 1} for all
{O, I} arbitrarily.
By the Chinese remainder theorem there is a u EN with (6.6.i) (6.6.j)
== 1 (mod a(K», u == cp (mod pd(p)) u
Then u(u - 1) (6.6.k)
== 0 (mod pd(p)
for pEP. for all PEP; i.e.
u(u - 1)
== 0 (mod y(K».
u(u - 1)
== 0 (mod a(K»,
Since
(6.6.1)
p E
Q.
XI. Asymptotic existence theory
800
it follows, by (1.6.il), that (6.6.m)
u(u - 1)
== 0 (mod {3(K».
By (6.6.a, i, h, j), A(U - f)
(6.6.n)
).(u - f)
== 0 both (mod y(K» and (mod a(K»; i.e. == 0 (mod {3(K».
Thus we have found u and Wilson's main theorem is proved. •
6.7 Remark. By Wilson's theorems, the necessary existence conditions for designs S,- (2, K; v) are asymptotically sufficient. However, the proofs presented by Wilson and in this book do not give an explicit constant vo(K, A) such that an S(2, K, v) exists for ill v :::: Vo (K, ).) satisfying the necessary conditions. In fact, providing such bounds is far from trivial and actually was considered hopeless in the first edition of this book. Recently, Chang (1995, 1996a, b) provided such a bound in the case of 2-designs by proving the following major result. No corresponding result has yet been obtained for designs S,- (2, K; v) in general. 6.8 Theorem. Let). and k :::: 3 be positive integers. Then the necessary arithmetic conditions for the existence of a design S,- (2, k; v), that is (6.8.a)
).(v - 1)
== 0 (mod k -
1)
and
AV(V - 1)
== 0 (mod k(k -1),
are sufficient whenever
(6.8.b) Wilson (1975) also proved the following asymptotic result concerning the number of isomorphism classes of 2-designs.
6.9 Theorem. Let k :::: 3 and J... be positive integers. Then there exist constants c(k, J...) > 1 andvo(k, J...) such that the number ofisomorphism classes ofdesigns v2 S,- (2, k; v) is at least c for all admissible v :::: Vo. • 6.10 Remark. Colboum and Rodl (1989) proved that Wilson's main theorem on the asymptotic existence of PED's remains true if one also prescribes the fraction Pi of pairs of points contained in blocks of given size ki E K; this holds both in an approximate sense, for Pi ± e, and also exactly which then of course gives rise to a further necessary condition, namely p;v(v - 1) == 0 (mod k i (k; - 1». Their proof uses the main theorem (several times) and the Chowla-Erdos-Straus Theorem X.5.4.
§7. The existence of resolvable block designs
SOl
§7. An Existence Theorem for Resolvable Block Designs In this section we shall present the theorem of Ray-Chaudhuri and Wilson (1973) that the necessary condition v
== k
(modk(k -1)
for the existence of a resolvable S(2, k; v) is sufficient for almost all v E N. Because of Theorems IX.6.5 and IX.7 .11, we may assume that k > 4.
7.1 Lemma. Let v
E RB(k)
n TD(k + 1). Then kv
E RB(k).
Proof. On each point class of a resolvable TD[k; v] construct a resolvable S(2, k; v). It is easily checked that the resulting S(2, k; kv) is resolvable . •
7.2 Lemma. Let q
(7.2.a)
== 1 (mod k(k
q = k(k - I)s
- 1» be a prime power, say
+1
with sEN. If q is sufficiently large, then
(7.2.b)
kq E RB(k) . •
7.3 Corollary. The closed sets RB(k) and R;; (cj. IX.6.2 and IX.6.4) are infinite. • 7.4 Proof of Lemma 7.2. First let us recall some notation from VII.S.I and
VII.S.l. Let w be a generator of the group (GF(q)*, .) and let H m =: H be the subgroup of index m = (~), generated by w m • Note that -1 E H. A (k + 1)choice Cis a mapping of the ordered pairs (i, j)ofintegerswithO :::: i < j:::: k into the set of cosets of H. Now let C have the following two properties: (a) among the CCi, j) with 1 :::: i < j :::: k each coset wI H (0 :::: I < In) occurs exactly once. (b) the k cosets CCO, j) with j > 0 are distinct. Assume that q is sufficiently large. By Theorem VII.S.2, there is a (k + I)-tuple (ao, at, ... , ak) E GF(q )k+l such that (7A.a)
aj -
ai E
C(i, j)
for i < j.
802
Xl. Asymptotic existence theory
This condition also holds for j < i, as -1 E H. Now put (7.4.b)
bj := aj - ao
for j E {1, 2, ... , k}.
Then (7.4.c)
bj
-
bi
E
CU, j)
for 1 ::s i < j
::s k,
and the b i (i = 1, ... , k) are in distinct cosets of H. Now the s base blocks
form a (q, k, I)-difference family. This was proved in Lemma VII.6.3. By construction, these s base blocks are mutually disjoint, and the assertion follows with Theorem VIII.2.2. •
7.5 Lemma. With notation as in Lemma IX.2.9, (7.5.a) ifk is even,
(7.5.b)
ifk is odd.
Proof. A necessary condition for each r
r
E
R;; is
== 1 (modk).
Hence k divides both a and {3, and if k is odd, 2k divides {3. Let (7.5.c)
a = xk,
{3
= yk
(x, YEN).
By Lemma 7.2 and Dirichlet's theorem, there is an sEN such that
By Lemma 7.1, also k 3 (k - l)s
+ k 2 E RB(k)
Hence a = xk divides gcd(k 2 s, k 3 s
(if s is large), i.e. k 3 s
+ k) =
Again there are infinitely many primes q
= k(k -
l)yt
+1
(t EN)
+k + 1 E
k, which implies x = 1.
R;"
§7. The existence of resolvable block designs
such that
kD/ = k
2
yt+ 1 E RZ. By Lemma 7.1, also
803
k:~~1 = k 3 yt + k + 1 E RZ.
Hence f3 = yk divides (k 3 yt + k + 1)(k 3 yt + k), which implies that y divides k + 1. Next we show that no odd prime p divides y. Otherwise, as p divides k + 1, p is relatively prime to k(k - 1), and by the Fermat-Euler theorem pn
== 1 (mod k(k - 1»
for all integers n which are divisible by rp(k2 - k). If n is sufficiently large, then Lemma 7.2 implies kpn E RE(k). Put pI! -1 t := -"-k(-k---l-)
Then k 2 t + 1 E RZ, and f3 divides (k 2 t + 1)k2 t. As p does not divide k nor t, p must divide k 2 t + 1 = p" + kt, a contradiction. This proves (7.5.b) in the case that k is even, as y divides the odd number k + 1. Assume now that k is odd. The integers 4k(k -1) and 2k(k -1) + 1 are coprime, so by Dirichlet's theorem there exist arbitrarily large primes q
= 4k(k -
l)t
+ 2k(k -
1)
+1
+ 2k2 + 1 E
RZ.
(t EN).
By Lemma 7.2
kq -1 - - = 4k 2t k-l
Thus f3, and hence y, divides [k 2 (4t + 2) + llk 2 (4t + 2). So y is not divisible by 4. Combining the previous results we obtain y = 2. •
7.6 Theorem. RE(k) contains almost all v
E
N with v
== k
(mod k(k - 1».
Proof. Use the necessary condition (7.6.a)
RZ S; kNo
+ 1,
Lemma 7.5, Theorem 4.1 (if k is even), and Corollary 5.5 (if k is odd). Thus RZ contains almost all kn + 1 (n E N) which implies the theorem. • While we only considered the asymptotic existence of resolvable designs with 1, the analogous result for arbitrary J.... also holds by a theorem of Lu (1984a); see Lee and Furino (1995) for an English version of Lu's paper where the presentation has been somewhat adapted "to be more consistent with contemporary J.... =
Xl. Asymptotic existence theory
804
techniques and notation". Thus the necessary conditions for the existence of resolvable designs are asymptotically sufficient.
7.7 Theorem. Let k be a positive integer. The necessary arithmetic conditions for the existence of an RS)..(2, k; v), that is
(7.7.a)
v
== 0 (modk)
and
A(V - 1)
== 0 (modk -1),
are sufficient whenever v is sufficiently large. •
We conclude this section with some related results. Nearly Kirkman systems were defined in Remark IX.12.8. Here one has the following asymptotic result due to Shen and Chu (1994).
7.8 Theorem. Let k be a positive integer. The necessary condition v == 1 (mod k(k - 1» for the existence of an NRS(2, k; v) is sufficient whenever v is sufficiently large.
•
Near-resolvable designs were defined in Remark IX.12.1 O. Here the necessary existence condition is also known to be asymptotically sufficient for all values of k, see Furino (1995). We also refer to the monograph by Furino, Miao and Yin (1996) and Section 1.6.10 of Colboum and Dinitz (1996a) for the following more precise result.
7.9 Theorem. The necessary condition v == 1 (mod k) for the existence of an NRB(v, k) is asymptotically sufficient, that is,for v:::: n(k) . • The following table gives the known bounds on n(k), where no(k) denotes the number of undecided cases below the bound n (k).
k n(k) no(k) k n(k) no(k)
6 151 2 17 464373 2647
7 610 14 18 103609 463
8 560 14 22 718057 2643
9 18145 142 24 58105 290
10 9701 100 26 1238849 4530
12 9061 53 28 121073 651
15 126331 1116 30 490141 1197
16 10033 158
805
§8. Some results for t 2: 3
§8. Some Results for t ;e: 3 Strong asymptotic results for the case t 2: 3 have been obtained in a series of papers by John L. Blanchard. We shall mention (but not prove) the most important of his results. The proof techniques employed are reminiscent of those of Wilson discussed in this chapter, but even more involved. We begin with an interesting existence result for Steiner systems given by Blanchard (1995b). 8.1 Theorem. Let q be a prime power. For every v, the necessary arithmetic conditionsfor the existence of an S(3, q + 1; vqn + l)for some n 2: 0 are (S.l.a)
v-I
== 0 (modq
-1)
and
v(v 2 -1)
== 0 (mod q 2 -1).
These conditions are sufficient whenever n is sufficiently large. •
Blanchard (1995a) obtained a similar result for orthogonal arrays of index 1 and arbitrary strength, cf. Definition VIllA. 1. 8.2 Theorem. Let q be a prime power; and let t and k be positive integers with 2::: t ::: k. For every positive integer s, there exists an OA(t, k, sqe) whenever e is sufficiently large. • Ray-Chaudhuri and Singhi (1988a, 1994) proved the existence of OA).(t, k, s) for all sufficiently large A. An analogous result for perpendicular arrays was given by Blanchard (1993a). A considerably stronger type of result is due to Blanchard (I993b, 1994, 1995d) who proved the following analogue of the Chowla-Erdos-Straus theorem (Theorem X.5A) for arbitrary strength t 2: 3. 8.3 Theorem. Let t and k be positive integers with 2 ::: t ::: k. Then there exists an OA(t, k; v) whenever v is sufficiently large. • The proof of Theorem 8.3 is rather involved; among other ingredients, it requires a generalisation of the approach given in §2 and an application of the sieves studied in number theory to construct certain representations of large integers, see Blanchard (1995d).
XII Characterisations of Classical Desig ns Denn die Sch5nheit, mein Phaidros, nur sie ist liebenswiirdig und sichtbar zugleich ...
(Thomas Mann)l
rIn this chapter we shall return to the starting point of our discussion by characte 1.2.12 ns Definitio ising the classical designs PGn - 1(n, q) and AG,,_1 (n, q), see ns and 1.2.15, partly by combinatorial properties, partly by transitivity conditio risacharacte on e on their automorphism groups. There is an 'extensive literatur tion problems, and we shall also report on some related results. § 1. Projective and Affine Spaces as Linear Spaces
B, E) 1.1 Definition. A projective incidence space is an incidence structure (V, with the following properties. (l.l.a) (l.1.b) (l.1.c) (l.l.d)
Any two points p, q are on exactly one line (= block) pq. If p, q, r, s are four distinct points and if pq and r s intersect, then pr and qs intersect. Every line has at least three points. There are two disjoint lines.
Young. The four properties (l.l.a, b, c, d) are called the axioms of Veblen and e 1.2 Theorem. Any projective incidence space is isomorphic to a projectiv nonbe may space PG 1 (n, F), where n > 2 is a cardinal and F is afield which e commutative. Any finite projective incidence space is isomorphic to a projectiv • space PGl (n, q), where n > 2 is an integer and q is a prime power. I
/LoLpav, werr' Confer Plato's "Phaidros": vvv OE K(O.).O, /LOVOV ravrT/v EerXE EK<pavEe rrarov ElVat Kat Epaer /Llclnarov .
806
§ 1. Projective and affine spaces as linear spaces
807
This is a classical theorem from projective geometry and therefore not proved in this book. See e.g. Veblen and Young (1916) or Lenz (1965).
1.3 Definition. An affine incidence space is an incidence structure (V, B, E) with the following properties. (1.3.a)
Any two points p, q are in a unique line (= block) pq.
(1.3.b)
(V, B, E) is resolvable, i.e. the lines are partitioned into parallel classes (such that distinct parallel lines are disjoint), and each parallel class covers all the points.
(1.3.c)
If A, B, C are lines, A II B oF A, if a E An C, bE B n C, and c E C are distinct points, and if x E A is a point, then there is a point d = B n ex.
(1.3.d)
If a, b, c are points, not on a line, then there is a point d such that ab II cd and ac II bd.
(1.3.e)
Any line has at least two points.
(1.3.f)
There are two disjoint lines which are not parallel.
1.4 Observations. (a) If one line of an affine incidence space has at least three points, then each line has. Proof. Let Land G be two lines, let a, b, c be points of L, and let G contain c and another point d. Let B the parallel line to ad through b. By (1.3.c) B intersects G in a third point x oF c, d. If H is a line with L n H = 0, then there is a line G intersecting both Land H. Hence by the above reasoning G and H have at least three points. (b) Any two lines have the same (possibly infinite) number of points.
Proof. Exercise. (c) If one (and hence each) line has at least three points, then (1.3.d) follows from the other properties in Definition 1.3.
Proof. Let a, b, c be three points, not on a line. Let A, C be the lines with a E A II be and c E ell ab, and let x oF b, c be a point of bc. By (1.3.c) ax intersects C in a point y oF c, and again by (1.3.c) yc intersects A in the desired pointd . • 1.5 Theorem. Any affine incidence space D is isomorphic to an affine space AG 1(n, F), where n > 2 is a cardinal and F is a (possibly non-commutative)
808
XII. Characterisations of classical designs
field. If D is finite, then D is isomorphic to AGI (n, q), where n > 2 is an integer and q is a prime power. • This is a well-known theorem in the foundations of affine geometry which is not proved in this book. See e.g. Lenz (1959) or Beutelspacher (1983).
1.6 Remarks. (a) The proof of the finite cases of Theorems 1.2 and 1.5 needs the theorem ofWedderbum that any finite field is commutative and basic knowledge on Galois fields. (b) It is convenient to prove Theorem 1.2 via Theorem 1.5 or a similar result which uses other axioms for affine spaces, e.g. incidence axioms using points, lines, and planes. An essential step is the proof of Desargues' theorem and of the existence of translations and dilatations. (c) It is quite easy to see that the finite projective and affine incidence spaces are Steiner systems SI (2, k; v). It is also easy to verify that PG 1(n, q) is a projective incidence space and that AG 1(n, q) is an affine incidence space. Now we proceed to characterisations of finite projective and affine spaces with the hyperplanes as blocks.
§2. Characterisations of Projective Spaces 2.1 Introduction. We begin by studying the designs obtained from projective planes and spaces. We recall that the projective planes of order n coincide with the designs S(2, n + 1; n 2 + n + 1); no purely combinatorial characterisation of the Desarguesian planes among these designs is known except for configurational conditions such as the Desargues configuration of Example I.3.6 or the Fano condition that any triangle generates a projective plane of order 2, see Gleason (1956). In contrast, a celebrated result of Dembowski and Wagner (1960) provides a combinatorial characterisation of finite projective spaces PGn-1 (n, q) with n > 2; see Theorem 2.10 below. We shall require some preliminary notions; but let us first observe that PGn - 1 (n, q) is not characterised by its parameters. As mentioned by Dembowski (1968), the following result is due to Bill Kantor. 2.2 Theorem. Let q be a prime power and n > 2 an integer. Then there exist . symmetric 2-designs with the same parameters as PGn - 1 (n, q) which are not isomorphic to PGn- 1 (n, q).
809
§2. Characterisations ofprojective spaces
= (V, B, E) be the symmetric design PGn-1 (n, q), and U a fixed block. Put A := {H n U : H E B \ {Un. Then (U,A, E) ';:!; PGn-z(n - 1, q). Letrr be a permutation ofA. We define a new incidence structureE = (V, C, E) by
Proof. LetD
(2.2.a)
e:= {U} U {(H \ U) U rr(U n H) : H
E B \ {Un.
Obviously ICI = IBI for any C E e, B E B. Now we show blocks of e intersect in the same number of points. If C E C = (H \ U) Urr(U n H), then IU n CI = Irr(U n H)I = C, DEB \ {U}, say C = (H \ U) U rr(U n H), D = (K \ U) with K, HE B \ {U}, then
that any two B \ {U}, say IU n HI. If U rr(U n K)
+ Irr(U n H) n rr(U n K)I UI + IU n H n KI = IK n HI.
Ie n DI = I(K n H) \ UI = I(K
n H) \
By Corollary II.3.3, E is a 2-design with the same parameters as D. Choose rr in such a way that the images of the qn-Z + ... + q + 1 2:: n blocks A E A containing a given point U E U have empty intersection. Let L be a line of D which intersects U in {u}, and choose two points a, bEL \ {u}. The blocks in E through a and b have exactly q points in common, namely L \ {u}, whereas in D the blocks through any two points have exactly q + I points in common. Hence D and E cannot be isomorphic. • Thus we will indeed require some new combinatorial concepts to characterise the designs PGn- 1(n, q), and these are natural generalisations of the concepts of line and plane. 2.3 Definitions. An incidence structure D = (V, B, l) is called cohesive if any two points are joined by at least one block. In this case, given two distinct points p, q, the line pq is defined as the intersection of all blocks containing p and q; formally (2.3.a)
pq:=
n
(B) = «p, q».
p,qEBEB
If p, q Est, then pq S; st. In general equality does not hold as easy examples show. Three points p, q, r are collinear if they are contained in a line. Otherwise {p, q, r} is called a triangle and (p, q, r) an ordered triangle. Triangles exist if there are blocks containing two points but not all the points.
810
XII. Characterisations of classical designs
q, r}. If A plane pqr is the intersection of all blocks containing a triangle {p, V. = pqr then r}, q, {p, ;2 there is no block B triangle A cohesive incidence structure with triangles is called smooth if every 0 is not = p that is contained in the same number p of common blocks. Note excluded. The set of lines will be denoted by L and the set of planes by P.
that 2.4 Lemma . Let D = (V, B, 1) be a cohesive incidence structure such propg followin the has D Then any two points are joined by exactly)... blocks. erties. (i) Any two points are contained in a unique line. (ii) Every line is contained in exactly)... blocks. (iii) For every line L ELand every block B I(B) nLI:::: 1.
E
B either L ~ (B) or
that Proof. Let p, q be any two distinct points of D, then p, q E pq. Assume blocks the)... of each in are they Then St. line a in p and q are also contained t s; pq, through sand t. As there are no other blocks through p and q we have s .• tely immedia follow (iii) and (ii) s assertion and hence st = pq. Now the
ng 2.5 Lemma . Let D = (V, B, 1) be a smooth incidence structure containi the Then blocks. )... precisely by joined are triangles such that any two points following assertions hold. (i) Any triangle is contained in a unique plane. (ii) Each plane is contained in exactly p blocks. line, or (iii) For each PEP and each B E B either P S; (B), or P n (B) is a Ipn(B) I::::l. containing (iv) For every line L and every point p ¢ L, there is a unique plane p and L. (v) p < ).... The p Proof. Let {s, t, u} be a triangle in the plane pqr. Then stu S; pqr. s, t, through blocks other no are there As u. t, s, blocks through p, q, r contain As (i). proves This stu. ~ pqr hence and and u, it follows that p, q, r E stu to left is s assertion other the of proof easy The there are triangles, (v) follows. the reader. •
SI1
§2. Characterisations of projective spaces
2.6 Lemma. Let D be a smooth 2-design SA (2, k; v). Then any line L of D has exactly m points, where (2.6.a)
A.k - pv m=---
A-P
Proof. Let L be any line and choose two points p, q E L. Counting all flags (x, B) with x i= p, q E B (and thus L s;; (B)) in two ways gives the equation
(ILl - 2)A + (v - ILI)p = A(k - 2) which implies the assertion. •
2.7 Lemma. Let D = (V, B, l) be a smooth 2-design with line set L, and let p E V be a point. Define an incidence structure D' (p) = (L p' B p' l) by putting Lp:= {L EL:p E L},
Bp:= {B EB:pIB},
and LI B iff L C (B). ThenD'(p) is a 2-design with parameters (2.7.a)
v-I v'=--, m-1
k-l k'=--, m -1
A'=p,
r' =A,
b' = r,
with m = ILl as in Lemma 2.6.
Proof. Exercise. • 2.8 Lemma. With notation as in Lemma 2.7, (2.S.a)
P::::
with equality
A(A - 1) r- I '
if and only if D' (p) is symmetric.
Proof. The assertion is trivial for A = 1. Now let A > 1. As there are triangles inD we have v > k, v' > k', hence Fisher's inequality r' ::: k' holds, cf. Corollary 1.8.6. Therefore the following mutually equivalent inequalities hold. k' (r' - A') :::: r' (r' - A'), r t2
e - rt2 + r'A' -
r' (r' (k' - 1))
k'A' ::: r'2k' - r'k',
+ r'A' -
k'A' ::: r'k' (r' - 1),
XII. Characterisations of classical designs
812
r'A' (v' - 1) + r'A' - k'A' ~ k'r' (r' - 1), A'k'(b' - 1) ~k'r'(r' - 1), , r'(r' - 1) A(A - 1) A=P< = . - b'-1 r-l This is inequality (2.8.a). Equality holds if and only if r' ll.3.3, this is true if and only if D' (p) is symmetric. •
= k'. By Corollary
2.9 Remarks. (a) Cameron (1974) calls a smooth 2-design "locally symmetric" if D' (p) is symmetric (for one and hence for each point p, by Lemma 2.8). We will consider such designs in §4 below. Note that the designs PGn-l(n, q) and AGn-l (n, q) are locally symmetric; in fact, here the designs D'(p) are isomorphic to PGn-z(n - 1, q). (b) Dembowski (1968) attributes Lemma 2.8 to Ryshpan. Using Lemma 2.8, Dembowski noted the equivalence of conditions 2.1 0 (i) and (v) below. The equivalence of the conditions (i) through (iv) is essentially due to Dembowski and Wagner (1960) who proved this result for the class of symmetric 2-designs. Later Kantor (1969d) remarked that the hypothesis could be somewhat weakened.
2.10 Theorem (The Dembowski-Wagner Theorem). Let D = (V, B, l) be a 2-design SA (2, k; v) with A > 1 and v > k + 1. Then the following conditions are equivalent. (i) D
~
PGn-l (n, q) for some prime power q and some integer n > 2.
(U) Every line meets every block.
(Ui) Every line has exactly ~=~ points. (iv) Every plane is contained in exactly A:~[ > 0 blocks.
(v) D is smooth and symmetric.
Proof. (a) We first note that PGn _ 1(n, q) satisfies conditions (ii) and (v). Next we shall show that conditions (ii), (iii), and (iv) are equivalent. Let L be any line of D. Then L is contained in exactly Ablocks and therefore meets precisely these and ILI(r - A) further blocks. Thus L meets every block iff ).,+ ILI(r -A) = b. This proves the equivalence of conditions (ii) and (iii). (b) We show that (iv) implies (iii). By (2.6.a) and (iv),
m=
Ak - pv Ak(v - 1) - (Ak - r)v -Ak + rv = =-c"--""'--:--A- P A(V - 1) - (Ak - r) r(k - 1) - (Ak - r)
bk - Ak k(r -).,)
=
b- A r - )., .
813
§2. Characterisations of projective spaces
(c) To prove the converse, we may also assume the validity of condition (ii) by the preceding argument. Let P = pqr be any plane of D and put L := qr. If P is contained in p blocks, then there are r - p blocks through p which do not contain L. Each of these blocks meets L in a unique point, by (ii). On the other hand, each point of L is on A - p such blocks, and therefore r - p = (A - p)ILI = (A - p)(b - A)/(r - A)
p=
= =
by (iii),
A(b - A) - r(r - A)
b-r A(b - A)(V - 1) - r(r - A)(V - 1)
--'..--'---,-::--~:---~----
(b - r)(v - 1)
(b - A)r(k - 1) - r(r - A)(V - 1)
(b - r)(v - 1) (b - r)(Ak - r)
Ak - r
(b-r)(v-l)
v-I
(d) Now assume that D satisfies the equivalent conditions (ii), (iii), and (iv). We will show that these conditions imply (i) by verifying the standard VeblenYoung axioms for projective spaces, i.e. (1.1.a, b, c, d), for the linear space (V, L). Thus (V, L) is isomorphic to PG 1(n, q) (n > 2 and q a prime power). Note first that any two points are on a unique line, by Lemma 2.4, which is the first axiom. The essential axiom is the following one: if four points p, q, r, s form a quadrangle and if the lines pq and r s intersect, then so do the lines pr and qs. Thus let t := pq n r s and consider the plane P := prt; then q, s E P. Now choose a block B with pr S; (B), but P % (B), which is possible as A > P by Lemma 2.5. Then Lemma 2.5 implies P n (B) = pro By (ii), the line qs intersects B in a point u. Thus u E P n (B) = pr, as qs S; P, verifying the second axiom. The third axiom requires that every line has at least three points. But otherwise every line would have exactly two points and thus b = 2r - A, by (iii). Then vr = k(2r -J,..), i.e. Ak = r(2k-v), andhencer(2k-v)(v-l) = Ak(v-l) = rk(k - 1), which implies (v - k? = v - k; i.e. either v = k or v = k + 1, contrary to hypothesis. Finally there are non-intersecting lines. Otherwise any four points would be contained in a plane. Let {p, q, r} be a triangle. Then x E pqr for all x E V. By (iv), pqr S; (B) for a block B E B; hence (8) = V, a contradiction. Thus
XII. Characterisations of classical designs
814
(V, L) is a projective space. Any block of D must be a linear subspace of (V, L). By (ii), it is a hyperplane of (V, L). As b 2::: v, all hyperplanes of (V, L) are blocks of D, and thus D is indeed isomorphic to PGn - 1 (n, q), as required.
It remains to show that (v) implies (iv). As D is symmetric, any two blocks intersect in A points. But then any two blocks of D' (p) intersect in ~=l points, where m is defined as in Lemma 2.6. Thus the dual structure of D' (p) is also a 2-design, and by Fisher's ineqUality D'(p) is symmetric. By Lemma 2.8 (case of equality), we obtain A(A - 1)
P=
r- 1
rCA - 1)
A(A - 1)
=
k- 1
=
v-I
Ak - r
= V-=!'
i.e. condition (iv) holds . •
2.11 Exercise. Show that the S7(2, 15; 31) = dev D, where D
= {1,2,4,S, 7,8,9,10,14,16,18, 19,20, 25, 28}
is the difference set of the non-zero squares in GF(31), has the same parameters as PG3 (4, 2) but is not isomorphic to PG3 (4, 2). Note that the three blocks D, D + 1, and D + 3 intersect in four points which is impossible for three hyperplanes of PG3 (2, 4). Both non-isomorphic 2-designs have a cyclic automorphism group. There are infinitely many examples for this phenomenon, see §V1.17. We next use Theorem 2.10 to give a characterisation of PGn -l(n, q) in terms of its "internal structures"; this is essentially due to Kantor (cf. Dembowski (1968, 2.2.14), when taking into account Mavron's Proposition II.8.14. In II.8.l3, 14 the block residue BD and the induced substructureD(B) of a symmetric 2-design D were defined by BD = (V \ (B), B \ {B}, I) and D(B) = «B), B \ {B}, l) for B E B.
2.12 Theorem. Let D = (V, B) be a symmetric 2-design with)", > 1 and v > k + 1. Then the following conditions are equivalent. (i) D :;:::; PGn - 1(n, q) for some prime power q and some integer n > 2. (U) D(B) is a multiple of a symmetric 2-designfor each block B E B. (iii) B D is an affine 2-design with parameter A' > 1 for each block B E B.
Proof. (ii) and (iii) are equivalent by Proposition II.8.14. Clearly (i) implies (ii): here D(B) :;:::; PGn - 2 (n - 1, q) for each block B E B. Finally, assume the validity of (ii) and (iii) and let B, C E B be any two blocks. Then a block X i= B
§2. Characterisations of projective spaces
815
contains B n C iff X \ B is parallel to X \ C in B D, as we have seen in the proof of Proposition II.8.14. Hence the line «B, C» of the dual design D* of D has preciselys+1elements,wheres = (v-k)j(k-J..) = (b-J..)j(r-J..)-1.Thus condition (iii) of Theorem 2.10 is satisfied for D*, and thus D* is isomorphic to some PGn - 1(n, q); but finite projective spaces are self-dual. • As a final consequence of Theorem 2.10 we present the second half of the original Dembowski-Wagner theorem characterising PGn - 1(n, q) by transitivity properties.
2.13 Theorem. Let D be a symmetric 2-design with J.. > 1 and v > k Then the following conditions are equivalent. (i) (ii) (iii) (iv)
+ 1.
D ;::: PGn-l (n, q)for some prime power q and some integer n > 2. AutD is transitive on ordered triangles. AutD is transitive annan-incident point-line pairs. AutD is transitive on unordered triangles.
Proof. (i) implies (ii), as ordered triangles correspond to ordered triples of linearly independent vectors in GF(q)n+l. Of course (ii) implies (iii) and (iv). On the other hand, if (iii) or (iv) holds, then obviously D is smooth, and, by Theorem 2.10, also (i) holds. •
2.14 Remarks. We have already seen that the existence of a regular automorphism group of a symmetric design is not enough to guarantee that this design is a projective space: difference sets provide many counterexamples. By Exercise 2.11, such counterexamples exist even with the parameters of a PGn - 1(n, q). Similarly, 2-transitivity on points is still not sufficient to force a symmetric design to be isomorphic to a projective space. As examples we mention the unique S2(2, 5; 11) (cf. 1.9.17 and IY.7.1O) and the complementary designs of the projective spaces. But here the following result of Kantor (1971a) shows that 2-transitivity together with the correct parameters forces a symmetric design to be classical.
2.15 Theorem. Let D = (V, B) be a 2-design with the parameters of some projective space PGn- 1(n, q). Then D is actually isomorphic to PGn- 1(n, q) if and only if Aut D is 2-transitive on points. • The proof requires some preliminary results.
816
XII. Characterisations of classical designs
2.16 Lemma. A line L of an SA (2, k; v) has at most ~=~ points. Proof. Any point x E L is on exactly r - A blocks not containing L. The number of all blocks not containing L is b - A, thus ILI(r - A) ::: b - A. Note that equality holds iff L meets every block. • The next two easy lemmas deal with permutation groups.
2.17 Lemma. Let G be a permutation group acting on the finite set X, and let P be a Sylow p-subgroup ofG.If pm divides IxGlfor some x E X and mEN, then pm divides IxPI.
= IGI and IxPI'IPxl Hence there is aCE N which is not divisible by p, such that
Proof. By Proposition Ill.3.lO, we have IxGI'IGxl
= IPI·
As IPxl divides IGxl, the assertion follows . •
2.18 Lemma. Let G be a permutation group acting transitively on two finite sets X and Y, where IXI and IYI are coprime. Then every stabiliser Gx(x E is still transitive on Y.
X)
Proof. By Proposition Ill.3.1O, both a := IXI and b := IYI divide IGI, say
IGI = abc with c E N. Then IGxl = bc and IGyl = acfor each y E Y. As G xy is a subgroup of both G x and G y' and as a and b are coprime, IG xy I divides c. Now the equation
which is derived from the natural action of G on X x Y, yields
I(x, y)GI ::: ab =
IX x Y!.
Therefore G is transitive on X x Y and G x is transitive on Y. •
2.19 Proof of Kantor's Theorem 2.15. By linear algebra, Aut PGn - 1 (n, q) is 2-transitive on points. Conversely, let Aut D be 2-transitive on points. In the case n = 2 the assertion follows by the theorem of Ostrom and Wagner (1959); see e.g. Hughes and Piper (1982) or Ltineburg (1980). Now assume n > 2.
§2. Characterisations 01 projective spaces Obviously all lines have the same size, say m Proposition 1.2.16 imply h ::: q.
817
= h + 1, and Lemma 2.16 and
We will show that actually h = q; then the assertion follows by Theorem 2.10. Choose a point x E V and two blocks B, C E B. There are (k - 1)1 h lines through x contained in Band (J.. - 1)1 h lines through x contained in B n C. Hence h divides k - J.. qn, and there is a prime p with q = pe, h pi for some e, 1 E N with 1 ::: e. We must show that actually e = I·
=
=
By Theorem IlIA.3, the group G is 2-transitive on blocks. Hence, for any block B, the stabiliser G B has two orbits on blocks. By Theorem IlIA. 1, G B has also two orbits on points. Thus G B is transitive on the k points x E B and also on the v - k points Y E V \ B. As k and v - k are coprime we can apply Lemma 2.18. Thus, for x E B, the stabiliser G B,x of both B and x is transitive on the v - k points Y E V \ B. Thus G is transitive on triples (B, x, y) with B E B, x E B, and y ~ B. Hence, given two distinct points x and y, it follows that G xy is transitive on the blocks B with x E Band y ~ B. Now there are k - J.. = qn-t such blocks. Hence qn-l divides IG xy I. Let X be the set of blocks B with x E B and y ~ B, and let P be a Sylow p-subgroup of G xy . By Lemma 2.17, qn-t divides IBPI for any B E X; i.e. P is transitive on X. Clearly P acts on the v - (h + 1) =q(qn-l + ... +q + 1) -hpointsz E V\xy. Now assume that e> I. Then ph divides q, and thus ph does not divide v - (h + 1). Hence there is a point z ~ xy with Iz PI = pi ::: h. Therefore, if B is a block with x E B, Y ~ B, then
IBP'I
= [Pz : Pz,B] = [P : Pz,B]/[P : Pz] = [P : PB][PB : pz.B]/[P :
Pzl
= IB P "ZPB I/lzPI 2:: IBPl/lzPI = qn-1lpi 2:: pqn-2. But as y ~ xz, there is a block B with fixes x and z, we have
x, z
E
Band y ~ B. As the group Pz
This contradiction proves the theorem. •
2.20 Remarks. (a) The 2-transitive collineation groups of finite projective spaces have been classified by Cameron and Kantor (1979) who also solved the harder problem of classifiying the antifiag-transitive collineation groups of these spaces. It is also interesting that all line-transitive collineation groups of
818
XII. Characterisations of classical designs
a finite projective space are already 2-transitive on points, with one exception (Kantor 1973). These results complement Theorem 2.15, but are much harder and will not be dealt with in this book. In this connection one should also mention Perin (1972) who has studied collineation groups of projective spaces acting transitively on the set of d-fiats for d > 1. (b) There are also characterisations of the designs PGn-l (n, q) in terms of special automorphisms, in particular of elations. Of course this needs a generalisation of the classical concept of elation. An elation is defined as an automorphism fixing some block pointwise and some point on this block blockwise. There are results saying that a symmetric design with many elations must be isomorphic to some PGn-l (n, q); see e.g. Kantor (1969d, 1970) and Liineburg (1961). More general symmetric designs still admitting comparatively many elations have been studied by Kelly (1979, 1984a, 1984b); his examples in fact all arise from the designs PGn-l (n, q) by some "twisting" and admit a nice characterisation. We finally mention Butler (1981) who considered symmetric designs with still fewer elations and the survey by Piper (1983). (c) There are some characterisations of other incidence structures related to the projective spaces PG(n, q). LeIevre-Percsy (1980) characterised the designs PGd (n, q) as those smooth 2-designs with the right parameters for which each line has size at least q + 1. Ray-Chaudhuri and Sprague (1976) characterised the incidence structures fO!1Tled by the d-fiats (as points) and the (d + I)-fiats (as blocks) in a projective space. (d) A Dembowski-Wagner type characterisation of the quasisymmetric design PG 2 (4, q) formed by the points and planes of PG(4, q) was given by Sane and
Shrikhande (1993). (e) Rahilly (1991) considered symmetric designs which possess (a spread of) lines of maximal size, that is, of size (v - A)/(k - A), cf. Lemma 2.16.
2.21 Exercise. Recall that a symmetric design is called skew if it has a (+1, -I)-incidence matrix A such that I + A is a skew-symmetric matrix. Clearly, any skew (Hadamard) difference set, cf. §VI.8, gives rise to a skew design; in particular, this holds for the classical Paley difference sets. Prove that the only smooth skew symmetric design is the projective plane of order 2. Hint: Use the Dembowski-Wagner Theorem 2.10; for an explicit proof, see Robinson (1995). The result discussed in Exercise 2.21 is contained in a more general result due to Cameron (1973b) which we now describe. A design D is called quasismooth, if
§2. Characterisations of projective spaces
819
any three points of D are in either Ihl or 1h2 common blocks (for some constants Ihl and 1h2). Then one has the following strengthening of Exercise 2.21; for a proof, see also Cameron and van Lint (1991).
2.22 Theorem. The only quasismooth skew symmetric designs are the projective plane of order 2 and the unique symmetric design S2(2, 5; 11) which belongs to the Paley difference set D = {I, 3, 4, 5, 9} in Z" . • We conclude this section with the following result due to Jungnickel (1984b) which shows that the classical symmetric designs are very exceptional structures indeed.
2.23 Theorem. The number of symmetric designs with the parameters of PGd-1 (d, q) grows at least exponentially. Froof. We first recall a method of constructing such designs which is due to Shrikhande (1951) and was already discussed in proving Proposition 11.8.15. Let A denote the classical affine design AGd-1 (d, q), and let T be any symmetric design with the parameters of PGd-2(d - 1, q). Note that the number of blocks of T equals the number of parallel classes of A. Given any bijection a from the set of blocks ofTto the set of parallel classes of A, define an incidence structure D = D(T, a) as follows. Enlarge the point set of A by adjoining the point set of T to it, and let these new points form a block Boo of D. For any block B of T, adjoin the points of B to all the blocks of A which belong to the parallel class B a and choose all these enlarged blocks as blocks of D. The reader is asked to check thatD is a symmetric design with the parameters of PGd_l(d, q); note that any symmetric design with these parameters which contains A as a residual design may be obtained in this way, cf. Proposition 11.8.15. In what follows, we call any such design a closure of A. Clearly, we may assume all closures of A to have the same point set V; in other words, all symmetric designs T used in the above construction are assumed to be defined on the same point set Boo. From the construction given above, (2.23.a)
D(T, a)
= D(U, fJ)
if and only if (T, a)
= (U, fJ),
since the blocks of T may be recovered from D(T, a) as the intersections of the blocks distinct from Boo with Boo. If we fix any specific T, we may construct from it distinct (though isomorphic) symmetric designs by applying arbitrary
XII. Characterisations of classical designs
820
permutations of Hoo. In this way, T gives rise to exactly (qd-l
+ ... + q + 1)! / IAutTI
distinct isomorphic copies. In view of (2.23 .a), we obtain the following auxiliary result. (2.23. b)
The number of distinct closures of A with point set V is precisely Lr«qd-l + ... + q + 1) !)2 /IAut where T runs over a complete system of representatives of symmetric designs with the parameters of POd-2(d - 1, q).
n
Now consider a fixed closure Do of A. We want to obtain an upper bound on the number of closures of A which are isomorphic to Do. Any such closure Di is the image of Do under a permutation of the point set V. Since Di has to contain A, the number of such closures is at most the cardinality of the set M of all those permutations y of V for which D~ contains A. But then AY-' is an affine design contained inDo, and hence IMI cannot exceed the number of affine designs contained in Do (which is clearly bounded by the number of blocks of Do) times the number of permutations y E M satisfyingAY =A. Since any such y induces an automorphism of A, there are at most IAutA I . (I Hoo J)! permutations Y EM satisfyingAY =A. Thus we have the following estimate (using Lemma III.S.9 and Proposition III.S.20): (2.23.c)
The number of distinct closures of A with point set V which are isomorphic to any given fixed closure Do is at most (qd-l + ... +q + I)! 'IArL(d, q)l. (qd + ... +q + 1) = (qd-l + ... + q + 1)! . IprL(d + 1, q)l.
Combining (2.23.b) and (2.23.c) gives the following estimate on the number Sed, q) of isomorphism classes of symmetric designs with the parameters of POd-lCd, q): (2.23.d)
Sd (qd-I+ ... + q +1)! ( , q) 2: IPrL(d, q)l· IPrL(d + 1, q)1 (qd-l
+ ... + q + 1)!
where q = pI, P prime. Using the power series representation for ek , one may show from (2.23.d) that the asymptotic rate of growth of Sed, q) is ek .1nk , where k = qd-l + ... + q + 1, which proves the assertion. The details are left to the • reader. •
§3. Characterisation of affine spaces
821
2.24 Remark. Kantor (1994) obtained considerably stronger results than Theorem 2.23 by proving that any given finite group G acts as the full automorphism group of a symmetric design with the parameters of PG(d, q) provided that q > 3 and that d is sufficiently large (meaning d > 50IGI 2 ); in fact, there are still exponentially many, namely more than [qO.8d]!, designs of this type. 2.25 Remark. As discussed in §VI.17, there is an infinite series of nonequivalent cyclic difference sets, namely the GMW-difference sets, which have the same parameters as the Singer difference sets; hence the associated symmetric designs have classical parameters. These GMW-designs were characterised in terms of their automorphism groups by Jackson and Wild (1997); the same authors also characterised a certain subclass of the GMW-designs (which actually consists of Hadamard designs) that can be obtained from quadrics in characteristic 2, generalising previous work of Maschietti (1992) and Jackson (1993) who had considered the planar case, where one uses hyperovals.
§3. Characterisation of Affine Spaces In this section we shall consider the designs obtained from affine planes and spaces. Recall that the affine planes are precisely the designs S(2, n; n 2 ). As in the projective case, the designs AG n - 1(n, q) are not characterised by their parameters alone; one obtains a characterisation similar to the DembowskiWagner Theorem 2.10. This is due to Dembowski (1964, 1967).
3.1 Definition. LetD = (V, B, l) be a resolvable 2-design and write B II C iff the blocks Band C are in the same parallel class of the given resolution. If L is any line and B any block of D, then we call L and B parallel (written L II B or BilL) iff there is a block C with L ~ (C) and B II c. Two lines Land L' are called parallel (again wiitten L II L') iff L is parallel to each block B with L' ~ (B). Note that resolvable incidence structures are simple, cf. 1.5.4; hence we identify B E B with (B). 3.2 Lemma. Any resolvable 2-design D has the following properties.
(i) Parallelism of lines is an equivalence relation. II L' implies L = L' or L L' = for all L, L' E L. (iii) Let p be any point and L any line of D. Then there is at most one line L' with pEL' II L. (iv) Two parallel lines are contained in at least one common plane. (ii) L
n
o
Proof. (i) Obviously parallelism of lines is reflexive. If L and L' are lines with L II L', then for each of the A blocks B containing L there is a block B'
822
XII. Characterisations of classical designs
containing L' and patallel to B. As parallelism of blocks is symmetric and transitive, the same holds for parallelism of lines. (ii) Assume L II L', L t L' and L n L' = {c}. There is a block B containing L but not L', say b E L'\B. There is a block B'II B containing h. As B and B' are disjoint, c
•
3.3 Lemma. Let D be a resolvable S)..(2, k; v), and let s = v / k be the size of the parallel classes ofD. Then each line has at most s points. Proof. Let L be a line of D and choose a block B not parallel to L, e.g. a block meeting but not containing L. Then L is not parallel to any of the s blocks parallel to B. Thus L intersects each of these blocks in at most one point. This implies the assertion, by Lemma 2.4 (iii). • 3.4 Theorem. Let D = (V, B) be a non-trivial resolvable S)..(2, k; v) with A > 1. FurthemlOre assume that s = v / k > 2 or that each plane ofD contains at least four points. Then the following conditions are equivalent. (i) (ii) (iii) (iv) (v)
D ~ AG n - 1(n, s)for some prime power s and some integer n > 2. Each line meets each non-parallel block. Each line has exactly s points. Each plane is contained in exactly p = A - r~1.. blocks. D is smooth and affine.
Proof. By classical affine geometry, (i) implies (ii), (iii), and (v). We next show that conditions (ii), (iii), and (iv) are equivalent. The proof of Lemma 3.3 shows that a line L has exactly s points iff it meets each block C in the parallel class of a block B not parallel to L. Hence (ii) and (iii) are equivalent. Now choose any plane P = pqt and consider the line L = qt. Let p be the number of blocks containing P, and assume the validity of (ii) and (iii). There are A - p blocks joining p and a given point of L but not containing L. Moreover, each block B containing L yields a unique. parallel block B' through p, and A - p of these blocks B' do not meet L, namely those
§3. Characterisation of affine spaces
823
blocks B' determined by blocks B with p et B. Every other block through p meets L by (ii). Thus altogether we obtain r = p + (A - p) + SeA - p) blocks through p. Hence p has the value stated in (iv). Conversely, assume the validity of (iv). By Lemma 2.6, each line of D has size
m=
Ak - pv 'A.-p
=
vCr - A) - AS(V - k)
r-A
As A(V - k) = (r - A)(k - 1) we obtain m = v - s(k - I) = s, i.e. condition (iii). Next we show that (v) implies (iv).Assume the validity of (v) and choose a point p of D. As D is affine, any two blocks through p intersect in fJ- - I further points of D. Thus any two blocks of the design D' (p) of Lemma 2.7 intersect in precisely (fJ- - I) / (m - 1) points of D' (p), where m is the line size of D, cf. Lemma 2.6. By Corollary 11.3.3, the design D' (p) is symmetric, and Lemma 2.8 implies p = A(A - 1)/(r - 1). As D is affine, we have r = AS + I by (11.8.7.a). Hence we get P=
A(A - 1) A-I r- A =--=A--r-l s s '
and thus (v) implies (iv). It remains to show that (ii), (iii), and (iv) together imply (i). This is the main part of the proof and will take several steps; cf. Jungnickel and Lenz (1985).
Step 1: 1Wo distinct lines of D are parallel if and only if they are contained in a plane of D and have no point in common. Proof" One part of the assertion has been proved in Lemma 3.2. Now assume that Land L' are two disjoint lines in a common plane P. If Land L' were not parallel, there would be a block B not parallel to L' but containing L. Then P ~ B and thus P n B = L, asD is smooth by (iv). By condition (ii), L' meets B in one point p, but then L' ~ P implies pEP n B = L, contradicting L n L' = 0. This proves our assertion.
Step 2: If (p, L) is a non-incident point-line pair, then there is exactly one line
G with pEG II L (Euclid's parallel axiom). Only the existence needs a proof; see 3.2 (iii). Choose a block B containing L but not p; thus P n B = L, where P is the plane determined by p and L. Furthermore, let C be the unique block through p parallel to Band
Proof"
824
XII. Characterisations of classical designs
let q be a point on L. Then q is on a line H C P with H -:f: Land p f'j H. To see this, let tEL \ {q} be a point. If tp contains a third point x -:f: p, t, then we may take H = q x. Otherwise D has constant line size s = 2. In this case P contains a point y -:f: p, q, t by hypothesis, and we may choose H = qy. Note that this is the only part of the proof where the additional condition for s = 2 is needed. By condition (ii), H intersects C in a unique point z -:f: p. Then pz = P n C, by Lemma 2.5. Since B n C = 0, we have pz n L = 0, i.e. pz is the desired parallel G of L through p.
Step 3: (V, L) is an affine space AG 1 (n, q) with n > 2. Proof' We shall verify properties (1.3.a)-(1.3.0. (1.3.a) and (1.3.e) follow from Lemma 2.4 and the definition of a line. By 3.2 (i) parallelism oflines is an equivalence relation. By step 2 above Euclid's parallel axiom holds. This implies (1.3.b). By Steps 1 and 2, each plane is an affine plane. As (1.3.c) and (1.3.d) hold in affine planes, they hold in (V, L). Note that planes do exist, as A > 1. Finally let us show that there are two non-parallel disjoint lines. It suffices to show that V is neither a line nor a plane. But in these cases D would be trivial. Hence there is an n > 2 and a prime power q such that v = qn. Moreover, s = q andk = vis = qn-I.
Step 4: Each block of D is a hyperplane of (V, L). Proof: If B is a block of D and L C B is a line, then B contains each line G parallel to L and containing a point x E B. Hence B is an affine subspace of (V, L) (i.e. a coset of a linear subspace). By property (ii), it is easily seen that B must be a hyperplane.
Step 5: Each hyperplane of (V, L) is a block of D. Proof: By (1.2.1 1. a) we have A(qn - 1)
== A(q -
1)
== 0
mod (qn-I - I),
-1 I • Therefore hence there is an hEN with A = h qn-J q-
b =A
v(v - 1)
k(k - 1)
=h
q(qn - 1)
q- 1
.
§3. Characterisation of affine spaces
825
As the number of hyperplanes in (V, L) is q(qn - 1)/(q - 1), Step 4 implies h = 1 and the assertion. Now Theorem 3.4 is proved. • 3.5 Examples. Note that the additional hypothesis for the case s = 2 of Theorem 3.4 is really necessary. Let D be any Hadamard 3-design (cf. Theorem I.9.9). Then D is affine resolvable (cf. Example II.8.9.(b» with s = 2, herice any line of D has exactly two points by Lemma 3.3. But D is not isomorphic to any AGn_1(n, q) if q is not apowerof2. In §I.9 and §V1.6 we constructed many examples of Hadamard designs for which this fails. In all these examples, some plane necessarily consists of exactly three points, by Theorem 3.4. Note that these examples nevertheless satisfy conditions (ii)-(v) of 3.4. In this context let us mention that the unique extension of the Hadamard design + 1,2); see Remark 1.9.10.
PGn-l (n, 2) is indeed isomorphic to AGn(n
3.6 Exercise. Let D be an affine 2-design with A > 1. Show that the following conditions are equivalent: (3.6.a)
D ~ AGn-l(n, q) for some prime power q and some integer
n:::: 3. (3.6.b)
Aut D is transitive on ordered triangles.
(3.6.c)
Aut D is transitive on non-incident point-line pairs.
(3.6.d)
Aut D is transitive on unordered triangles.
Hint: Use similar arguments as in the proof of Theorem 2.13. 3.7 Remarks. (a) There is also an affine analogue of Theorem 2.12, but this is more involved. We will obtain the result in question as a corollary to a more general result of Kantor, see Theorem 5.20. Pascasio, Praeger and Raposa (1996) gave a characterisation of the classical affine designs within the class of quasisymmetric designs. (b) There are also a few other incidence structures related to AG(n, q) which have been characterised combinatorially. We mention two examples. LerevrePercsy (1980) characterised the designs AGd (n, q) as those smooth designs with the correct parameters for which each line has size at least q. Buekenhout (1969) proved that a linear space which is not an affine plane and which has a line of size at least 4 is isomorphic to some AG(n, q) if and only if each of its planes is affine. This is rather deep, but the corresponding result for projective geometries is almost trivial (given the Veblen-Young characterisation of projective spaces).
826
XlI. Characterisations of classical designs
(c) The condition on the line size in Buekenhout's result discussed under (b) is in fact necessary. Hall (1960) constructed the first examples of Steiner triple systems which are not affine spaces but for which any three non-collinear points generate an affine plane. Hall (1967) proved that any such triple system may be described by a commutative Moufang loop of exponent 3; cf. the correspondence between Steiner triple systems and certain quasigroups discussed in Observation IX.1.6. By a result of Bruck (1958), each such loop is centrally nilpotent and thus the cardinality of such a triple system is a power of 3. For more recent results on such structures, we refer the reader to Ray-Chaudhuri (1981), Beneteau (1983), Roth and Ray-Chaudhuri (1984) and the papers cited there. A brief summary can also be found in §IY.25 of Colbourn and Dinitz (1996a). (d) Teirlinck (1975) proved that a linear space in which every plane is either affine or projective either has all its planes affine or all its planes projective and hence belongs to one of the cases already discussed. The situation changes if one also allows dual affine planes. Of course, such a geometry cannot be a linear space (since a dual affine plane contains points which are not joined by a line); so we should consider partial linear spaces in this context. Such partial linear spaces were first studied by Hale (1975,1977) and Farmer and Hale (1980). The partial linear spaces in which every plane is a dual affine plane were classified by Hall (1988, 1989). A partial linear space in which all planes are affine planes or dual affine planes is called a generalised Fischer space, since the case of lines of size 2 was introduced by Fischer (1971) for studying groups generated by 3-transpositions. The generalised Fischer spaces were classified by Cuypers (1990), Cuypers and Shult (1990,1993); see also Shult (1995) for a nice survey on this topic and some related areas. The 2-transitive automorphism groups of affine spaces were considered by Key (1983). The following stronger result is due to Buekenhout, Delandtsheer and Doyen (1988).
3.8 Theorem. Let G be a line-transitive automorphism group of AG(n, q), where n ?: 3. Then G contains the translation group of AG(n, q), unless (n, q) = (3,2) and G is isomorphic to PSL(4, 2). • We conclude this section with the following affine analogue of Theorem 2.23 which shows that the classical affine designs are very exceptional structures indeed.
3.9 Theorem. The number of affine designs with the parameters of AGd_1 (d, q) grows exponentially.
§3. Characterisation of affine spaces
827
Proof. Since the proof is quite analogous to that of Theorem 2.23, we will only give a brief sketch; the reader may either supply the details as an exercise or consult the original source, namely Jungnickel (1984b). This time one uses a method of constructing the desired designs which goes back to Shrikhande (1964) and was already discussed in Lemma X.6.7. LetD denote the classical affine designAGd_1 (d, q), and letS be a classical symmetric net obtained from D by omitting all blocks through two fixed points, see Proposition I. 7.18. Now let T be any affine design with the parameters of AGd_2(d - 1, q), and note that the number of points of T equals the number of point classes of S. Given any bijection a from the point set of T to the set of point classes of S, define an incidence structure A = A(T, a) as follows. For any block B ofT, denote by B OI the union of the qd-2 point classes p" with p E B, and adjoin all these blocks as new blocks to S; the result is an affine design with the parameters of AGd -I (d, q). In what follows, we call any such design a completion of S. One easily shows (3.9.a)
A(T, a) = A(U, fJ) from Tonto U.
if and only if afJ- 1 is an isomorphism
This implies the following auxiliary result. (3.9.b)
The number c(d, q) of distinct completions of S is precisely c(d, q) := LT(qd-1 !)/IAut TI, where T runs over a complete system of representatives of affine designs with the parameters of AGd-2(d - 1, q). In particular, c(d, q) 2: (qd-I)!/IArL(d - 1, q)l.
Next, one proves the following estimate: (3.9.c)
The number of distinct completions of S which are isomorphic to any given fixed completionAo is at most IArL(d, q)l.
Combining (3.9.b) and (3.9.c) gives the following estimate on the number A(d, q) of isomorphism classes of affine designs with the parameters of AGd_1 (d, q).
(3.9.d)
Ad (qd-I)! ( , q) 2: IArL(d, q)1 . IArL(d - 1, q)1
828
XII. Characterisations of classical designs
where q = qt, P prime. Using the power series representation for ek , one concludes that the asymptotic rate of growth of A(d, q) is e k .lnk , where k = qd-l .
•
3.10 Exercise. By a result of Kantor mentioned in the book of Dembowski (1968), there are at least two non-isomorphic affine designs with the parameters of AGd - 1 (d, q) unless (d, q) = (3,2). Of course, Theorem 3.9 gives a much stronger result, even for comparatively small values of the parameters. However, it does not cover some of the really small cases. Prove Kantor's result by using the method of Theorem 3.9 to construct an affine design with a line of size 2 (for q =1= 2) or with a plane of size 3 (for q = 2); see also Jungnickel and Lenz (1985). We remark that the affine design AG2(3, 2) is uniquely determined by its parameters, even though there are four non-isomorphic designs S3(2, 4; 8); see Gibbons (1976).
§4. Locally Projective Linear Spaces In this section we consider linear spaces S(2, k; v) which behave "locally" like a projective geometry. This will yield a common characterisation of the designsAG I (n, q) and PGI (n, q) with n :::: 4 among the 2-designs with A = 1. This result which is due to Doyen and Hubaut (1971) will be used in the next section to prove a common characterisation of the designs AGn-l (n, q) and PGn - 1 (n, q) with n 2: 4 which is due to Kantor (1969d).
4.1 Definition. Let D be an S(2, k; v), and let p be a point of D. Denote by D / p the linear space (L(p), P(p), c), where L(p) is the set of lines L of D with pEL and P(p) is the set of planes P of D with pEP. D is called locally projective iff each D/ p is isomorphic to a projective plane S(2, n+ 1; n2+n+ 1) or to a projective space PG l (m, q). Thus the lines and planes through p satisfy the Veblen-Young axioms (1. La, b, c). 4.2 Example. The designs AG1 (n, q) and PG l (n, q)(n :::: 3) are locally projective. To see the connection to the kind of axioms we studied in the preceding sections and for use in the next section, we state the following fact.
4.3 Observation. Let D = (V, B) be a smooth S)..(2, k; v) and assume that each design D' (p) is isomorphic to some PGn - l (n, q) or to some projective plane; cf. Lemma 2.7. Then (V, L) is a locally projective S(2, m; v), where L denotes the line set and m the line size of D; cf. Lemma 2.6.
§4. Locally projective linear spaces
829
4.4 Lemma. Let D be a locally projective S(2, k; v). Then there are positive integers nand q such that D / p is a projective geometry of dimension nand order q for each point p.
Proof. Let P be any plane of D and p a point in P. Then P is a line of D / p and therefore contains q p + 1 points of D / p, where q p is the order of D / p. Thus P contains (qp + 1)(k - 1) + 1 points of D. If p' is another point of D, choose P to contain both p and p'. This shows that qp does not depend on p, since the same argument also implies IPI = (qp' + 1)(k - 1) + 1. Similarly, writing v in terms of the total number of points of D / P shows that the dimension of D / p does not depend on p . • In view of Lemma 4.4, the following definition makes sense. 4.5 Definition. Let D be a locally projective S (2, k; v) and let q and n be as in Lemma 4.4. Then q is called the order and n + 1 the dimension of D. Note that always n ::: 2, by Definition 4.1. 4.6 Lemma. Let D be a locally projective S(2, k; v) of dimension n + 1 and order q, and let P be a plane of D. Then the points of P and the lines of D contained in P form an S(2, k; (q + 1)(k - 1) + 1). If(p, G) is a non-incident point-line pair of P, then there are exactly t := q + 1 - k lines of P through p and disjoint from G. So, in particular, q ::: k - 1. Moreover, the intersection of two distinct planes does not contain a plane.
Proof. Exercise. • 4.7 Proposition. Let D be a locally projective S(2, k; v) of order q and dimension n > 2. Assume k ::: 3 and t = q + 1 - k ::: 1. Then D is isomorphic to PG 1 (n, q) ift = 0 and to AG 1 (n, q) ift = 1. Proof. First assume t = O. Then each plane of D is a projective plane, since k ::: 3. It is then easily seen that D satisfies the Veblen-Young axioms (1.1.a, b, c). Moreover, D contains disjoint lines, since it is not a projective plane. Hence D is isomorphic to PG 1 (n, q) by Theorem 1.2. Next assume t = 1; this time we will verify the axioms of Definition 1.3 and use Theorem 1.5. The validity of (1.3.a) and (1.3.e) is trivial. To prove (1.3.b), we first define a parallelism on the line set in the usual way: G and L are called parallel (written GilL) iff they are contained in a common plane P and either G = Lor GnL 0. Clearly II is reflexive and symmetric. We have to show that II is transitive. Thus assume G II Hand H II K for three pairwise distinct lines
=
830
XII. Characterisations of classical designs
G, H, K. If all of G, H, K are contained in a common plane P, the G II K, since each plane of D is an affine plane, by Lemma 4.6. Thus assume that G, H, K are not contained in a common plane. Choose a point p of G and denote by P and Q the planes determined by p and H and by p and K, respectively. Also let U be the subspace generated by p, Hand K. Then U is a plane of Dip; thus the lines P and Q of this plane intersect in a point of Dip, i.e. in a line L of D through p. We claim that L n H = L n K = 0. Otherwise we may assume the existence of a point r E L n K. Then let S be the plane containing the parallel lines H and K. Thus S is the plane generated by H and the point r of K. Obviously, rEP n Q n S. By the last assertion of Lemma 4.6, we obtain P n S = H, Q n S = K, and hence r E H n K, a contradiction. Thus L n K = 0 and, similarly, L n H = 0. Now both G and L are lines through p in the affine plane P; thus both G and L are parallel to H and therefore G = L. Hence G II K. To verify (1.3.b), we finally have to show that each parallel class partitions the point set of D. But if (p, G) is a non-incident point-line pair of D, then the affine plane determined by p and G contains a parallel of G through p. Finally, (1.3.c, d) hold as each plane of D is affine, and (1.3.f) holds as D itself is not an affine plane. Now Theorem 1.5 yields the assertion. •
4.8 Theorem (The Doyen-Rnbaut Theorem). Let D be a locally projective S(2, k; v) of dimension n :::: 4 and order q. If k :::: 3, then D is isomorphic to PGl (n, q) or to AGI (n, q).
Proof. In view of Proposition 4.7, we have to showthatt = q+l-k ::: 1. Thus we may assume that t i= 0 and have to show that t = 1. Denote by s the number of points in a plane of D (i.e. s = (q + 1)(k - 1) + 1) and by u the number of points in a three-dimensional subspace of D (i.e. u = (q2 + q + 1) (k - 1) + 1, as such a subspace is the union of q2 + q + 1 lines of D through p forming a two-dimensional subspace of Dip). The proof now requires several steps. Step 1: k divides q.
Proof Choose a plane P and a point pEP. Denote by U the subspace of D generated by P and p. As t:::: 1, there is aline G withp E G c U andGnp = 0. Then the sets P n Q (where Q runs over all planes with G c Q c U) partition -P, as each point x of P determines a unique such Q. Now any two distinct planes Q and Q' of U are either disjoint or intersect in a line. For assume r E Q n Q'; then Q and Q' are lines contained in the projective plane U of D I r; thus Q and Q' intersect in a point of D I r, i.e. in a line of D. This shows that k divides IPI = s; i.e, k divides (q + 1)(k - 1) + 1 and therefore k I q.
§4. Locally projective linear spaces
831
Step 2: s divides u. Proof" This is similar to the proof of Step 1: choose a three-dimensional subspace U of D (i.e. the union of the lines through a point x which are the points of a two-dimensional subspace of D / x) and a point p rt U. This is possible, since n ::::: 4. Choose a line L in U and let G be a line disjoint from L and passing through p, which is possible, as t =1= O. By the last assertion of Lemma 4.6, we obtain GnU = 0. Thus pEG c V, where V denotes the subspace of D generated by U and p, and GnU = 0. Then the sets un Q (where Q runs over all planes with G C Q c V) partition U. If all these planes Q would in fact meet U, then wewouldhaveu = (q2+q+ l)k, since each plane Q meeting U necessarily meets it in a line and since the number of planes Q with G c Q c V equals the number oflines through a point in the three-dimensional projective space V ofD/ p.Butalso u = (q2+q+ l)(k-l)+ 1, a contradiction. Thus we may choose a plane P with G c P C V and P n U 0. Then the sets W n U partition U, where W runs over all three-dimensional subspaces of D with PeW c V (Le. all planes of D / p containing the line P of D / p). Moreover, each W with W n U =1= 0 meets U in a plane of D (since U and W are intersecting planes of the three-dimensional projective subspace V of D / x for a point x E W n U). We conclude that s divides u.
=
Step 3: t = 1. Proof" By Step 1, there is a positive integer a with q = ak. Thus
= q + 1 - k = (a - l)k + 1, s = (q + 1)(k - 1) + 1 = k(ak t
a + 1),
and
u
= (l + q +
1)(k - 1) + 1 = k(as - a + 1).
Now Step 2 implies s I k(a - 1), i.e. ak - a + 1 divides a-I. Assume a =1= 1. Then ak S 2(a - 1), contradicting k > 2. Thus a = 1, q = k + 1, t = 1. • 4.9 Corollary. Let D = (V, B) be a smooth S)..(2, k; v), and assume that each design D' (p) is isomorphic to some PGn - t (n, q) with n ::::: 3. Then D is isomorphic to PGn(n + 1, q) or to AGn(n + 1, q). Proof. Use Observation 4.3 and Theorem 4.8 . •
4.10 Remark. In view of Observation 4.3, it makes sense to call a smooth design with)", > 1 locally projective if each design D' (p) is a projective plane
832
XII. Characterisations of classical designs
or isomorphic to somePGn _ 1 (n, q). Corollary 4.9 then partially classifies the locally projective designs. A more general notion has been studied by Cameron (1974). We will conclude this section with a discussion of his results, but we will not give proofs. Cameron's results also give some restrictions on the structure of smooth designs which are locally projective planes (one example being S(3, 6; 22)); but no complete classification of such structures seems to be known.
4.11 Definition. LetD be a smooth SA (2, k; v) with k < v -1. ThenD is called locally symmetric if each design D' (p) is a symmetric 2-design; cf. Lemma 2.7. 4.12 Theorem. Let D be a locally symmetric SA(2, k; v) with line size m = s + 1, and assume that any three non-collinear points ofD are contained in t common blocks. Then one of the following cases occurs:
(i) D ~ PGn-l(n, q)forsome n :::: 3; ~ AGn_ 1 (n, q) for some n :::: 3; (iii) D is a Hadamard 3-design, i.e. s = 1, v = 4(t + 1), k = 2(t A = 2t + 1; (iv) v = (1 + st)(2s2 + 2s + 1 + (3s + 2)s2t + s4t2), k = (1 + st)(1 + s + s2t), A = 1;' (2s + 1)t + s2t 2; (v) t = 1, v = (s + 1)4(s3 + 2S2 + 3s + 1), k = (s + 1)2(s2 + s A = s3 + 3s 2 + 4s + 3; (vi) s = 1, t = 3, v = 496, k = 40, A = 39. • (ii) D
+ 1),
+ 1),
4.13 Remark. We have proved several special cases of Theorem 4.12 in this book: the Dembowski-Wagner Theorem 2.10 is the case whereD is symmetric; Dembowski's Theorem 3.4 is the case where D is affine; the Doyen-Hubaut Theorem 4.8 (resp. Corollary 4.9) is the case where D is locally a projective space of dimension at least 3; and Cameron's Theorem II.7 .12 is the case where D is an extension of a symmetric design (i.e. s = 1). It may be remarked that only one example for cases (iv)-(vi) is known, i.e. the unique S (3, 6; 22); cf. Proposition IV.8.1.
4.14 Remark. Cameron (1974) also considered locally affine designs, i.e. smooth SA (2, k; v)'s where each design D'(p) is an affine 2-design. The result corresponding to Theorem 4.12 is much simpler in this case: D is locally affine if and only if it is an inversive plane (i.e. an extension of an affine plane). This is not too difficult and the reader might try to prove this result as an exercise. Only divisibility considerations and Fisher's inequality are required.
§5. Good blocks
833
§5. Good Blocks
In this section we study so-called "good" blocks of incidence structures. The main result will be a common characterisation of the designs AGn-1 (n, q) and PGn-l (n, q) for n ::::: 4, which is due to Kantor (1969d). We will use the Doyen-Hubaut Theorem 4.8 for its proof. The term "good" has been introduced by Kimberley (1971) in the study of certain Hadamard designs. Other authors (e.g. Mavron (1972a» speak of "prime" blocks instead. 5.1 Definition. Let D = (V, E, 1) be an incidence structure. A block B of D is called good if the following condition is satisfied. (5.1.a)
Given any block C distinct from but intersecting B and any point p ¢ (B) n (C), there is a block D containing both p and (B) n (C).
5.2 Examples. (a) All blocks of AG,,_I (n, q) and PGn_l(n, q) are good. (b) LetD be the (unique) S(3, 6; 22); cf. Proposition IV.8.l. If Band C are any two intersecting blocks of D, then I(B) n (C)I = 2, by Example IV.2.2. As D is a 3-design, (5.1.a) is trivially satisfied. Thus all blocks of D are good. Before stating and proving Kantor's theorem mentioned above, let us consider good blocks in symmetric and affine designs. The symmetric case is closely related to Proposition 11.8.14; it is due to Mavron (1973). 5.3 Theorem. Let B be a block of a symmetric design D. Then the following conditions are equivalent. (i) 1J is good. (ii) BD is resolvable. (iii) BD is an affine 2-design. (iv) D(B) is a multiple of a symmetric design.
Here the notation BD and D(B) is that of Exercise Ii.8.B and Proposition II.8.14. Proof. Let B be a block of D. We call blocks C and D of B D parallel if B n C = B n D in D. Clearly distinct parallel blocks are disjoint when considered as blocks of B D. Moreover, condition (5.l.a) holds iff each parallel class of blocks of BD covers all points of BD. Thus conditions (i) and (ii) are equivalent. Now assume the validity of (ii). As BD is a residual design, its parameters k', r' and
834
XII. Characterisations of classical designs
A' satisfy r' = k' + A', by Exercise IT.8.13. Thus Bose's Theorem IT.8.7 implies that BD is in fact affine; i.e. (ii) implies (iii). The converse implication is trivial. Finally, conditions (iii) and (iv) are equivalent by Proposition IT.8.14. • An immediate consequence of Theorem 5.3 and Theorem 2.12 is the following.
5.4 Theorem. Let D be a symmetric 2-design with A > 1 and v > k + 1. Then D is isomorphic to some PGn - 1(n, q) with n 2': 3 if and only if all blocks of D are good. • We will obtain a similar characterisation of the designs AGn-l (n, q) in Theorem 5.20 below. Let us consider good blocks in affine designs first. Here we have the following result of Mavron (1972a).
5.5 Theorem. Let D be an affine 2-design AJ.L(s), and let B be a block ofD. Then the following conditions are equivalent. (i) B is good. (ii) D(B) is a multiple of an affine 2-design, where we define D(B) as the induced substructure (B, {C
nB
: C -It B}, /).
Proof. Let B be a good block of D. We call blocks C and D of D(B) quasiparallel if Bnc = B nD inD. Clearly distinctquasiparallel blocks are disjoint when considered as blocks ofBD, whereas usual BDdenotes (V\B, B\{B}, J). As B is good, each quasiparallel class covers all points of B D and thus contains exactlysblocks,sincev-k = S2t-t-St-t = s(st-t-t-t) = s(k-t-t);cf. Theorem IT.8.7. Hence D(B) is an s-fold multiple of a simple incidence structure A. We show that A is an affine design AJ.L'(s) with t-t' = t-t/s. Clearly A is an S)..,(2, k'; v') with k' = t-t, v' = st-t, and A' = (A - 1)/s. Note that k' + A' = t-t+A' = (st-t-1)/(s-l) = A'(st-t-1)/(A-l) = A'(v'-I)/(k'-I) =: r';thus Bose's Theorem IT.8.7 will show that A is affine, provided that A is resolvable. Now each parallel class of blocks of D, distinct from the class of B, induces a parallel class of D(B). This gives a resolution of A , if we can show that any two of the s parallel classes of D determined by the quasiparallel class of a given block C with C II B induce the same partition of the points of B. This means the following: given any block C' with C' II C and C' =1= C and any block D with B n C = B n D, we have to show the existence of a block D' II D with B n C' = B n D'. But otherwise D would intersect all s blocks X with B n X = B n C', and also B; hence D would contain s t-t + t-t points, a contradiction.
§5. Good blocks
835
Conversely, let D(B) be a t-fold multiple of an affine design AIL,(s'), say F. Then F has s /-L = k = s,2/-L' points and block size /-L = k' = s' /-L'; hence s' = s and /-L' = /-L/s. Moreover, A' = ().. - 1)/t = (S/-L - s)/(s - l)t; on the other hand, A' = (s'/-L'-I)/(s'-1) = (/-L-l)/(s-l)andthereforet =s.Hence,for any block C of D with C {t B, there are exactly s blocks X with B n C = B n X. These blocks cover all v k = s 2/-L - s /-L = s (s /-L - /-L) points of B D; hence B is a good block. • 5.6 Remark. Using Dembowski's Theorem 3.4 and applying induction, it is now not too difficult to prove that an affine design is isomorphic to some AGn - 1 (n, q) if all its blocks are good; cf. e.g. Beutelspacher (1983). We will, however, consider more general incidence structures for which all blocks are good, in order to prove the result of Kantor already mentioned. The result on affine designs just stated is then a simple consequence of these considerations. We begin by stating the hypotheses assumed in what follows.
5.7 Assumption. Let D = (V, B) be a simple incidence structure satisfying the following conditions: (S.7.a)
All blocks of D are good.
(S.7.b)
There is a positive integer /-L such that any two intersecting blocks of D have precisely /-L points in common.
(S.7.c)
There are v points and every block has k points, where v - 2:::: k > /-L.
(S.7.d)
Given two distinct points p and q of D, there is a block containing p but not q.
5.8 Exercise. Show that PGn - 1 (n, q) andAGIl _ 1 (n, q) satisfy (5.7.a, b, c, d), and that the equation (/-L - 1)(v - k) = (k - /-L)2 holds for n = 3. Verify that the S(3, 6; 22) also satisfies all these conditions. 5.9 Lemma. D· satisfies Assumption S.7 with /-L = 1 S(2, k; v) with v > k + 1.
if and
only
if D
is an
Proof. Exercise. • 5.10 Lemma. Let D satisfy Assumption 5.7. Define a flat of D to be a set B n C, where Band C are distinct intersecting blocks. Then the following
L
XII. Characterisations of classical designs
836 statements hold.
(i) If distinct blocks Band C contain aflat F, then F = B n C. (ii) Given a flat F and a point p ¥ F, there is precisely one block B(p, F) containing both p and F. (iii) Every flat F is contained in precisely P = (v - p,)/(k - p,) blocks. (iv) Given anyftat F, there exists a block B with F n B =f:. 0 and F % B. For any such B, one has (5.lD.a)
Proof. (i) and (ii) are obvious from (5.7.a, b). Let F be a flat contained in PF blocks. Because of (ii), we have
v
= IFI + PF(k - IF!)
yielding PF = (v - p,)/(k - p,), since IFI = p,. This proves (iii). The existence assertion in (iv) is immediate from (5.7 .d) and p, > 1. For such a block B we have
k
= IBI =
IF n BI + p(p, - IF n BD;
using (ii), (iii) and P > 1, this yields (5.lD.a) . •
5.11 Lemma. Let D satisfy Assumption 5.7 with p, > 1 and let B be a block ofD. Denote by D'(B) the incidence structure (B, {B n C: C E Band
C n B =f:. 0}, E);
thus the blocks of D'(B) are the flats ofD contained in B.2 ThenD'(B) satisfies Assumption 5.7 with v(B) = k, k(B) = p,
and
p,(B) = p, - (k - p,)2/(v - k) < p,.
Proof. Let F and G be distinct flats of D contained in B, say F = B n C and G = B n D for suitable blocks C and D. If F n G =f:. 0, then C n D %B, for
otherwise F = B nc nD = G, by 5.10 (i). Thus F nG = B n (Cn D) and D'(B) satisfies (5.7.b) with p,(B) instead of p" by 5.10 (iv). If pis a point of B not contained in F n G and if B' is a block containing both p and C n D, then
2
Note that D' (B) is the reduced structure (D(B)) of D(B); cf. Proposition 11.8.14 and Definition III.2.12.
837
§5. Good blocks
B n B' is a fiat containing both p and F n G; thus D' (B) satisfies condition (5.7.a). If we had v(B) = k(B) + 1, then k - 1 = JL and /1(B) would not be integral, a contradiction. HenceD'(B) also satisfies (5.7.c). Finally, the validity of (5.7.d) for D' (B) follows from the corresponding property for D . •
5.12 Lemma. Let D satisfy Assumption 5.7. Then the number r of blocks through a point p does not depend on p. Proof. We use induction on JL. The case JL = 1 follows from Lemma 5.9. Thus assume /1 > 1 and let B be any block of D. By Lemma 5.11 and induction, we may assume thatD'(B) has rB blocks, i.e. fiats FeB, through any point. Now let p be a point contained in B. As each block C through p with B ::j C yields a fiat B n C through p, one sees that p is on exactly rp = prB + 1 blocks of D, by 5.10 (iii). Hence rp is independent of the choice of pin B, and rB is independent of the choice of B through p. As any two points of D are joined we conclude that r B does not depend on B and r r p does not depend on p. •
=
5.13 Proposition. Let D satisfy Assumption 5.7 with JL > 1. If D does not contain disjoint blocks, then D is isomorphic to some PGn - 1(n, q) with n 2: 3. Proof. By Lemma 5.12, the dual of D is a 2-design. The assertion now follows from the dual of the Dembowski-Wagner Theorem 2.10, since the designs PGn-l (n, q) are self-dual. (Note that the dual of (5.7.a) is the assertion that every line meets every block.) • Note that the dual of Proposition 5.13 is a strengthening of criterion (ii) in the Dembowski-Wagner Theorem, since the assumption that D is a design has been weakened. This can be strengthened further; cf. Theorem 5.17 below. We shall require some further auxiliary results first.
5.14 Lemma. Let D satisfy Assumption 5.7. Then there is at least one block incident with each ofrnin{/1 + 1, 3} given points. Moreover, all lines have the same size m, and any two points determine a unique line. Proof. Both assertions are trivial for /1 = 1, by Lemma 5.9. Thus assume /1 > 1 and let p and q be any two points ofD; then p and q are on a common block B of D. By Lemma 5.11, we may apply this argument toD'(B) to obtain a common fiat through p and q. Any further point r then is on a common block with p and q by (5.7.a). This proves the first part of the assertion. Next, let p, q, r be
838
XII. Characterisations of classical designs
any three distinct points and choose a block B containing these points and a flat in B containing p and q. Then the line pq contains the same points when regarded as a line of D or as a line of D' (B), and similarly for pr. By induction, we may assume Ipq I = Ipr I in D' (B), hence in D. Thus all lines of D have the same number m of points; so any two points of D determine a unique line. •
5.15 Lemma. Let D satisfy Assumption 5.7 with 11 > 1 and let p be a point of D. Define the incidence structure D' (p) as in Lemma 2.7. Then either all D' (p) are isomorphic projective spaces PGn - 1 (n, q)for some n ?: 3 or allD'(p) are projective planes of the same order. Proof. We verify the hypotheses of Proposition 5.13 for D'(p). As 11 > 1, any two blocks of D'(p) meet. The validity of (5.7.a) for D readily implies that for D'(p). By Lemma 5.14, the number of lines of D through p is v' = (v - l)/(m - 1), and the number oflines of D through p contained in a block through p is k' = (k - 1)/(m - 1). Similarly, the number of lines through p contained in a flat through p is 11' = (11- 1)/(m - 1). This proves(5.7.b) and part of (5.7.c) for D' (p); moreover, we see that the parameters of D' (p) do not depend on p. Clearly 11' < k', as 11 < k. Also v' ?: k' + 2; to see this, choose any block B of D, distinct points q, r ¢ B and a point s E B with s ¢ qr. This is possible, as v?: k+2 and IB nqrl ::: 1 < k. Then v' ?: k' +2 for D'(s) (and hence for every D'(p», since neither qs nor rs is in B. Thus D'(p) satisfies (5.7.c). Finally, (5.7.d) is immediate from the last assertion of Lemma 5.14. Thus D' (p) is isomorphic to some PGn - 1(n, q) by Proposition 5.13, provided that 11' > 1. But for 11' = 1, D'(p) is an S(2, k; v) by Lemma 5.9; as any two blocks of D' (p) meet, D' (p) is a projective plane in this case. • 5.16 Lemma. Let D satisfy Assumption 5.7 with 11 > 1. Then D' (B) is a 2design for each block B, and the parameters of D' (B) do not depend on the choice of B. Denote the number offlats in B by b'; then the number of blocks of D incident with a given point p ¢ B and disjoint from B is r b' and therefore independent of the choice of Band p. Proof. By Lemma 5.11, D'(B) has constant block size k' = 11. Now choose any two distinct points p and q of D' (B). The number of blocks of D joining p and q equals the number of blocks of D' (p) containing the point pq of D' (p); thus x does not depend on the choice of p and q by Lemma 5.15. Then p and q are joined by exactly I = (x - 1)/ p blocks of D(B) by 5.1O(iii). Hence D(B) is a 2-design SI(2, 11; k) for each B. The final assertion is now immediate using Lemmas 5.12 and 5.1 O(ii). •
§5. Good blocks
839
We now proceed to present the strongest version of the Dembowski-Wagner Theorem known at present, due to Kantor (1969d).
5.17 Theorem. LetDsatisfyAssumption 5.7 withM > landassumethatsome block ofD meets every other block. ThenD is isomorphic to some PGn- 1(n, q) with n :0:: 3. Proof. From Lemma 5.16, we see that any two blocks of D meet. Now the assertion follows from Proposition 5.13 . • We now prove the main result of this section, i.e. Kantor's common characterisation of affine and projective spaces of dimension at least 4.
5.18 Theorem. LetDsatisfyAssumption5.7withfL > Iand(fL-I)(v-k) =f. (k - fL)2; cf Exercise 5.8. ThenD is isomorphic to some PGn-1 (n, q) or some AGn_1(n, q) withn:o:: 4. Proof. The hypothesis (M - 1)(v - k) =f. (k - fL? implies fL(B) > 1, by Lemma 5.11. Let p be any point of D and B a block through p. Furthermore, let D'(p, B) be the design (D'(B»'(p), i.e. the design formed by the lines of D' (B) through p and the fiats of D through p contained in B. By Lemmas 5.11 and 5.15, D'(p, B) is a projective geometry of dimension :0::2. Hence it is clear that D' (p) is a projective geometry of dimension :0::3, again by Lemma 5.15. Moreover D has constant line size m by Lemma 5.14. If m > 2, then the assertion follows from the Doyen-Hubaut Theorem 4.8. Thus assume m = 2 and choose a fixed point p. Then the points of D' (p) may be identified with the points r =f. p of D, since pr = {p, r} for each r. Under this identification, D' (p) is isomorphic to D p; i.e., D is an extension of a design isomorphic to some PGn_l(n, q) with n ::: 3. By Cameron's Theorem 11.7.12, this is only possible for q = 2. By Remark 1.9.10, we then have D ~ AGn(n + 1,2) .
•
We conclude this section by considering an analogue of Theorem 5.17 for affine spaces; this is also due to Kantor (1969d).
5.19 Theorem. Let D satisfy Assumption 5.7 with fL > 1 and assume the existence of a non-incident point-block pair (p, B) such that there exists precisely one block C with pEe and B n C = 0. Then D ~ AGIl - 1(n, q) for some n :0:: 3 and some q.
840
XII. Characterisations of classical designs
1= (k - IL)2, then the assertion is immediate from Theorem 5.18, asD obviously is not aPGn-l (n, q). Otherwise we have IL(B) = 1 by Lemma 5.11; then each D'(B) is a 2-design with A = 1 by Lemma 5.9. Using the fact that any two points of D are joined and Lemma 5.10(iii), it is now easily seen thatD itself is a 2-design (with joining number p). By Theorem 5.17 and our hypothesis, D is an affine 2-design; thus D' (B) is an affine plane by Theorem 5.5. But the planes of D are just the blocks and thus the conditions of Proposition 4.7 are satisfied (with t = 1; cf. Lemma 4.6), provided that m 2: 3. Hence in this case D ~ AG2(3, q) with q 2: 3. Finally, for m = 2 one obtains D ~ AG2(3, 2) as in the last part of the proof of Theorem 5.18 . Proof. If (IL - 1) (v - k)
•
The following result is an obvious consequence of Theorems 5.19 and 5.5; it is essentially due to Kimberley (1971).
5.20 Theorem. Let D be an affine 2-design AI' (s) with IL> 1. Then the following conditions are equivalent. (i) D ~ AGn-l(n, q)forsome n 2: 3. (ii) Each block ofD is good. (iii) D'(B) is an affine 2-designfor each block B ofD. • McDonough and Mavron (1995) considered quasisymmetric designs with good blocks and obtained the following two interesting results which should be compared with Theorems 5.3, 5.5 and 5.18. The proofs use techniques which are similar to those already presented.
5.21 Theorem. Let D be a quasisymmetric design 8A (2, k; v) with intersection numbers 0 and y which has a good block. Then D is either a symmetric or affine design or a linear space or else has one of the following two parameter sets:
(5.21.a)
v = y4(l -
i + 2y -
1),
k=
i(i - y + 1),
A=l + y+ 1; (5.21.b)
v = y(y4 - l + y2 - Y + 1),
k = y(y2 - Y + 1),
A=i+l. In the last two cases, D(B) is a multiple of an 8(2, k; y)for any good block B .
•
§6. Concluding remarks
841
5.22 Theorem. LetDbeaquasisymmetricdesignS),.(2, k; v) with intersection numbers 0 and y, where A > 1. Then all blocks of D are good if and only if either (5.22.a)
D is isomorphic to AGn - 1(n, q) or PGn-1 (n, q) with n ~ 3
or (5.22.b)
D has parameters as in (5.21.a) or (5.2l.b), all lines ofD have size y and D p is a projective plane of order y3 + y or y2, respectively.
There is a unique example for (5.22.b) in the case y = 2, namely the Witt design S(3, 6, 22) . •
§6. Concluding Remarks In this final section of Chapter XII we report on some further characterisation results. Though we have concentrated in the preceding sections on results characterising the classical affine and projective spaces, there are many similar results for other interesting types of incidence structures. A common characterisation of finite projective spaces and affine planes was given by Beutelspacher and Delandtsheer (1981), see also Delandtsheer (1983b). Delandtsheer (1983a) gave a common characterisation of affine spaces, projective spaces, boolean lattices and the Witt design S(3, 6; 22), extending and correcting the work of Kantor (1974b). A common characterisation of affine and projective spaces and Hadamard 3-designs is due to Beutelspacher (1982b). We now report on some results involving transitivity conditions on linear spaces. The following fundamental result of Kantor (1985b) characterises the Steiner systems S(2, k; v) which admit an automorphism group acting doubly transitively on points. Its proof uses the classification of doubly transitive groups which was discussed in §Y.3.
6.1 Theorem. Let D be a Steiner system S(2, k; v) which admits an automorphism group acting doubly transitively on points. Then D is one of the following structures: (1) AG1 (d, q) or PG 1 (d, q)for some d ~ 2. (2) The affine plane over the near-field of order 9, see, for instance, Andre (1955).
XII. Characterisations of classical designs
842
(3) The affine plane of order 27 constructed by Hering (1969). (4) A hermitian unital as described in Remark VIII.5.31. (5) A non-classical unital associated with a Ree group. (The Ree groups are a family of simple groups due to Ree (1961); a short description of these groups may be found in Huppert and Blackburn (1982b). For the corresponding unitals, the reader is referred to Laneburg (1966) and Tits (1960).) (6) One of two designs S(2, 9; 729) constructed by Hering (1986). • The following result of Key and Shult (1984) and Hall (1985) is partially contained in Theorem 6.1; it gives more precise information for the special case of Steiner triple systems.
6.2 Theorem. Let D be a Steiner triple system admitting a doubly transitive automorphism group G. Then one of the following occurs: (1) D (2) D (3) D
PGI (d, 2) and PSL(d + 1,2) ::: G. PGI (3,2) and G ~ A 7 . ~ AG 1 (d, 3) and G contains the translation group of AG(d, 3) . •
~ ~
We also mention two results due to Delandtsheer (1984b, 1986b), respectively, which use Theorem 6.1.
6.3 Theorem. Let D be a non-trivial linear space and assume that Aut D is transitive on pairs of intersecting lines as well as on pairs ofdisjoint lines. Then D is either a Desarguesian affine plane AG(2, q) or a Desarguesian projective space PGl (d, q) . • 6.4 Theorem. Let D be a non-trivial linear space containing a proper subspace U which contains at least two lines. If Aut D is transitive on pairs of intersecting lines, then D is a Desarguesian affine or projective space. • For further similar results, we refer the reader to De1andtsheer (1984a, 1984b, 1986b, 1992a). The 2-homogeneous (but not 2-transitive) linear spaces have also been classified; this is due to Delandtsheer, Doyen et al. (1986).
6.5 Theorem. Let D be an S(2, k; v) admitting a group G which acts 2homogeneously but not 2-transitively on the points of D. Then v = pn for some prime p == 3 (mod 4), where n is odd. Moreover, one has one of the following two cases:
§6. Concluding remarks
843
(1) D is an affine space over a subfield of GF(pn); (2) D is a Netto triple system, k = 3 and p == 7 (mod 12). •
6.6 Remark. The Netto triple systems are, for instance, described on page 98 of Dembowski (1968) who seems to have coined this term (even though they are different from the examples due to Netto (1893), as noted by Robinson (1975». They are the derived triple systems of the quadruple systems of Liineburg (1965b) which were exhibited in Example Ill.6.17; see also Liineburg (1963). Theorems 6.1 and 6.5 completely classify the linear spaces admitting a 2homogeneous group. Thus one should now study weaker transitivity properties, of which the most natural are flag-transitivity and line-primitivity. Some general results on designs admitting a group of this type were already discussed in §Ill.4. Major progress was obtained by Buekenhout, Delandtsheer and Doyen (198586, 1988); one of their results is the following fundamental theorem showing that the flag-transitive linear spaces split into two large classes.
6.7 Theorem. Let G be ajlag-transitive automorphism group of an S(2, k; v) D, where 2 < k < v. Then one of the following two cases occurs: (6.7.a)
("Affine case") G has an elementary abelian minimal normal subgroup T acting regularly on the point set of D.
(6.7.b)
("Almost simple case") G has a non-abelian simple normal subgroup N such that N ::: G ::: Aut N . •
Based on this theorem, the classification of flag-transitive linear spaces is now reasonably complete, see Buekenhout, Delandtsheeretal. (1990). These authors have completely settled the almost simple case, with partial results being due to Delandtsheer (1986a) and Kleidman (1990).
6.8 Theorem. Let D be an S(2, k; v) admitting ajlag-transitive automorphism group G. In the almost simple case (6. 7.b), one of the following possibilities occurs (up to isomorphism): D = PG(d, q) with d ?: 2, and N = PSL(d + 1, q). D = PG(3, 2) and N = A7. D is a hermitian unital S(2, q + 1; q3 + 1) with e ?: 1, and N = PSU(3, q). D is a Ree unital S(2, q + 1; q3 + 1) with q = 32e +1for some e ?: 1, and N is the corresponding Ree group. (5) D = W(2 e ), cf Remark 6.9, with e ?: 3, and N = PSL(2, 2e ) • • (/) (2) (3) (4)
844
XII. Characterisations of classical designs
6.9 Remarks. (a) Regarding the examples in case (5) of Theorem 6.8, one way of describing them is as follows. Define W (2 e ) to be the Steiner system S(2, 2e - 1 ; 2e - 1 (2 e - 1» the points and blocks of which are the lines of P := PG(2, 2e ) disjoint from a regular hyperoval H (that is, a conic together with its nucleus) and the points of P outside of H, respectively; cf. Exercise VII1.5.17 and Remark VIII.5.34.(d). W(2e ) admits a group isomorphic to prL(2, 2e ) as its full automorphism group which is the stabiliser of H in Aut PG(2, 2e ); the observation that W(2 e ) is flag-transitive seems to be due to Kantor (1975a). This family of examples goes back to ideas of Bose, Shrikhande and Witt; we refer the reader to Buekenhout, Delandtsheer and Doyen (1988) for an extensive discussion of and references for the possible approaches to W(2 e ). Recently, the 2-rank of these designs was shown to be 3e - 2 e by Carpenter (1996), using results of Blokhuis and Moorhouse (1995). More generally, as noted in Exercise VIII.5.17, any hyperoval in a projective plane of even order q can be used to construct both an S(2, !f; (D), see Bose and Shrikhande (1973), and a Hadamard 2-design on q2 - I points, see Maschietti (1992). (b) The affine case of Theorem 6.7 presents considerably more difficulties, among them the classification of flag-transitive affine planes and the possible (if unlikely) existence of non-desarguesian sharply flag-transitive projective planes, cf. Theorem VI. 7 .24. Kantor and Williams (1996) produced huge numbers of flag-transitive planes of even order; in view of these results, a complete classification of such planes is impossible. (c) For some results on line-primitive linear spaces, see Delandtsheer (1988, 1989). Camina and Siemons (1989) studied block-transitive S(2, k; v)'s and characterised these in the case where k = 4 and the given block-transitive group G is solvable. Huiling (1995) proved that for k = 4 in the non-solvable case G necessarily is flag-transitive. Combining this with the work of Buekenhout, Delandtsheer et al. (1990) gives a classification of block-transitive S (2, 4, v),s. (d) For more on flag-transitive linear spaces, see also Zieschang (1992, 1993). The reader might also want to have a look at the survey by Doyen (1989) on designs and their groups and at the relevant parts of the survey by Delandtsheer (1995). In connection with the affine case of Theorem 6.8, we mention the following beautiful result of Hiramine (1990).
6.10 Theorem. Anyfinite affine plane with a point-primitive collineation group is a translation plane. •
§6. Concluding remarks
845
In a special case, one can say even more. The following result was independently obtained by Gluck (1990), Hiramine (1989) and R6nyai and Szonyi (1989).
6.11 Theorem. Afinite affine plane ofprime order p with a transitive collineation group is isomorphic to AG(2, p). • The next few results concern linear spaces embedded into projective planes. The hermitian unitals as described in Remark VIII.5.31 can be characterised in several ways.
6.12 Theorem. Let D be a unital S(2, m + 1; m 3 + 1) embedded in a finite projective plane P of order m 2, and assume that D is invariant under a group G of collineations of P. Then each of the following conditions implies that P is desarguesian and that D is hermitian: (6.12.a)
G is flag-transitive on D.
(6.12.b)
G is line-transitive on D.
(6.12.c)
G is point-primitive on D.
Moreover, PSU(3, m 2) :::: G .:::: prU(3, m 2 ) provided that m :::: 4. • Here the characterisation via (6. 12.a) is due to Kantor (1971 b); then (6.12. b) follows, since the line-transitivity of G implies that G is actually flag-transitive by Lemma III.4.21. Finally, (6.12.c) is due to Biliotti and Korchmaros (1989a, b), who actually study the more general case where G is point-transitive. As the occurrence of the Ree unitals in Theorem 6.1 shows, the condition thatD is embedded in some plane is essential. The following interesting characterisations of classical unitals which are assumed to be embedded in PG(2, q2) were obtained by Lerevre-Percsy (1982a), Faina and Korchmaros (1983) and Blokhuis, Brouwer and Wilbrink (1991).
6.13 Theorem. LetD bea unital S(2, q+ 1; q3+ 1) embeddedintoPG(2, q2). Then the following conditions are equivalent: (6.13.a)
D is classical.
(6.l3.b)
Each line ofD is a line of a Baer subplane of PG(2, q2).
(6.13.c)
D is contained in the GF(p)-code spanned by the lines of PG(2, q2), where q is a power of the prime p . •
846
XII. Characterisations of classical designs
Further characterisations of the classical unitals are due to Tallini-Scafati (1966), Kantor (1971b), O'Nan (1 972b), Wilbrink (1983), Hirschfeld, Storme and Thas (1991), Thas (1992) and Barwick (1994a). De1andtsheer and Doyen (1990) gave the following classification of the maximal arcs (which were discussed in §VIII.5) admitting a line transitive group; a similar question (regarding "semi-ovals") was dealt with by Delandtsheer (1991 b). 6.14 Theorem. Let P be a finite projective plane of order q, and let S be a maximal (k, n)-arc in P. If G is a collineation group of P fixing S which acts line-transitively on the linear space S (2, k; kq + k - q) D which is induced on S, then one of the following cases occurs: (1) q is a prime power, k = q, D is a line-transitive affine translation plane, and G contains the translation group ofD. (2) q = 2e for some e :::: 3, k = q /2, P = PG(2, 2e ), D = W(2 e ), and G contains PSL(2, 2e ), cf Remark 6.9. (3) q = 4, k = 2, S is a hyperoval in P = PG(2, 4), and G = A6 or S6.
Moreover, in each afthese cases, except for the sporadic case (3), G stabilises a line P disjoint from s. • 6.15 Remark. The Steiner systems W (2 e ) were already discussed in some detail in Remark 6.9.(a). It can be seen that W(2 e ) forms a maximal arc in the dual projective plane P* which is, of course, isomorphic to P. We also note that the maximal arcs of Denniston (1969) which we have constructed in Theorem VII1.5.20 were characterised by Abatangelo and Larato (1989) as those maximal arcs in PG(2, q), q even, which are invariant under a cyclic subgroup of order q + 1 of PGL(3, q). Linear spaces in which all planes are affine or projective were discussed in Remark 3.7. A more general result was obtained by Teirlinck (1977). We require a definition: An affino-projective plane is a linear space obtained by deleting a set of collinear points from a projective plane; one has the following result. 6.16 Theorem. Let D be a finite linear space whose planes are all affinoprojective and assume that at least one plane of D has order ::::4. Then D is isomO/phic to a projective geometlY from which part of a hyperplane has been deleted. •
§6. Concluding remarks
847
There is also some work on linear spaces having all their planes isomorphic to a given linear space rr; these are often called rr-spaces. Some papers the reader might like to consult are Buekenhout and Deherder (1971), Delandtsheer (1982, 1983c) and Leonard (1982). Laskar (1974) introduced certain threedim~nsional analogues of the (s, r; I)-nets which she called "3-nets". These are partial linear spaces whose planes are nets with {), = 1. Let us give the precise definition.
6.17 Definition. Let V, L, and P be sets, whose elements are called points, lines and planes, respectively, and let (V, L, II), (L, P, Iz) and (V, P, h) be incidence structures. Then the 6-tuple D := (V, L, P, h, h h) is called a 3net with parameters r, s, and b (where r, s, b are integers :::2) provided that the following axioms are satisfied.
(6.17.a)
phL and LIzP imply phP.
(6.17.b) (6.17.c)
If two lines intersect, then they are contained in a common plane. The points and lines of any plane form an (s, r; 1)-net.
(6. 17.d)
The planes are partitioned into b classes such that each class partitions V and such that any two planes from distinct classes intersect in the points of a line.
(6.17.e)
Each line is on at least one plane.
6.18 Exercise. Let P be a plane of a three-dimensional finite projective space P, put A := P \ P, and let Q be a subplane of P. Choose as points, lines and planes of D respectively the points of A, the lines of A meeting P in a point of Q, and the planes of A meeting P in a line of Q, with incidence induced from P. Show that this yields a 3-net. The following result of Sprague (1979) shows that Exercise 6.18 essentially provides all possible 3-nets.
6.19 Theorem. Every finite 3-net with r > 3 is isomorphic to a 3-net constructed as in Exercise 6.18. • 6.20 Remarks. (a) For generalisations of these results to dimensions d ::: 3, that is, for d-nets, we refer the reader to Laskar and Dunbar (1978) and Sprague (1981a); see also Sprague (1978, I981b) and Ray-Chaudhuri and Sprague (1979).
XlI. Characterisations of classical designs
848
(b) A rather general embedding theorem for planar spaces is due to Metsch (1989), and an interesting characterisation of the three-dimensional projective spaces among all planar spaces was given by Durante and Metsch (1996). (c) Extensive surveys on the classification of highly transitive "dimensional" linear spaces and on dimensional linear spaces in general are given by Delandtsheer (1994, 1995). Regarding extensions of AG(2, q), Thas (1990a, 1990b, 1994) settled a longstanding conjecture by proving the following result.
6.21 Theorem. The Mobius plane S(3, q + 1; q2 + 1) constructed in Theorem III.6.9 is the only extension of AG(2, q), provided that q is odd. • We now collect some characterisation results concerning designs with)" > 1. The 2-transitive symmetric designs were classified by Kantor (1985a).
6.22 Theorem. Let D be a 2-transitive symmetric design SJ.(2, k; v), where v > 2k. Then D is one of the following structures: (1) A classical design PGd-l (d, q). (2) The unique S2(2, 5; 11); (3) A unique symmetric design Sl4(2,50; 176) admitting the Higman-Sims
group, see Higman (1969) and Calderbank and Wales (1982). (4) A unique design with v = 22m, k = 2m - l (2 m -1) and).. = 2m - l (2m - l _l),
where m 0:: 2, see Kantor (1975b). •
6.23 Remarks. (a) In this connection we mention Lander (1988a) who among other results - classifies all biplanes (i.e., symmetric designs with ).. = 2) admitting an automorphism group G fixing some block B and acting 2-homogeneouslyon B. Biplanes admitting an automorphism group G isomorphic to PSL(2, q) such that G fixes some block B and acts on B in its usual permutation representation were considered by Kelly (1981) and Fanning (1991). (b) A symmetric design is said to have the symmetric difference property (and is, for short, called a symmetric SDP-design) if the symmetric difference of any three blocks is either a block or the complement of a block. Kantor (1975b) proved that every symmetric SDP-design has parameters of the form (6.23.a)
v
= 22m , m
).. = 2
-
k = 2m - l (21)> - 1) and l (2m - l - 1), where m 0:: 2.
§6. Concluding remarks
849
The designs occurring in case (4) of Theorem 6.22 are in fact SDP-designs. Kantor (1984) proved that the number of isomorphism classes of symmetric SDP-designs with parameters (6.23.a) grows exponentially. Both symmetric and quasisymmetric SDP-designs (which may be defined in a similar way) have found considerable interest, since they are connected to difference sets, "bent functions" and interesting classes of codes. We refer the reader to Rothaus (1976), Dillon and Schatz (1987), Wolfmann (1977), Assmus and Key (1992b), Iungnicke1 and Tonchev (1991a, 1992), Tonchev (1993), Parker, Spence and Tonchev (1994), McGuire (1997) and McGuire and Ward (1998). A summary of results on SDP-designs can be found in §V.1.9 of Colbourn and Dinitz (1996a). (c) The following analogue of Theorem 6.22 for affine designs is due to Pfaff (1991). His result shows that the property of admitting a 2-transitive group more or less characterises the classical affine designs, up to five sporadic exceptions; the situation in the symmetric case discussed in Theorem 6.22 is somewhat different, since there is a second infinite series in this case. Some interesting remarks on 2-transitive and flag-transitive designs may be found in Kantor (1993).
6.24 Theorem. Let D be an affine design admitting an automorphism group which acts 2-transitively on points. Then D is one of the following designs: (1) A classical designAGd_l(d, q).
(2) (3) (4) (5)
The unique Hadamard 3-design S2(3, 6; 12), see Theorem IV.7.12. The affine plane over the near-field of order 9. The affine plane of order 27 constructed by Hering (1969). One of two non-isomorphic affine designs with s = J1- = 3 which are not isomorphic to AG2 (3, 9), see Pfaff (1993a). •
Another class of geometries that should be mentioned here is the class of Hadamard designs. Kantor (1 969b ) characterised the Paley-Hadamard designs belonging to the Paley difference sets, see Theorem VI. 1. 12. His result is as follows.
6.25 Theorem. The Hadamard designs admitting automorphism groups which are transitive on incident point-block pairs but which are not 2-transitive on points are precisely the Paley-Hadamard designs. • We also mention a related result due to Norman (1968).
850
XII. Characterisations of classical designs
6.26 Theorem. LetD be a Hadamard design SA(3, 2A+2; 4),,+4), where).. is even.lfD admits a 3-transitivecollineationgroup G, then).. = 2 and G ;;: M ll .
•
6.27 Remark. Automorphism groups of Hadamard matrices were considered by Kantor (1969c). Norman (1973) considered automorphism-free Hadamard designs; and non-isomorphic Hadamard designs were studied by Norman (1976). Automorphism groups of Hadamard 3-designs were also studied by Kimberley (1973). The 2-transitive Hadamard matrices were classified by Moorhouse (1995). Next let us consider symmetric (s, f.L )-nets. The classical examples constructed in Proposition 1.7.18 may be characterised as follows, with appropriate modifications in the definitions of lines, triangles, planes, and smoothness; as there are points which are not joined, the definitions of §2 cannot be used exactly as stated there. The following result is due to Mavron (1981a) and Jungnickel (1981c).
6.28 Theorem. Let D be a symmetric (s, f.L)-net with s > 2 and f.L > 1. Then the following conditions are equivalent: (6.28.a)
D is isomorphic to a classical example as constructed in 1.7.18.
(6.28.b)
AutD is transitive on ordered triangles.
(6.28.c)
Aut D is transitive on point-line pairs (p, L), such that p is joined to every point of L.
(6.28.d)
D is smooth.
(6.28.e)
Every line of D has precisely s points. •
6.29 Remark. Jungnickel (1981 c) also exhibited symmetric (pi, pj )-nets with a group acting transitively on ordered pairs of joined points for each prime p and all positive integers i and j. The classical symmetric nets may be obtained from projective spaces by deleting all points on a hyperplane H and all hyperplanes through a point p for a flag (p, H). A corresponding characterisation of the divisible designs obtained from projective spaces by the same kind of deletion for a non-incident point-hyperplane pair (p, H) was given by Jungnickel and Vedder (l984b). 6.30 Remark. We finally mention two topics which are related to the material discussed up to now, even though they do not really belong to design theory (but rather to finite geometry). We begin with the polar spaces introduced by
§6. Concluding remarks
851
Veldkamp (1959, 1960). Two classical families of examples are obtained as follows. Let V be a vector space over a commutative field, f a non-degenerate quadratic form on V, and consider the point set S (f) of the quadric determined by f; then S(f) and its totally singular subspaces form a polar space. Let rr be a polarity of a projective geometry over the field K and assume that rr is not orthogonal if K has characteristic 2; then the point set S(rr) of all absolute points of rr together with all totally isotropic subspaces forms a polar space. Three important references on polar spaces are Tits (1974), Buekenhout and Shult (1974) and Cameron (1982). A nice treatment of projective and polar spaces is due to Cameron (1993). Polar geometry is only a small part of the broad area of the geometry of buildings and diagrams; this very active and important field which has close connections to the theory of finite simple groups has been initiated by the fundamental book of Tits (1974). Another fundamental reference is Buekenhout (1979a). The reader is also referred to Tits. (1981a, 1981b) and Buekenhout (l979b, 1981, 1982, 1983). A systematic exposition of diagram geometries was given by Buekenhout and Buset (1988). We also mention the monograph by Pasini (1994) and the following recent surVeys for diagram geometry, buildings, polar spaces, and related topics: Buekenhout (1995), Buekenhout and Pasini (1995), Scharlau (1995), Cohen (1995) and Rohlfs and Springer (1995).
XIII Applications of Designs
10
0 5
2 7 6
3 8 4
Du muSt verstehn! Aus Eins mach' Zehn, Und Zwei laB gehn, Und Drei mach' gleich, So bist du reich. Verlier' die Vier! Aus Fiinf und Sechs, So sagt die Hex', Mach' Sieben und Acht, So ist's vollbracht: Und Neun ist Bins, Und Zehn ist keins. Das ist das Hexen-Einmal-Eins! (Johann Wolfgang von Goethe)!
§1. Introduction From a historical point of view the title "Applications of Designs" is misleading. The development of the theory of combinatorial designs happens to be one of the few areas of mathematics which was initiated by questions arising in applications. When Leonard Euler was employed by the Prussian King Friedrich II, called "Der GroBe", he carried out a lot of applied science, ranging from the computation of layouts for draining channels for swamps to compiling handbooks for the management of sailing ships. Amongst these, the famous Euler problem, concerning the arrangement of officers of different ranks and troops in the form of a pair of orthogonal Latin squares of order six, was rather academic, but can still be considered as a problem in applied mathematics, which may have triggered most of the work described in this book. Even though magic squares have been used and discussed much earlier in history, this seems to be the first appearance of a type of problem
1
This solution of the "Hexen-Einmaleins" can be found in the Faust museum at Knittlingen, Gennany. For this most recent version of this quote and possible interpretations see Kracke (1992).
852
§1. Introduction
853
which developed into a systematic theory of designs of statistical experiments about two centuries later.
1.1. Design Theory Influenced by Applications While 19th century problems, like Kirkman's schoolgirl problem were an outgrowth of the so-called "Unterhaltungsmathematik" (recreational mathematics), regular combinatorial structures became important concerns in the geometric-group-theoretic schools of research towards the tum of the century, cf. Aigner (1990), page 97. Beginning with the 1930s, the area which is nowadays called design theory, underwent rapid development due to three influences: • the influence of synthetic finite geometry; • the beginning development of universal algebra in terms of which Latin squares are an example of algebraic systems with binary operations of some interest; • last, but not least, actual demands from an area of applications, namely the design of statistical experiments. The investigation of the connections of finite geometry and design theory with group theoretic means was initiated by Baer (1939, 1940, 1947) and his school, eventually providing as an inner mathematical application one of the tools of the classification of sporadic simple groups, cf. Gorenstein (1982). Methods and results from representation theory and algebraic number theory derived since Turyn (1965) and later Ott (1975, 1981), see also Pott (1995), have created the powerful link between the geometric algebra of designs and coding theory, providing one of the most successful applications of mathematics of this century. Fisher's and Yates' early statistical investigations established a systematic connection between this area of combinatorics with the then rapidly developing theory of statistical planning and inference. The successful application of the combinatorial methods to statistical experiments arising in such areas as agriculture, biology, medicine, industrial engineering, etc., created a considerable impact on the development of combinatorial theory during the last 50 years. This was especially influenced by the marriage of set-theoretic combinatorics with algebra. The applications of designs described in the subsequent sections use two important properties of designs: • balancedness and • orthogonality.
XlII. Applications of designs
854
The former refers to a purely combinatorial aspect described in the category of finite sets' of incidence structures and occurs in applications like packing, covering, partitioning and testing. The latter is concerned with the representation theoretic aspects of the modules generated by the incidence matrices. It is used mainly in coding theory, but also in instrumentation techniques such as measuring, weighing, correlations and projections. The success of design theory in many of the classical areas is well documented in Chapter V of the comprehensive handbook by Colbourn and Dinitz (1996a).
In the following ten sections we shall present many of the known, cf. Colbourn and van Oorschot (1989), and several new applications of designs and geometries. The applications range from communications engineering to physics, from computer engineering to cryptography and from statistical experiments to games. Thus we conclude this introduction by an example from gambling, in view of the 2000 year old motto: panem et circenses (Juvenal, Satire X.81).
1.2. Designs in Games and Gambling It has been known for a long time that designs are natural settings for the arrangements for matching and matches, Rev. T. P. Kirkman's schoolgirl problem (l850b) being one of the earliest examples of this type. Resolvability of designs has been a celebrated field for applications of designs. It has been used for arranging tournaments, for ball games (such as soccer, golf, etc.), or for card games (whist, bridge, etc.). The reader is referred to parts V.7.8 of the CRC Handbook, Colbourn and Dinitz (l996a), where an extensive listing of such applications is given. Even for gamblers, designs play a significant rOle in planning an "optimal" game strategy, cf. Colbourn (1996b). We give an example, cf. Oberschelp (1972), of this kind of application in the German national Lotto Block, where k = 6 randomly chosen numbers have to be predicted out of N =49. One way to predict the winning numbers is to choose all of the = (;i) = 13983816 possibilities or blocks. But in this strategy the total stake will be much higher than the winning prize.
G)
The Lotto company offers several predefined systematic betting patterns, which are solutions of the following problem. Choose a set of v different numbers (v 2: k) out of the numbers I, ... , N. Then the systematic patterns consist of b different k-subsets of the v chosen numbers. The b subsets have to satisfy the additional condition, that if one correctly predicts t :5 k numbers in the v-subset of
Table 1.1. YEW 22 of the German national Lotto Block fJ16
f3l7
x
x
x
X
X
fh fh fh f34 f3s f36 f37 f3s f39 f3lo f3ll f312 f313 f314 f3IS al a2 a3 a4 as a6 a7 as a9 alO all al2
x x x x x x
x x x
x x
x x
x x
x
x
x
x
x
x x x
x x x x
x x x x
x x
x x
x
x x x
x x
x x x
x
x x
x x x
x
x x
x
x
x x x x
x x x x x
x x
x x
X
X
x
x
x
x
x x X
X
X
x
x
x x
x x
x x
x x x
x x x x x
f3IS
f319
f320
x
x
x
X
X
x x x
x x x
x x
x
x
x
f322
x x x
x x
x
x
f321
x x x
x x x x x x
XIII. Applications of designs
856
chosen numbers, then At of the b subsets have to contain these t numbers. Thus one just needs at-design S,- (t, k; v).
It should be mentioned that in the German national Lotto Block a prize is given only in the cases 3 ::5 t ::5 6. One of the most popular patterns, the system YEW 22 ("verktirzte engere Wahl") is shown in Table 1. 1. Using this system a player chooses v = 12 numbers a1, ... , a12. The system YEW 22 defines b=22 combinations ({3i)i=l...22 of k = 6 of the chosen numbers aI, ... , a12. This means that if three of the six winning numbers are among the twelve numbers of the gambler's choice, then he hits them exactly twice if he tips all the 22 "blocks" of the YEW 22, each of them exactly once. Analogously for the following other predefined systematic patterns. These patterns facilitate tipping but do not improve the gambler's winning chance, such that in the case of three correct predicted numbers, two of the b subsets contain the correct predicted numbers. A list of predefined systematic patterns with corresponding parameters is listed in Table 1.2. The pattern YEW 132 has been generated using the Mathieu group Mh2, cf. ChapterIV. YEW 66 is a residual design (cf. II.S.13) of YEW 132, obtained by ignoring one element ai and all the blocks which contain ai. YEW 77 is based on the Mathieu group M zz , cf. Chapter IV. It should mentioned that the design corresponding to the pattern YEW 30 was found by the "Niedersachsische Lotto-Gesellschaft" and was not known in the mathematical literature previously; cf. Oberschelp (1972). Table 1.2. List of YEW systems of the German national Lotto Block, cj. Oberschelp (1972) Name
v
VEW22 VEW30 VEW66 VEW77 YEW 130 VEW 132
12 10 11 22 26 12
3 3
4 3 3 5
b=AO
Al
A2
A3
22 30 66 77 130 132
II 18 36 21 30 66
5 10 18 5 6 30
2 5
A4
8
3
1 1 12
4
§2. Design of Experiments In this section, we shall see that incidence structures provide a suitable method of reducing the number of statistical tests in experiments. This is based
§2. Design of experiments
857
on the underlying model of linear regression analysis. We illustrate this by a simple heuristic from agriculture: suppose there are v different artificial fertilisers (in general called treatments) whose effectiveness on b types of plants has to be examined. A complete experiment would require b plots each subdivided into v areas for the different applications of fertilisers. Even for relatively small values for v and b, the cost of the required area can exceed the available resources. For this reason, an incomplete experiment can be designed in which every type of plant is tested with k < v different fertilisers such that any two fertilisers are tested on ).. different types of plants. The reasoning behind balancing the occurrences of pairs of treatments on exactly).. of the b blocks of size k is that, heuristically speaking, the regular appearance of pairs of fertiliser treatments on the same plant provides a sufficiently strong coupling between the effects so that a complete correlation analysis of variance can be carried out. Note that this experimental design corresponds to a SA(2, k; v)-design - a balanced incomplete block design (BIBD). By using BIBD's, e.g. in agricultural tests, the underlying complexity measurearea (of plots) and time (for growth and harvest) - can be reduced significantly compared to a complete experiment. 2 Such incidence structures lead to efficient statistical tests, because the parameters to be determined can be estimated by variance analysis using an incomplete set of lists without a deterioration in quality. The undesired side-effects of experimental factors inherent in physical observations can be averaged out by taking the mean values of the occurring correlations of the first, second or third order, depending on the number of factors applied in the actual experiment. The mathematical setting for this model is given by the theory of linear models, and especially by the multiway classification models. In Section 2.2 ff. we give a formal derivation relying on the very detailed and informative book by Pukelsheim (1993).
2.1. Linear Models A statistical linear model is represented by an affine relationship, cf. Pfanzagl (1966),
(2. 1. a)
2
The above complexity measure was used long before a similar measure for the evaluation of VLSI circuitry in computer science had become fashionable.
XIII. Applications of designs
858
in which x T is a so-called regression vector, e E lI~k is the mean parameter vector to be estimated and the observed value Y E jR of an experiment is disturbed by an error e subject to a probability distribution p. With the usual moment assumptions on the mean parameter
and variance
a sequence Yi of observations (2.l.b)
xi
n
with different regression vectors in the i -th run gives a system of equations to determine the k + 1 parameter system consisting of the vector e and the variance parameter (J'2. In more compact notation we therefore write linear models as (2.1.c)
Y= X
·e+e,
with the response vector Y E jRn, the model matrix X e E jRn.
E jRnxk
and the error vector
In a general linear model with the usual assumptions the random vector Y is distributed according to (2.l.d)
xe
around the multidimensional mean vector and its dispersion matrix cr 2 y with the (n x k)-mode1 matrix X and the non-negative definite (n x n)-matrix y of known type. In general, if s linear forms of e given by Kt e with a (k x s )-coefficient matrix K are to be estimated from the linear model (2.l.c), a linear estimator LY for Kte is needed. Here L denotes an (s x n)-matrix fulfilling the unbiasedness condition (2. I.e)
LEU:= {U
E jRsxn
I UX = K}.
An optimal linear estimator [y for Kt e is obtained by minimizing the corresponding dispersion matrices cr 2 L YL t over the set U.
§2. Design of experiments
859
It is one of the more beautiful insights provided by mathematics that this analytical minimisation task can be characterised geometrically and combinatorially by the technique of LOwner ordering and can be solved algebraically by applying the Gauss-Markov Theorem.
2.2. Optimal Estimators As the dispersion matrices in question are non-negative definite matrices, we recall some facts about the geometry of these sets. For mEN the set NND(m) C Sym(m) of non-negative definite symmetric (m x m)-matrices is a pointed closed convex cone whose interior is the set PD(m) of positive definite matrices, with respect to the set Sym(m) of symmetric (m x m) matrices. Sym(m) is endowed with a partial order relation, the so-called L6wner ordering 2: by the definition (2.2.a)
A ::': B
{=:}
A- B
(2.2.b)
A > B
{=:}
A - B E PD(m).
E
NND(m)
The geometric properties of NND(m) are respected by the Lowner ordering through the necessary antisymmetry, reflexivity and transitivity to form a partial order. In addition we have (2.2.c)
VA. E jR+, A 2: 0:
(2.2.d)
VA, B 2: 0:
A+B::,:O
(2.2.e)
lim Al 2: 0:
for all sequences (Al)/ in NND(m).
/->0
A.A 2: 0
With this notation we are now prepared to formulate the Gauss-Markov Theorem, cf. Pukelsheim (1993), p. 20, which is fundamental to what follows. (2.2.f)
Gauss-Markov Theorem.
Let X be an (n x k)-matrix and V be a non-negative definite (n x n)-matrix. Suppose U is an (n x s )-matrix. A solution L of the equation LX = U t X attains the minimum of LV L t, relative to the £Owner ordering and over all solutions L of the equation LX = U t X,
LvLt=
min
LelR"":LX=U t X
if and only if LVRt =0,
LVLt,
860
XIII. Applications of designs
where R is the projector given by R = In - X G for some generalised inverse G of X, i.e. G E x-:= {G E jRkxll I XGX = X}. A minimising solution L exists; a particular choice is UtUn - V Rt H R), with any generalised inverse H of RV Rt. The minimum admits the representation
min
LVLt = Ut(V - VRt(RVRt)-RV)U
LelR"":LX=ul X
and does not depend on the choice of generalised inverses involved. •
With the notion of an optimal estimator being a minimum variance unbiased linear estimator in a classical linear model, cf. Equation (2.1.c), the GaussMarkov Theorem therefore implies, cf. Pukelshcim (1993), 1.23, that • the optimal estimator for the mean vector a 2 p and
xe is PY with dispersion matrix
(2.2.g) i.e. the orthogonal projector onto the range of X; • the optimal estimator for e is (2.2.h) with dispersion matrix a 2(Xt X)-l. While in the first case the optimal estimator P Y only depends on the range of the model matrix, i.e. the space (Xi liE [1, '" , n]) spanned by the regression vectors, in the latter case when estimating the parameter directly the choice of the regression vectors Xi influences the optimal estimator and its dispersion matrix.
e
The optimal estimator for e as given in (2.2.g) coincides with the least square estimator which minimises the sum of squares of deviations (2.2.i) With respect to each el this is defined by a solution (2.2.j)
VI E [1, ... , k]:
as ael
= 0
eof
§2. Design of experiments
861
given by (2.2.k)
2.3. Greenhouse Agriculture The following practical example, adapted from Chapter 7 of Mead (1988), illustrates the different concepts of optimal parameter estimates by least squares models allowing a trade-off of the number of tests against a controlled increase of the variance. To optimise the yield in greenhouse agriculture, an experiment is to be designed to investigate the effects of heating, lighting and carbon dioxide on the growth of green paprika peppers in a glasshouse. For technical reasons the number of possible treatments is six. Each observation on a treatment combination requires a greenhouse compartment. There are 10 greenhouse compartments available in two sets of five compartments. As compartments in different sets are likely to produce rather different yields, the problem is to design an experiment in two blocks, each of five units, to compare the six treatments. The primary idea behind blocking is that identifying blocks of homogeneous units allows a more precise comparison of treatments, by eliminating the large differences between units in different blocks from the comparison of treatments. It follows that information from blocked experiments is predominantly based on the comparisons that can be made between treatment observations in the same block. If two treatments do not occur in the same block, it is still possible to make a valid comparison between the two treatments if each treatment occurs in a block with a common third treatment. Essentially, a comparison between two treatments p, q E [1, ... ,6], within a block, has variance 20'2, where 0'2 is the variance of units within a block. If each of p and q occur in one block with treatment r then the p ___ q comparisons can be calculated via the intermediary by (p ___ r) and (r ___ q) comparisons, each of the two component comparisons being made within a block, and the variance of the implicit p ___ q comparison is 2(20'2). If the indirect comparison of p and q can be made through several different intermediaries, then the precision of the p ___ q comparison is improved. With two blocks of five units each available to compare six treatments, then, provided each treatment appears at least once, and no treatment appears twice in a block,
XIII. Applications of designs
862
a simple arrangement for this is shown in Figure 2.1. We now can set up the linear model, cf. Equation (2.1.c), for this problem.
Block
~ I:
r
s
u
r
s
u
Figure 2.1. Experimental plan for comparing six treatments in two blocks of five units (taken from Mead 1988). The yields Yij from a compartment in block i E {I, il} under treatment j E [1, ... , 6] are reasonably assumed to behave like Yij = i-L + b i + tj + eij with an expected mean i-L and deviations depending on the block bi or the treatment tj only with an error eij distributed with E(eij) = 0, Var(eij) = (12 and normalised constraints bI + bil = 0 and LjE[I, ... ,6] tj = O. The (10 x 9)model matrix X representing the experimental plan, cf. Figure 2.1, with respect to the vector
(2.3.a)
B=
E JJl9,
is given by 0 1
(2.3.b)
X=
0
0
0
0
0 0
0 0 0
1 0 0 0
0 0 0
0
1 0
0 0 0
0
0
0
1 0
0 0 0 0 0 0 0 1 0 0 0 0
0
0 0
1
0 0
0
0
0
0
0 0
0
0
1 0 0
0
0
1 0 0 0 0 0
0
1 0
863
§2. Design of experiments From this we compute 5
5 1
1
2
2
2
2
5
5
0 1 0
1
1
1
1
5 I
0 5 0 1 1 I 1 1 I 0 1 0 0 0 0 0
1
0
1 0
1 0 0 0 0
2
1
1 0
0 2
2
1
1 0 0 0
2
0 0
2
1
1 0
0
0
2
2
1
1 0
0 0 0 0 2
10
(2.3.c)
xtx=
0
0 0 0 0
giving the least square equations (2.2.i). These can be solved by Gaussian elimination giving e.g. the following estimates of the differences: (2.3.d)
A
A
tp - tq
=
1
4(4YIl -
YI3 - YI4 -
YIS -
YI6 - 4YII2
. + YII3 + Yn4 + Yus + YII6) (2.3.e)
A
A
tp - tr
1
= g(8 Yll
-
5YI3 - YI4 - YI5
- 3YII3 + Yn4 +
YIIS
YI6
+ Y1I6)
with (52
'iq) = 16 (2· 16 + 8· 1) =
(2.3.f)
Var(tp -
(2.3.g)
Var(tp - tr )
A
A
(52
5(52
2' 13(52
= 64 (64+25 +9 +6 ·1) = -8-'
We have given the details of this example from Mead (1988) here, as it shows a diagrammatic method of obtaining the linear combination of yields for complex least squares estimates, illustrated in Figure 2.2. The method is essentially just writing each total in (2.3.f) as the sum of the individual yields. The visualisation, however, provides a structure, reducing the scope for mistakes. The separate figures in each "unit" derive from the totals in (2.3.f); the circled figures represent the total coefficient for each unit in (2.3.g). 3
3
The hatched squares do not indicate broken windows of the greenhouse, but rather refer to the treatment not applied.
XIII. Applications of designs
864
Treatment 4 3
5
-1
-1
(]
(]
(]
@
6
-1
-1
Block +1
@
@
Figure 2.2. Diagrammatic representation of estimate of t p -
+1
+1
+1
@ tr
@
(taken from Mead 1988).
2.4. Linear Regression and Experimental Designs Following from this rudimentary example, we observe that in general the matrix X in fact fonns a list (table) of n regression vectors Xi. which we shall henceforth call an experimental design, even though it is in general not a t -design. The precision matrix of this design (2A.a) being the inverse of the dispersion matrix, therefore has to be maximised to give From Section 2.2 it is clear an optimal estimator for the parameter vector that for a given set S of regression vectors (which due to physical requirements can only be realised), the so-called regression range, each experimental design X is a sequence over S. For the computation of X only the multiplicity n(x) of the vector x in X is important, since
e.
xt
(2A.b)
~XiXr = ~n(x)xxT. XES
With the sample size n defined by (2A.c)
n = ~n(x) XES
of X we obtain the nonnalised moment matrix (2A.d)
M(X)
"n(x) T = 'L.t xx XES
n
of the design X. Thus the standardised precision grows directly proportionally to the sample size and decreases inversely proportionally to the model
865
§2. Design of experiments
variance 0- 2 . Even though the numbers ~ are rationals, by abuse of notation we define the convex hull (2A.e)
x=
conv(S)
as the set of all possible designs supported by S. Maximisation of M(X) is possible due to the following lemma. (2A.f)
Lemma (Pukelsheim 1993)
The set M (X) of moment matrices of all designs is the convex hull conv({xx T : XES})
of rank 1 matrices in the space Sym(k). If the regression range S is compact, then M(X) is a convex and compact subset of the cone NND(k) . •
e
If for the problem of estimating a parameter subsystem K t of the mean parameter vector a design x E X with moment matrix M = M (X) is used, the quality of the estimate (e.g. in the classical sense of least squares approximation) can be described by the generalised information matrix MK of the design X for Kte. MK is well defined by the mapping (2.4.g)
(2.4.h)
NND(k)
~
Sym(k)
due to the geometry and another application of the Lowner/Gauss-Markov argument. A moment matrix is called LOwner optimal for Kte in a set M C M(X) of competing moment matrices if (2A.i)
This, however, implies (cf. Pukelsheim 1993, Cor. 4.7) that no moment matrix is Lowner optimal for Kte in the set M(X) unless rank(K) = 1. While at first glance this result seems very pessimistic, it just means that in a class of designs conforming to practical realities and restricted to rational conditions, it may well be a daunting task to investigate special types of designs, for which the set M of competing moment matrices is more restricted. By means of geometric results of the Lowner/Gauss-Markov type, a bound for the number
866
XIII. Applications of designs
of points from S supporting a design can be derived, cf. Pukelsheim (1993), Theorem 8.1. This theorem states that every (k x k)-matrix M is achieved by + 1 different regression vectors. If the a design supported by at most (k x s )-coefficient matrix K has (full) rank s, the (s x s )-generalised information matrix can already be realised by a design supported by at most (s~l) + S . (rkM - s) vectors.
C!l)
Motivated by these results, somewhat resembling Fisher inequalities and rank formulae (cf. II.2.6), we now show how the class of block designs, forming the central topic of this book, is obtained from this general setting of operator theory and statistical estimation theory.
2.S. Two-Way Models and Block Designs We now describe the two-way classification model (with no interaction), which is the prototype of the type of design problems this book is concerned with. Suppose an experiment consisting of two factors gives responses
Yi,j,/
=
Cii
+ /3j + ei,j,/,
i E [1, ... , a], j E [1, ... , b], l E [1, ... , nij],
consisting of the mean effect Cii corning from so-called level i of the first factor A := [1, ... , a], i.e. the effect of treatment number i, plus the mean effect /3j stemming from the level j of the second factor B := [1, ... , b], i.e., the mean effect of the j-th block. Finally ei,j,/ denotes an error term coming from the l-th test replication and the replication number ni,j counts how often the "treatment" i is applied to "block" j. We encode these numbers in a rational matrix (2.S.a) which weighs the proportional contribution of observations that use a level i of factor A and level j of factor B. With the regression range given by (2.S.b)
S= ei
lei
liE A} x {fj
= (Oi/)/eA
E]Ra
Ij
and
E B} C ]Ra+b, /;
= (0 j/)/eB E ]Rb,
§2. Design of experiments
867
this leads to regression vectors of the form (2.S.c)
Xi,j
=(
i
e j) ' f
(i, j) E A x B,
with (2.S.d) defined as in Equation (2.4.d). With this regression set fixed, the design X can be uniquely represented by the (a x b)-weighing matrix W. With the following notation (2.S.e)
r := Wlb E JRa
(2.S.f)
s := WT la
(the vector of treatment replication numbers)
E JRb (the vector of block sizes)
the matrix W leads to a block design with the incidence matrix
N=n·W (having positive integer coefficients possibly exhibiting multiple blocks being multisets) corresponding to the experimental design X. The moment matrix M := M(X) can then be written in the form (2.S.g)
M
= L n(x)xx = L T
xeS
ieA
jeB
eie Wij
(
d
T
.;r
)
I
where for any mEN and x E JRm, Cl x denotes the m x m diagonal matrix formed by the vector x. In this model the parameter vector e has the form e = (p) where a = (al,"" aa)T and f3 = (f3I, ... , f3b)T. As MC~:) = 0, the optimal estimator (nM)-1 XTy for e cannot exist. We have to interpret this fact as saying that
868
XIII. Applications of designs
only certain subsystems mated.
Kt e of the mean parameter vector
e can be esti-
Indeed, the subvector, for instance a = (aI, ... , aal, can only be estimated when corrected by the mean effect (cf. Pukelsheim 1993, 3.20 and 3.25) (2.5.h)
a=- La; 1
a ;
giving the subsystem (al -
(2.5.i)
a, ... , aa -
Kaa = Kte with K
a)T of so-called centred contrasts,
= (~a)
defined by the centring matrix (2.5.j) and the averaging matrix (2.5.k) forming an orthogonal sum decomposition oflRa , which has special applications in measuring quantum systems (cf. Section 4.4). The generalised information matrix for the centred treatment contrast given by
(2.5.1)
!::"r MK= ( o
Kaa is
W!::";wt 0) 0
essentially being the Schur complement of !::"S in M, (2.5.m) here called the contrast information matrix of the block design
w.
We can now quote (cf. Pukelsheim 1993, Claim 4.8) the basic fact:
Let n E N+ and r E lRa be a given treatment replication with nr; E No, t! .r = 1. Then the product designs W = rsT are the only LOwner optimal
§2. Design of experiments
869
designs for the centred contrast Kj a in the set T (r) of block designs with blocksize vector r, having contrast iriformation matrix b. r - rrT . • With these results, we see that the incidence matrices of complete designs which test every unit against every other unit give optimal estimators. Here the following question arises: Amongst the many choices of possible matrices W in the class T of block designs of the form (2.5.a) with a . b tests, would it be possible to derive sufficiently many relations in order to determine the parameters or structure of the design? However, a close look at the dimensions involved shows that a block design with sample size n < a . b is sufficient even though it has incomplete support to the effect that some treatment-block combinations will not be observed.
2.6. Symmetry and BmD's For this we assume with some justification that the quality of a parameter estimation should be invariant with respect to the labelling of the treatments and blocks. More specifically, the concept of Lowner optimality has to be extended suitably, to remain invariant under permutations of the experimental domain; i.e. the sets A and B in the case of the two-way model. To simplify the problem, we consider a closed subgroup G < O(k) ofthe orthogonal matrices acting by left multiplication on the regression set, simultaneously acting by conjugation on the set M c NND(k) of moment matrices. Furthermore, this group affords a representation H < GL(s) on the group of invertible (s x s)-matrices acting by conjugation on the set MK C Symes) of corresponding information matrices for Kte. The design problem for Kte in M is called G-invariant, if (2.6.a)
\/g E G:
gMgt = M
and
g(imK) = imK.
The related information matrices are made H -invariant by defining the socalled Kiefer ordering (cf. Pukelsheim 1993, Chapter 14) as an extension of the Lowner ordering. Under this group theoretic-eombinatorial notion for optimality criteria, one sees that a moment matrix M is (Kiefer) optimal for Kte in M, if and only if Mis Lowner optimal for Kt() and fixed under the action ofG. With these preliminaries we can go back to the two-way classification problem. Applying the results about the centred contrasts (2.S.i) we consider now the
XIII. Applications of designs
870
maximal parameter system
(2.6.b)
where the dots in the index of the value x represent sums over the corresponding variables (e.g. Xi .. = Lj Lz xijz), with (2.6.c)
K
= (la 1b
Ka 0
0)
E ffi,(a+b) x (a+b+l) ,
Kb
with
-
1
1a:= -la. a
Within the moment matrix set M = M(T) of all block designs and the group G (2.6.d)
being the direct product of permutations of all treatments resp. all blocks represented by block diagonal permutation (a + b) x (a + b) matrices (2.6.e)
the identity
(2.6.f)
G~)K K(o1 0 0) =
0
0 0 a
p
holds as Ka and Kb are symmetric. Thus the group H afforded by G on MK is isomorphic to G. Now, we conclude that the uniform design (2.6.g)
W
= -1a ·l-bt
is the best block design for the two-way classification problem. Its realisation would imply that the sample size n must be a multiple of a . b, quite often an unfeasibly large number. A sample size n < a . b implies incompleteness, in so far as some combinations (i, j) E A x B are not observed. By requiring the other properties we arrive at the definition of a balanced incomplete block design, if W is
871
§2. Design of experiments
• a uniform (a x b)-weight matrix with
wij E
{O,
*}
• equiblock sized, WTl a = lb' • balanced in the sense that its contrast information matrix C (W) is completely symmetric and non-zero, i.e. of the form (2.6.h)
X/a
+yJa ·
A short calculation reveals that this is identical to the well-known formula for incidence structures of balanced incomplete block designs (2.6.i) with positive integers ~, b- and A = ~ . (1l~~;1. One can see from Equation (2.6.h) and the requirement that N and NT is to be a positive matrix, that this implies Fisher's inequality. We conclude this section by saying that by using functional analysis applied to linear estimation theory, the concept of a balanced incomplete block design can be derived in a purely functional analytic setting, producing the parameters of balanced incomplete designs as defined in Chapter 1.
2.7. Testing Speech Scramblers Similar questions about efficient experiments occur in many other areas of science. As a classical example we mention the problem of comparing the effects of different drugs in the vast area of medicine and pharmaceutics. Here we present another example concerning a rather unusual application of design theory to the design of cryptographic devices, namely quality tests for voice scramblers based on different permutations, cf. Beth, HeB and Wirl (1983), and Beker and Piper (1982). These scramblers are realised by a so-called time division multiplexing of the input voice data by dividing a segment of transmitted or recorded speech of length t = 600 ms into n blocks, which are shuffled by certain permutations, the scrambling keys. Such a scrambler can be viewed as a cipher, permuting the n blocks of a segment into a signal which sounds like noise rather than voice. The lack of objective criteria for the evaluation of the desired "non-understandability" of voice scrambling schemes, and the associated scrambling keys, necessitates subjective tests to validate a given metric used for the permutations.
872
XIII. Applications of designs
This technical goal has required that a series of psycholinguistic hearing tests be perfonned. During each test seven sample texts To, ... , T6 are used. With the exception of text To each text is scrambled by a fixed pennutation. Text To is scrambled by continuously varying pennutations. The permutations used for text TI-T6 are pairwise different. During the test, subjects listening to the scrambled texts have the task to "descramble" texts using their auditory system, i.e. by listening. To prevent errors caused by loss of concentration, each subject was asked to listen to three texts only. For planning the experiment a Steiner system 3(2, 3; 7) is used which leads to seven schemes (the blocks of the Steiner systems) each containing three texts. Each of the seven texts is contained in three schemes and occurs exactly once at the first, second and third position in a scheme (see Table 2.1). The outcome of these tests has been successfully applied to evaluate the quality and the design of voice scramblers, cf. Beth, HeB and Wirl (1983).
Table 2.1. Experimental scheme testing voice scramblers, which resembles the well-known incidence matrix of the Fano plane 3(2, 3; 7), cf Example 1.1.3 Subject
1st test
2nd test
3rd test
0 1 2
To TI Tz T3 T4 Ts T6
Tl Tz T3 T4 Ts T6 To
T3 T4 Ts T6 To Tl Tz
3 4
5 6
To solve this ranking problem by a regression analysis it is assumed that on a logarithmic scale the success Vtp of subject p in descrambling text t can be described by the linear model
where ILt denotes the log (understandability in % of text) and D!p describes the ability of person p on a logarithmic scale. The desired metric for nonunderstandability is then provided by the linear order of the number (- ILt), so that higher numbers represent higher crypt(ograph)ic quality. In analogy to
873
§2. Design of experiments
Example 2.3 the regression vector in this case is !J,l
(27) .. a
0 =!J,7
with model matrix
1 0 0
0 0
1 0 0 0 0
1 0 0
0 0
0
0 0 0
0
1 0 0 0
0
1 0 0 0 0 0 0
0
1 0 0 0 0 0 0 0 0
0
1
0 0 0 0 0 0 0 0 0
0
0
1 0 0 0 0
0 0
1 0 0 0 0
1 0
0 0
0
0
X=
1 0 0 0 0 0 0
1 0 0 0 0 0 0
0 0
(2.7.b)
0 0 0 0
0
0
0 0
1 0 0 0 0 0 1 0 0 0
0
1 0 0
1 0 0
1 0 0 0 0 0 0 0 0 0
1 0
0 0 0
1 0 0 0
0
0 0
1 0 0 0
0
1
0
0
0
0
0
0
0
0
0 0 0
1 0 0 0 0 0
0
0 0
0
0
0 0
0
0 0
0
1
0
0 0 0
0
0
1
1 0 0 0 0
0
0
1 0 0
0
1 0
0
0
0 0
0
1 0 0
0
0
0
1 0 0 0 0 0 0
0 0 0
0
0
1 0
0 0
0 0 0 0 0 0
0 0
0
0
0
0
0
0
1
1 0
1 0 0 0 0 0 0
1 0 0
1 0 0 0
0
0
1 0 0 0 0 0 0
1
0 0 0 0
1 0
0
0
1 0 0
0
0
0
0
0
1 0 0 0 0 0 0
0 0
1 0 0 0
0
1 0 0 0 0
XIII. Applications of designs
874 so that
300 0 000
101
000
o
I
3 0
0
0 0
0
0
I
0
003
000
0
0 0
I
0
I
000
3 0 0
0
000
I
0
000
I
0000300 000
0
0
000 000
100 0
I
0
3
0
o
3 0
o
I 0
0
0
0
0
100 3 0 0
0
0
000
000
0 1
000 000
0 0 0 000
o
I 0
1 103 000
000
o
0
000
3 0
(2.7.c)
o 0 o
0
300 0
100 000
0 3
0
0
0
1 000
0
003
0
o
000 0
3
1 0 0
has the form (2.7.d) where b. r = r . Iv, b. k = k . h, W is the incidence matrix of a proper 2-design Sl (2, k; v) with r = k = 3, v = b = 7, i.e. the Fano plane, as planned.
§3. Experiments with Latin Squares and Orthogonal Arrays
In Section 2, the theory of linear regression analysis has been described leading to the properties of balanced incomplete block designs for the so-called twoway classification models. However, a more general theory of multiway classification models could be envisaged. Here several factors of classification would have to be included leading to multidimensional tensorial incidence structures which are not dealt with in this context. However, the linear model of estimating three qualitative factors without interaction, cf. Witting and Nolle (1970), leads to a reduced complexity estimator for a linear theory of this type. First we give a simple example.
875
§3. Latin squares and orthogonal arrays
3.1. Three Factors on Four Tyres An example is the assessment of the wear of car tyres (see Mead 1988). Because there are performance differences between the four possible wheel positions of a car, which are also influenced by differences between cars, the experiment has to eliminate these influences. In the case of four different tyres, four cars can be used and the tyres are allocated to the cars so that one tyre of each brand is fitted to each car in the four different possible wheel positions distributed over the four tyres of the cars. This leads to the concept of Latin squares (cf. Table 3.1).
Table 3.1. Latin square for testing tyres.
Front left Front right Back left Back right
BMW
Mercedes
Porsche
Audi
Michelin Goodyear Conti Pirelli
Goodyear Conti Pirelli Michelin
Conti Pirelli Michelin Goodyear
Pirelli Michelin Goodyear Conti
We recall from Observation 4.5 in Chapter VIII that a Latin square of order n is an n x n array with entries 1, ... , n, such that each of these n elements appears exactly once in each row and each column. The corresponding 4 x 4 Latin square for our tyre experiment is given in Table 3.1. The formal method underlying this solution is taken from the theory of analysis of variance. It is described in Section 3.2.
3.2. Testing Three Qualitative Factors without Ihteraction Let Xijl be stochastically independent random variables with normal distribution D(Xijl)
= N(/-L
+ Vi +Kj + PI, ()2),
with (/-L,
V;, Kj, PI, ()2) E]R4 X ]R+,
i = 1, ... , m; j = 1, ... , m; l = lei, j), m 2: 3 where lei, j) describes a Latin square of order m. This property is obviously symmetric in the variables i, j and l so that summing over i for fixed j is equivalent to summing over the range of l.
876
XIII. Applications of designs
The mean vector a can be written in the form (3.2.a)
a -- fA' lie •..
+ '" L-t v·e· + '" i...J K·e J .J.. + "'Pie ..I , l
~
l..
I
j
with ei .. := Lj LI eijl, where eijl represents the unit vector of]Rn with a 1 at position (i, j, I) describing whether a treatment is or is not applied to a unit; e.j., e ..1 and e ... are defined in a similar way. This means that different treatments lead to additive effects during the experiment. In Equation (3.2.a) J1,e ... represents the effect of treatments which are performed for all experimental units, whereas Li Viei.. describes the effect based on differences in the treatments on a subset of units. The so-called averaged treatment effect J1,e ... is chosen so that the means V., K., p. of Vi, K j and PI are equal to O. Since the vector ei .. fixes one of the m dimensions, Li Viei .. spans an (m - I)-dimensional subspace of]Rn and because of the orthogonality of ei .. , e.j. and e ..I, the (k = 3m - 2)dimensional subspace of]Rn described by Equation (3.2.a) can be divided into three three-dimensional and one one-dimensional pairwise orthogonal subspaces (cf. Witting and Nolle 1970). A linear hypothesis H, which covers an h = 2m - 1 dimensional subspace of]Rn , and its alternative hypothesis K, which covers a k-dimensional subspace, can now be given by (3.2.b)
H : Vi = ... = Vm
= 0,
K:
I>l > o.
The estimator T*(x) is determined by a variance analysis which leads to
(3.2.c)
T* (x)
_l_m "(x. - X )2 = _---,;--_. . . . ._:.::m'--.:,.l--.:L..-=i~_I .._ _.._._ _ _ _ __ (m_l)1(m_2) Li Lj L/Xijl - Xi .. - X.j. - X .. I + 2x.. J2'
where dots in the index of the mean value x represent sums over the corresponding variables (e.g. Xi .. = L j LI Xijl), cf. Equation (2.6.b). The numerator can be interpreted as the projection of x onto the space of the hypothesis normalised by its dimension h = m - 1. Similarly the denominator can be viewed as the normalised projection of x onto the n - k = (m - I) (m - 2) dimensional space of the alternative hypothesis; cf. Equations (2.2.g) and (2.2.h). To prevent systematic bias, the Latin square is randomly chosen from the set of all possible Latin squares. The following two examples show the necessity of using layer families of Latin squares, so-called orthogonal arrays, for the design of experiments.
§3. Latin squares and orthogonal arrays
877
3.3. Four Factors on Four Tyres Suppose in addition to the situation given in Example 3.1 four differen t treads W, X, Y and Z (see Figure 3.1) are available for every brand, which could also influence the results of the experiment. So the treads have to be taken into consideration. On the one hand, a Latin square could be constructed ignoring the dependency on the car brand; on the other hand, the wheel position could be ignored. To overcome this situation two Latin squares have to be found which can be superimposed in such a way that all combinations of treads and tyre brands are considered. This leads naturally to the concept of orthogo nal Latin squares, as provided by the notion of orthogonal arrays in Chapter VIII.
w
~
W w
X
I
Y
Z
tt
~
~ yy
Figure 3.1. Correspondence between tread names and treads.
So, one Latin square corresponds to a parallel class in a column- row partition ed scheme. Every element is connected with another element not from the same row or column, a relationship which we could call a block decomposition, as described in the previous example of tyre positions. Another orthogon al Latin square would then allow for the putting in a different setting of an orthogon al factor, where no two units that have already been tested will be tested together on the same object. By contrast this, the total dimension of projectio ns carried out on the Hilbert space to estimate linear regressions, can be increase d as described by Pukelsheim (1993), see Chapter 1.22 f. Thus a sequenc e of mutually orthogonal Latin squares can be used to approximate the balanced incomplete block design experiment that comes from an affine plane, as describe d in Chapter Vill.4. Going back to Example 3.3, the Latin square in Table 3.1 is chosen in a Z4 circulant form. By Corollary X.l2.4, this Latin square has no transver sal and hence no orthogonal mate. Therefore, to solve the four-tyre problem for the orthogonal feature "tread" as well, different Latin squares (which are based on EA(4» have been chosen, corresponding to the parallel classes of lines in AG(2,2).
XlII. Applications of designs
878
It should be noted that the different behaviour of the constructions given in
Tables 3.1 and 3.2 is a consequence of the underlying algebraic structures. As this example shows, the choice of these algebraic structures can determine failure or success of an experimental design.
Table 3.2. Latin square for testing tyres with attention to different treads BMW Front left Front right Back left Back right
W Michelin Y Goodyear Z Conti X PireIIi
Mercedes Goodyear Michelin Y Pirelli W Conti
X Z
Audi
Porsche Y Conti W Pirelli X Michelin Z Goodyear
Pirelli Conti W Goodyear Y Michelin Z
X
In our example, obviously not all combinations between tread, brand of tyres, car type, and wheel position are tested. For this reason, we call the experiment incomplete. But each brand of tyres is tested with each tread exactly once, so that the experiment is called balanced. Special sets of MOLS, which are generated using abelian groups, correspond to translation nets (see Section X.9). Bailey and Jungnickel (1990) mentioned these for statistical applications which naturally lead to the use of group theory. In Chapters VIII and X, conditions for the existence and construction of sets of mutually orthogonal Latin squares are given. Most recently another application of design theory and algebra to the construction of higher factorial designs has been presented by Robbiano (1998). He describes the set of treatments of a full factorial design as a direct product of several factors, each factor being a finite discrete rational variety, of say integers. The defining ideal of this variety can be constructed as being generated by the canonical multinomial basis (almost analogously to Equation (6.2.d)). The problem of the design of experiments (DoE) to build up all the necessary inferences, cf. Yates (1937), from smaller fractions of the full design, cf. Street and Street (1987), such as sets of MOLS, is here reduced to solving an ideal theoretic problem by the use of Grobner bases, cf. Buchberger (1985). With Robbiano's approach another door to research in design theory may have been opened, as methods from algebra and geometry become more and more applicable through the use of advanced computer algebra packages, cf. Section 7.4.
880
XIII. Applications of designs
3.4. DNA Analysis A high-tech application of orthogonal Latin squares has been used recently in the design of so-called gene chips. By exposing a carrier plate (1 cm2 of specially coated silicon, "blank") to an on/off array of light patterns M while lying in a solution of one of the few nucleotides N E {adenine, guanine, thymine, cytosine} a 400 x 400 matrix of microplots of 25 /-tm x 25 /-tm with entries left blank or the nucleotide attached to the underlying surface can be formed. If subsequently alternating the patterns M and the choice of the nucleotide N, a 400 x 400 matrix of nucleotide chains (D ij )i,jE[I...400] E {A,G,T,C}* can be built up; cf. Figure 3.2. Each Dij is designed to pair with a special substring of a DNA sequence, due to the wellknown complementary docking reaction A ~ T and C ~ G in the genetic code. The position of an instance of a docking reaction is then read out by laser light of suitable wavelength giving fluorescent reactions at the special binding sites. With a suitably chosen orthogonal array both the production of the gene chips and the read-out pattern can be optimised.
§4. Application of Designs in Optics Amongst the many applications of design theory to the development of electronics communications, computer technology and optics which have been reported during the last three or four decades, one of the most surprising and successful examples is that of Hadamard transform optics used in the design and construction of modem imaging systems. In this chapter we dedicate a short section to this topic by presenting two examples. Firstly, even though the book Hadamard Transform Optics by Harwit and Sloane (1979) became famous amongst insiders after its publication about two decades ago, the results contained in this book seem not to have been adequately recognised in the areas of discrete mathematics, statistics and communications engineering. Even in optics it has remained a rather specialised topic, which is mostly used in the area of digital optics and advanced image reconstruction techniques in coherent optics, laser optics, speckle holography, etc. (Cf. Beth, Klappenecker et al. 1995.) As the intrinsic use of balanced incidence structures (i.e. designs) is a key tool in these applications, we give a brief summary of this application. The key to Hadamard transform techniques in optical processing and transform techniques lies in the fact that the signal-to-noise ratio of receivers and
§4. Application of designs in optics
881
measurement instruments can be enhanced considerably by applying special orthogonal transforms. As shown in Table 4.4 below, the principle idea behind these techniques for extracting information from quantities which are composed of several signal functions can be demonstrated easily by the example of weighing designs, which we define as follows.
4.1. Improving the Signal-to-Noise Ratio by Hadamard Matrices A number of physical quantities1{rl, ... , 1{rn, henceforth called objects (with unknown weights), are to be measured by a measuring device (4. 1. a)
P:1{rI-+(j"
which we call a weighing scale with output values or measurement (4.1.b)
disturbed by errors ej. We assume the stochastic process e = (ei)i, i E I, to be independent of the actual weight 1{rj and identically independently distributed with (4.1.c)
Vi
E
I:
E(ei)
=0
and
V(ei)
= 0- 2 •
If a weighing scale could measure the objects 1{r1, ... , 1{rn separately in a straightforward manner then, even though the mean error distribution of the actually measured values (j, is (4.1.d)
the mean square error in each scale equals (4. 1. e)
Thus the signal-to-noise ratio, defined by (4.l.f) is equal to the energy of the j -th object divided by the noise variance, that is
(4.l.g)
SNR(j) = 1{rJ2 . 0-
XIII. Applications of designs
882
This means that the signal quality is reduced by an average factor of degradation of 1 - ~ or, as in the language of communications engineering, going to a logarithmic scale the signal-to-noise distance is (in dB) (4.1.h) Shannon (1948) proved a celebrated result about the capacity of the communication channel, which in our case reads that the information transmission rate through a noisy channel can be increased as close as possible to channel capacity
(4.1.i)
1 C = -log2(1
2
bit ], + SNR) [measurement
with arbitrary small error by the use of random codes (cf. Chambers 1985 and McEliece 1977). This is, of course, related to the ideas of Yates (1935) who demonstrated that the mean square error can be reduced, by applying weighing strategies of several objects at once, i.e. by putting objects together in blocks. In this way the signal-to-noise ratio can be improved considerably, cf. Tables 4.3 and 4.4 below, as exemplified in the following. Suppose the same set of n physical quantities described above can be weighed in groups using a chemical balance having two scale pans, so that the right and the left pan action can be expressed by plus and minus signs. For the sake of simplicity we consider the case n = 4 to describe a weighing strategy. We call this a weighing design. Objects weighted with a plus sign have to be put into the left-hand side pan; objects weighted with a minus sign have to be put into the right-hand side pan. Additional standard weights 11i have to be added to the up going pan until balance is achieved. In this way we get 11 (111, ... , 114)T with
=
(4.1.j)
111 =
lh + 0/2 + 0/3 + 0/4 + e1
+ 0/3 - 0/4 + e2 + 0/2 -0/3 -0/4 + e3 0/1 -0/2 -0/3 + 0/4 + e4·
112 = 0/1 - 0/2 113 = 0/1 114 =
The weighing matrix W of this design is given by
(4.1.k)
W =
(~-~ 1
1 1 -1
1 -1
-~)
-1
-1
1
L
883
§4. Application of designs in optics
which is obviously a Hadamard matrix H4 of order four, hence W- I small calculation shows that for the vector (4.1.1)
=
~ W. A
I] = W1jr +e
the quantities of interest (4.1.m)
,
1 4
1jr = -WI]
are of the form (4.1.n)
, 1jr
1
== 1jr + -We. 4
For the mean value of the various {r j so derived, we obtain a mean square error (4.1.0)
which leads to an improved signal-to-noise ratio (4.l.p) Hence using a Hadamard matrix Hn improves the signal-to-noise ratio by a factor of ../ii, which in our example equals two, and this gives an improvement in dB by a factor of 10../ii = 20. Note that this improvement was obtained just by using a different weighing matrix W = Hn , instead of the trivial weighing matrix W = In. This shows that the use of designs is not only suitable for reducing the number of experiments while maintaining the same fidelity (as shown in the examples in Sections 2 and 3), but also will give a highly improved fidelity if the appropriate measurement strategy (i.e. weighing design) is chosen. The main idea behind this method of analysis of variance is based on blocking techniques as pointed out by Yates (1935). It is the nowadays well-understood idea of developing signal systems into suitable sets of orthogonal functions which has led to the theory of signal processing and fast signal transforms (cf. Section 5.4 of Blahut (1985) and Beth (1984». Special modulation techniques in communication engineering led to the widely known principles of soft decoding, etc. (cf. Clark and Cain 1982; Wozencraft and Jacobs 1965; Lazic and Senk 1992). These, after all, resemble an application of design theory which has been considered in the area of coding theory, see Section 5, where the codes generated by incidence matrices of designs have been studied; see Section 11.11 and Assmus and Key (1992a). In piuticular, the theory of modular codes and
XIII. Applications of designs
884
the associated modular representation theory plays an important role obtaining non-trivial combinatorial and algebraic results which can be applied in various settings such as, for instance, tomography; see Section 6.
4.2. Weighing Designs and Hadamard Transform Optics
Extending the weighing pattern of Example (4.1.j) to n = 8, (4.2.a)
T/I = VI + V2 + V3 + V4 + V5 + V6 + V7 + V8 + el ~=~-~+~-~+~-~+~-~+~
~=~+~-~-~+~+~-~-~+~ ~=~-~-~+~+~-~-~+~+~
~=~+~+~+~-~-~-~-~+~ ~=~-~+~-~-~+~-~+~+~ m=~+~-~-~-~-~+~+~+~ ~=~-~-~+~-~+~+~-~+q
the weighing matrix W becomes 1
-1
(4.2.b)
W=
1
-1
-1
-1
-1
-1
-1
1
-1
-1
-1
-1
-1
-1
1 -1
1
-1 -1
1
-1
-1
1
-1
-1
-1
-1
-1
-1
1 -1
1
-1
1
-1
1
1 -1
Even though the incidence matrices of designs over fields of characteristic zero are algebraically of no interest, since they have full rank, for decoding purposes they have been successfully used for the so-called Reed-Muller codes in early Mars missions, cf. Section 5.4. However, in signal processing square matrices of full rank, especially unitary matrices, play an important role for signal reconstruction as the above examples show. The following example does not use a (1, -1)-generalised incidence matrix of a Hadamard design, but rather a proper (O,1)-incidence matrix, which will apply to the same problem in a surprising manner.
885
§4. Application of designs in optics
If one uses a spring balance, which just weighs the total weights hung on it, the corresponding weighing matrix is obviously an (O,I)-matrix if we put those elements that have been hung on the spring at each weighing instance into common blocks. Thus they define an (O,I)-incidence matrix of a weighing design which in the case of n = 7 given below is the incidence matrix S7 of a PG(2, 2). This is equivalent to the matrix obtained from the Hadamard matrix Hg (see Example (4.2.b) above) by omitting the first row and the first column of H8, and changing the entries +1 into 0 and the entries -1 into I. After suitable column and row permutations, S7 becomes a circulant giving the measurements (4.2.c)
1J
= S71/r +e.
Note, that
-1 1 -1 (4.2.d)
•
-1
1/r=S7 '1J=
-1
-1
-1 -1
1 -1
1 -1 -1
-1
1 -1
1
1
-1
-1
1 '1J 1 -1
-1
-1
-1 1
-1
-1
-1
holds. Owing to the fact that the Hadamard matrix
H2k
1
-1 is obtained via the
AG(k,2) design related to the corresponding projective geometry with Singer
cycle, cf. Section 5.2, Sn and S;;' show a similar circulant structure, which can successfully be applied to optical technology. The properties of the optical design matrices are obtained by a theorem due to Harwit and Sloane (1979), which we summarise as follows. (4.2.e)
Theorem.
Hadamard matrices make the best chemical balance weighing designs, or the best masks when entries + 1, 0 and -1 can be used (Table 4.1). S-matrices make the best spring balance weighing designs, or the best masks when entries 0 and 1 can be used (Table 4.2).
If there is a number n of unknowns for which a Hadamard matrix of order n exists, then a weighing matrix ofHadamard type reduces the mean square error in each unknown by a factor of n. If the number of unknowns is nand an S-matrix of order n exists then the corresponding weighing S-matrix for
§4. Application of designs in optics
887
is, corresponding to (-1, 1) and (0, I)-matrices) are derived naturally. For instance, the example of (0, I)-matrices corresponds to a mask brought into the optical axis of an imaging system (cf. Figure 4.1), where an entry 0 describes a non-transmissive element at the corresponding matrix position, while an entry 1 means a fully transmissive hole in a so-called amplitude mask. While these masks are amplitude elements, cf. Section 9, the behaviour of phase elements - in which -I would correspond to a phase shift by Jr - is discussed e.g. by Aagedal, Beth, et al. (1996) for applications in diffractive optics or by Schroeder (1986) for acoustics.
. \. . . . . . . . . ~.,,:::::-....,".. . . ..
:.\
'· ·. · .Jti:;)\ I //'
1
·1
pinhole
...... _"::.
~.,::\.....
. . i/><.·. .:.ir.·:·.·.I.·:·::::::::>4I '
phase grating
",
I '-/
Hadamard mask
detector
Figure 4.1. Spectrometer with a Hadamard mask. From the left the light which has
to be analysed passes the pinhole and the first collimating lens and is dispersed by the phase grating. The Hadamard mask selects certain frequencies, the intensity of which is measured in the detector. From the preceding example, we see that a weighing design with a large symmetry group can be helpful in the experimental set-up. The S-matrices corresponding to cyclic projective planes (cf. Section VI.7) can easily be set up as a rotating "movie" of (0, I)-slits which correspond to a periodic (0, I)-sequence with period n -1 as compared to the (17 -1) X (n -1) = 0(n2) mask positions needed if a matrix is applied. So, there is either a space division or time division mUltiplexing, which leads to optimal measurements. Historically the first applications, e.g. in medical imaging, are described in the book by Harwit and Sloane (1979) in Chapters 3, 4 and especially 7. Generalised weighing matrices have been studied by Berman (1978). The actual details are of a rather technical nature and the reader is referred to the original source cited there.
4.3. Hadamard Transform Imaging in Raman Spectrometry The second reason for elaborating on the Hadamard transform in this book is due to rather recent developments in the area of quantum optics, which have gained some importance in applications of high precision measurement, where digital methods of the described type seem likely to playa role. These methods also
§4. Application of designs in optics
887
is, corresponding to (-1, 1) and (0, I)-matrices) are derived naturally. For instance, the example of (0, I)-matrices corresponds to a mask brought into the optical axis of an imaging system (cf. Figure 4.1), where an entry 0 describes a non-transmissive element at the corresponding matrix position, while an entry 1 means a fully transmissive hole in a so-called amplitude mask. While these masks are amplitude elements, cf. Section 9, the behaviour of phase elements - in which -I would correspond to a phase shift by Jr - is discussed e.g. by Aagedal, Beth, et al. (1996) for applications in diffractive optics or by Schroeder (1986) for acoustics.
. \. . . . . . . . . ~.,,:::::-....,".. . . ..
:.\
'· ·. · .Jti:;)\ I //'
1
·1
pinhole
...... _"::.
~.,::\.....
. . i/><.·. .:.ir.·:·.·.I.·:·::::::::>4I '
phase grating
",
I '-/
Hadamard mask
detector
Figure 4.1. Spectrometer with a Hadamard mask. From the left the light which has
to be analysed passes the pinhole and the first collimating lens and is dispersed by the phase grating. The Hadamard mask selects certain frequencies, the intensity of which is measured in the detector. From the preceding example, we see that a weighing design with a large symmetry group can be helpful in the experimental set-up. The S-matrices corresponding to cyclic projective planes (cf. Section VI.7) can easily be set up as a rotating "movie" of (0, I)-slits which correspond to a periodic (0, I)-sequence with period n -1 as compared to the (17 -1) X (n -1) = 0(n2) mask positions needed if a matrix is applied. So, there is either a space division or time division mUltiplexing, which leads to optimal measurements. Historically the first applications, e.g. in medical imaging, are described in the book by Harwit and Sloane (1979) in Chapters 3, 4 and especially 7. Generalised weighing matrices have been studied by Berman (1978). The actual details are of a rather technical nature and the reader is referred to the original source cited there.
4.3. Hadamard Transform Imaging in Raman Spectrometry The second reason for elaborating on the Hadamard transform in this book is due to rather recent developments in the area of quantum optics, which have gained some importance in applications of high precision measurement, where digital methods of the described type seem likely to playa role. These methods also
888
XIII. Applications of designs
play an important part in communications and computing. In current technology optical transmission on high-speed glass fibre links relies on advanced methods of digital optics and optical switching. In the future, optical computers will have to rely on high-precision optical measurements, especially in the newly developing area of quantum computing which has generated much interest recently (cf. Bollinger et al. 1996). Most of the physics and computer science aspects of quantum computing is concerned with the design of fast algorithms and the error-correcting stabilisation of quantum states (cf. Calderbank and Shor 1996 and Beth and Grassl 1996) based on methods similar to those described here. The problem of measuring quantum states in a suitable way is crucial and will remain so for fundamental reasons. In this context measurement techniques which stem from advanced imaging processes may play an important role. A recent publication (cf. Hammaker, Bohlke et al. 1996) gives a summary of the usage of Hadamard transform imaging techniques in quantum optics. A very recent application is in Raman spectrometry where, following a drawing from Hammaker, Bohlke et al. (1996) in Figure 4.2, a Hadamard transform spectrometer is being used. Here it can be seen that the movable mask of the cyclic rotating film gives the different measurements in the detector summing the values up according to the S7 matrix. Raman spectrometry is not only of high interest in advanced spectroscopy for the purpose of physics, but forms the decisive tool for high-precision time measurements. These will be available with the next generation of high precision quantum mechanical clocks, which are replacing the well-known atomic clocks currently used. With the large market for such precision measurements, this will be a basis for a new industry of II detector
6
movable mask
II 12 13 14 Is 16 17 DI = (1) II
+ (I) 12 + (1)13 + (0)14 + (1) Is + (0)16 + (0)17 + E
Figure 4.2. HT spectrometer (taken from Hammaker, Bohlke et al. 1996), where the shaded areas do not permit the radiation to pass through the slit. Therefore the intensity reaching the detector is equal to II + h + 13 + IS plus an additional noise E.
889
§4. Application of designs in optics
Table 4.3. Weighing designs and optical multiplexing (taken from Hammaker, Bohlke et al. 1996) Instruments and characteristics Balance Optical instrument Weighing design matrix (w is an integer) Matrix elements
Dual-detection
Single-detection
Double pan Two detectors Hadamard matrix nxn n =2W G) entries + 1
Single pan Single detector S-matrix NxN N =n -1 = 2w-l (Nil) entries +1 (N-I)
(1) entries -1
2
..(ii
Increase in SNR
entries 0
(N+I) "'"
2.jN
:IJ!.. 2
frequency standards and time measurements, where the fidelity of precision to as compared to the present-day technology (cf. will be improved from Bollinger et al. 1996), where N is the number of particles involved.
IN
-k
That these experimental conditions will lead to slightly more involved set-ups than just the use of (0, 1)-designs can be seen in the article by Hammaker, Bohlke et aI. (1996), p. 194; cf. Table 4.3. Here the hysteresis behaviour of switchable optical elements or the hysteresis of the measuring picture processing unit leads to patterns of matrices, where the variables change between 0 and 1 according to a fractional transmissiveness as indicated in Figure 4.3. As these are still
DI
I
D2
!
D3
I
1 1
0
0
0 4 D -
... ..--0
I11III
0
0 0 ",v-
Ds ~I~~~~I o 0
0
0
0
1
1
1
I I
I
110
__IL-~"a. o
1.11
•• 010
Figure 4.3. Schematic demonstration of the effect of finite temporal response of the mask on the distortion of the weighing design using a left circulant Srmatrix (taken from Hammaker, Bohlke et al. 1996).
Table 4.4. Comparison o/various encoding schemes. Notation: HTS = Hadamard transform Spectrometer, M = total number measurements, p = total number o/unknowns, e = average mean square error in the estimates, (j2 = variance, SNR = root mean square signal-to-noise ratio, cf Sections 4.1,4.2. (Taken/rom Harwit and Sloane 1979.) Name
Encoding
Masks
M
p
Equation
Monochromator
None
None
n
n
T/ =1fr+e
2
HTS (or imager)
Single
Hn
n
n
3
HTS
Single
On
n
4
HTS (or imager)
Single
5
HTS
Single
&
;;r
SNR
T/
1
.,fo
n
= Hn1fr+e T/ = On1fr +e
S"
n
n
T/=S,,1fr+e
n 1n
Hm,n
m
n
T/
6
HTS
Single
Sm.n
m
n
= Hm •n1fr +e T/ = Sm.n1fr + e
7
HTS, 2 detectors
Single
Hn
2n
n
T/=W1fr+e
8
Imaging spectrometer
Double
Hm,HlI
mn
mn
9
Imaging spectrometer
Double
Sm, Sn
mn
mn
= Hm 1frH'[ +e T/ = Sm1frSI +e T/
11
~
fi .Jri
1..
,fiii
m
.i
m ~ n
2
Fm
'2
fi
...L
mn
-/mil
1§.
../mii
mn
4
e
Name
Encoding
Masks
M
p
Equation
10
S-matrix grill spectrometer
Double
Sn, Sn
n
n
TJ = SnSI1fr
+e
nr
11
Double encoded HTS
Double
Sm, Sn
n
n
TJ= USI1fr+ e
1§.
12
Double encoded HTS
Double
Sn, SIl
n2
n
TJ =
SIl1frS;f +e
nr
13
Double encoded HTS
Double
Sm, Sm
m2
n
TJ = Sm1frS,~ +e
1§.
14
Interferometer 1 detector
Fourier
None
n
n
TJI(n =
~
11fro -
16
mn 22
Hooo
SNR
n
<[
.;mn
-4-
Fn
4.7
.;mn
mn
-4-
Jl. n
[i
1fr1 (v) cos(41fv~) dv 15
Interferometer 2 detectors
Fourier
None
n
n
i
16
Interferometer wide aperture
Fourier
None
n
n
m'n
Fn
n
"2
8
m[i
XIII. Applications of designs
892
circulant matrices, identities from group algebras (cf. Chapter VI) provide a suitable tool to cope with these problems by designing the optical masks accordingly to cover up the hysteresis behaviour in advance. Such problems have to be solved for each application separately, but the pattern of applying the circulantdiagonal Fourier correspondence, cf. Jungnickel, Beth and Geiselmann (1994), which initially had been described by MacWilliams and Sloane (1977) in the context of coding theory, is extensively used in the area of diffractive optics, where special mapping set-ups occur (Aagedal, Beth et al. 1996).
4.4. Preparing Quantum States by S-matrices Even though this example is very special it may be worthwhile to mention it for historical reasons, as very recently an application of H / S-matrices was presented by Laflamme, Knill and Zurek (1997). In the historically first experiment to prepare a so-called GHZ-state, a triply entangled quantum state generalising a Schrodinger cat state (cf. Rae 1986), they have applied the regular action of the Singer cycle on SrI being the derived matrix of H2k for n = 2k - 1, to a nuclear magnetic resonance (NMR) experiment, carried out on 4 July, 1997 at Los Alamos National Laboratories. The resulting state (4.4.a)
IGHZ)
1
i
= .J2IOOO) + .J21111)
has been measured in a corresponding density matrix p, cf. Mahler (1995), which was directly decomposed into a signal matrix PGHZ of an approximately pure state and a noise matrix PNoise obtained by averaging the noise effects by a quantum group representation of the Singer cycle acting regularly on the 2k - 1 remaining components.
§5. Codes and Designs The close interrelations between design theory and coding theory have provided an important and successful tool in mathematics and its applications. An extensive account of this connection is given in the book by Assmus and Key (1992a); see §II.ll for an introduction. While the purely combinatorial set-theoretic approach to the construction and description of designs has considerably profited from the representation theoretic methods of coding theory, there has been an enormous impact of geometry and design theory on the theory
893
§5. Codes and designs
and practice of coding and communications engineering. For this we refer to the books by MacWilliams and Sloane (1977), and Lin and Costello (1983). In the introduction (Sections S.1-S.3) we will illustrate the multiple aspects of coding theory and features of geometric designs and of experimental designs.
5.1. Basic Notions of Error-Correcting Codes In theory, an error-correcting code C is a k-dimensionallinear subspace of a vector space lFn, n = k+r (k, r EN). The systematic encoder can be described by a matrix G : lFk -+ lFn, the elements of lFk are called messages, the elements of C = im(G) are called codewords, obtained from the messages by appending r parity check coordinates. The characteristic parameters of this code C are its rate R = ~ and its minimum weight w, defined by
= min {wgt(c) ICE C, c =1= OJ,
(S.1.a)
w
(S.1.b)
x=
(S.1.c)
supp(x) := {i E [0, ... , n - I]:
(XO, ... , XII-I)
E lFn,
where for
wgt(x):= Isupp(x)l, Xi
=1=
with
O}.
In constructing error-correcting codes, besides solving the parametric optimisation problem to maximise the rate of C and its minimum weight it is most important to design an efficient decoding algorithm at the same time. In principle, a decoder can be built from the fact that the dual code Cl. of C, being the (n - k)-dimensional subspace oflFn orthogonal to C, is generated by any matrix (S.l.d)
H : lF n -
k
-+ IF''
with
im(H)
= Cl..
Obviously we have (S.1.e)
C = kerHT
and (S.U)
G·H T =0.
Any codeword c E C is transmitted through a channel, which is assumed to add the so-called error vector e E lFn to the codeword so that a noisy vector u = c + e is received. From the syndrome (S.1.g)
§5. Codes and designs
895
Hr = kercF(2)H; where Binary(l)
(S.2.a)
Binary(j)
Binary(2r - 1)
is the (O-l)-matrix of rows of the binary expansions of the numbers j E [1, ... , 2r - 1]. Obviously any two of these are linearly independent over ]F2, thus providing a 1-error-correcting code of rate R = ~ = 1 - 2' ~! ' as the rank ofH;isr. The specification of a l-error-correcting decoder for this code is a direct consequence of Equation (S.2.a). This is a prime example of so-called "automatic algebraic algorithm engineering" (A3E), cf. Beth, Klappenecker et al. (1995). If the transmitted codeword c of length n arrives at the receiver side as It with at most one erroneous bit in an unknown position, i.e. (S.2.b)
u
= c +e
with
wgt(e):::: 1,
then obviously the syndrome is given by (S.2.c)
uH T =eH T = r
r
o {Binary(l)
if no error occurred, if I is the error location.
Thus the simple circuit given in Figure 5.1 computes the correct codeword c = u EEl el wheree/, the unit-vector with a I-entry atthel-thposition, is provided
as output of the (l-n)-demultiplexer upon input s.
c
Figure 5.1. Decoder for Hanmling code.
The beauty of this construction becomes fully apparent when the row vectors of HrT are interpreted as the projective coordinates of the points of PG(r I, 2)
§5. Codes and designs
895
Hr = kercF(2)H; where Binary(l)
(S.2.a)
Binary(j)
Binary(2r - 1)
is the (O-l)-matrix of rows of the binary expansions of the numbers j E [1, ... , 2r - 1]. Obviously any two of these are linearly independent over ]F2, thus providing a 1-error-correcting code of rate R = ~ = 1 - 2' ~! ' as the rank ofH;isr. The specification of a l-error-correcting decoder for this code is a direct consequence of Equation (S.2.a). This is a prime example of so-called "automatic algebraic algorithm engineering" (A3E), cf. Beth, Klappenecker et al. (1995). If the transmitted codeword c of length n arrives at the receiver side as It with at most one erroneous bit in an unknown position, i.e. (S.2.b)
u
= c +e
with
wgt(e):::: 1,
then obviously the syndrome is given by (S.2.c)
uH T =eH T = r
r
o {Binary(l)
if no error occurred, if I is the error location.
Thus the simple circuit given in Figure 5.1 computes the correct codeword c = u EEl el wheree/, the unit-vector with a I-entry atthel-thposition, is provided
as output of the (l-n)-demultiplexer upon input s.
c
Figure 5.1. Decoder for Hanmling code.
The beauty of this construction becomes fully apparent when the row vectors of HrT are interpreted as the projective coordinates of the points of PG(r I, 2)
897
§5. Codes and designs Bach of the codewords
Co, •.• , Cl3
represents an incidence vector of a line in
PG(2, 2) or its complement. In geometric language, these codewords Ci can be
characterised by the property that (5.2.f)
'v'C
E
H.3
38 E
{O, I}
"Ii E [0, ... ,3]: Isupp(c) n sUPP(11i ))1
== 8
mod 2;
in other words (5.2.g) H.3 = Co U CI with (5.2.h)
Co
= {c 1"Ii
E
CI
= {c 1"Ii
E [0, ... , 3] : (c
[0, ... , 3] : C ·li
= 0 mod 2}
and (5.2.i)
+ 1) ·Ii = 0 mod 2},
where x . y denotes the standard inner product of x and y. This property is extensively used in Section 5.4 for the construction of Reed-Muller codes, cf. also the properties of H- and S-matrices in Section 4. Also note that the code Co ::: = H.3, a property which has recently proved to be essential for the construction of quantum error-correcting codes, see Section 5.7, Calderbank, Rains et al. (1996, 1997a), Beth and Grassl (1996).
ct
Thus the Hamming code H.3 is the IF 2-row space of the incidence matrix of PG(2, 2). This can be used for a geometrical decoding procedure, which we
demonstrate by the example given in Figure 5.2. Suppose in the codeword (0,1,0,1,1,1,0) (Figure 5.2(a)) the first bit was changed during transmission (see Figure 5.2(b)). To remove the error the decoder has to obtain the same parity (5.2.h) or (5.2.i) by changing at most one point in supp(u). Figure 5.2(b) shows that the lines 11,/2 ,13 define a majority of consensus 0 already, so that only the intersection with 10 has to be corrected. The only possibility is to remove the incidence of point Po. In an obvious manner this example carries over to a general decoding procedure. Decoding procedures and further applications of this geometric type (cf. Beth and lungnickel 1982; Beth 1993) will be discussed in Section 5.6.
5.3. Codes in Signal Space In practice, however, the use of error-correcting codes (BCC) cannot be based solely on the combinatorial model described above. In real channels, codewords are modulated to form signals on a physical transmission line, where they are exposed to an additive stochastic noise process.
897
§5. Codes and designs Bach of the codewords
Co, •.• , Cl3
represents an incidence vector of a line in
PG(2, 2) or its complement. In geometric language, these codewords Ci can be
characterised by the property that (5.2.f)
'v'C
E
H.3
38 E
{O, I}
"Ii E [0, ... ,3]: Isupp(c) n sUPP(11i ))1
== 8
mod 2;
in other words (5.2.g) H.3 = Co U CI with (5.2.h)
Co
= {c 1"Ii
E
CI
= {c 1"Ii
E [0, ... , 3] : (c
[0, ... , 3] : C ·li
= 0 mod 2}
and (5.2.i)
+ 1) ·Ii = 0 mod 2},
where x . y denotes the standard inner product of x and y. This property is extensively used in Section 5.4 for the construction of Reed-Muller codes, cf. also the properties of H- and S-matrices in Section 4. Also note that the code Co ::: = H.3, a property which has recently proved to be essential for the construction of quantum error-correcting codes, see Section 5.7, Calderbank, Rains et al. (1996, 1997a), Beth and Grassl (1996).
ct
Thus the Hamming code H.3 is the IF 2-row space of the incidence matrix of PG(2, 2). This can be used for a geometrical decoding procedure, which we
demonstrate by the example given in Figure 5.2. Suppose in the codeword (0,1,0,1,1,1,0) (Figure 5.2(a)) the first bit was changed during transmission (see Figure 5.2(b)). To remove the error the decoder has to obtain the same parity (5.2.h) or (5.2.i) by changing at most one point in supp(u). Figure 5.2(b) shows that the lines 11,/2 ,13 define a majority of consensus 0 already, so that only the intersection with 10 has to be corrected. The only possibility is to remove the incidence of point Po. In an obvious manner this example carries over to a general decoding procedure. Decoding procedures and further applications of this geometric type (cf. Beth and lungnickel 1982; Beth 1993) will be discussed in Section 5.6.
5.3. Codes in Signal Space In practice, however, the use of error-correcting codes (BCC) cannot be based solely on the combinatorial model described above. In real channels, codewords are modulated to form signals on a physical transmission line, where they are exposed to an additive stochastic noise process.
§5. Codes and designs
899
retrieves the vector u = (Xl + Z\, ... ,Xn + Zn), which a posteriori is observed as the output of a Gaussian channel of noise variance (52 = ~o and input energy s: per symbol. The capacity of this channel is given by -
(5.3.g)
C=
1
(
ST ) [
2" log2 1 + 2 nNo
bits ] symbol'
cf. Chambers (1985) and McEliece (1977). Taking the channel bandwidth W into account (based on the sampling theorem, cf. Babovski, Beth et al. 1987; Blahut 1992), which allows transmission of if = 2W symbols per second under the assumption that the total noise power be constrained to No W we obtain
(5.3.h)
C = W log2 ( 1 +
~).
This is Shannon's (1948) celebrated capacity formula. We conclude this elementary introduction by mentioning that the combined theories of combinatorial coding and communications engineering have produced in the last 50 years a wealth of increasingly involved and improved coding and modulation procedures, such as trellis and block coded modulations, concatenated codes and so-called turbo codes with soft decoding. These methods of matching combinatorial and modulation techniques, which are described in detail e.g. by Ungerboeck (1982), Blahut (1987), Beth, Lazic and Senk (1990), Bossert (1992), Lazic and Senk (1992) and Friedrichs (1995) are based on the interleaving of set theoretic and representation theoretic methods as outlined at the beginning of this chapter. The mathematical progress achieved in this classical context has on the other hand produced one of the most successful synergies between geometry, groups. sphere packing, lattice theory and number theory cf. the book of Conway and Sloane (1993), definitely driven by the original questions of design theory and coding theory. For the sake of completeness it must be mentioned that the advances stemming from this coevolution of discrete mathematics and communications engineering have provided high-level state of the art techniques in the design and production of consumer electronics, as well as in satellite communication and broadband multimedia technology, which have become commonly used worldwide.
§5. Codes and designs
899
retrieves the vector u = (Xl + Z\, ... ,Xn + Zn), which a posteriori is observed as the output of a Gaussian channel of noise variance (52 = ~o and input energy s: per symbol. The capacity of this channel is given by -
(5.3.g)
C=
1
(
ST ) [
2" log2 1 + 2 nNo
bits ] symbol'
cf. Chambers (1985) and McEliece (1977). Taking the channel bandwidth W into account (based on the sampling theorem, cf. Babovski, Beth et al. 1987; Blahut 1992), which allows transmission of if = 2W symbols per second under the assumption that the total noise power be constrained to No W we obtain
(5.3.h)
C = W log2 ( 1 +
~).
This is Shannon's (1948) celebrated capacity formula. We conclude this elementary introduction by mentioning that the combined theories of combinatorial coding and communications engineering have produced in the last 50 years a wealth of increasingly involved and improved coding and modulation procedures, such as trellis and block coded modulations, concatenated codes and so-called turbo codes with soft decoding. These methods of matching combinatorial and modulation techniques, which are described in detail e.g. by Ungerboeck (1982), Blahut (1987), Beth, Lazic and Senk (1990), Bossert (1992), Lazic and Senk (1992) and Friedrichs (1995) are based on the interleaving of set theoretic and representation theoretic methods as outlined at the beginning of this chapter. The mathematical progress achieved in this classical context has on the other hand produced one of the most successful synergies between geometry, groups. sphere packing, lattice theory and number theory cf. the book of Conway and Sloane (1993), definitely driven by the original questions of design theory and coding theory. For the sake of completeness it must be mentioned that the advances stemming from this coevolution of discrete mathematics and communications engineering have provided high-level state of the art techniques in the design and production of consumer electronics, as well as in satellite communication and broadband multimedia technology, which have become commonly used worldwide.
§5. Codes and designs
901
is converted into the real (row) vector (5A.b)
by replacing a O-entry in I by a I-entry in F and a I-entry in I by a ( -1)entry in F. Modulation is then achieved by transmitting the step function in the interval [0, 2m - 1] defined by 2171_l
(5.4.c)
F(t) =
L FBin(i)I[i.i+IJ(t). ;=0
Note that the sum (5.4.d)
fro =
10
2m
F(t) dt
of all coefficients of F is the number of O-components minus two times the number of I-components in I, thus (5A.e)
fro = 2m
-
2 wgt(f).
We shall illustrate this by the example of the first-order Reed-Muller code RM(1,3). The codewords of the first order Reed-Muller code RM(1, m) with m = 3 of length 8 can be regarded as incidence vectors of special subsets of points of AG(3, 2). The table (5A.f) 1 = (1,1,1,1,1,1,1,1)
corresponding to the whole space
II = (1, 0, 1,0,1,0,1,0) corresponding to the 2-flat (*, *,0)
12 =
(1, 1,0,0,1, 1,0,0)
corresponding to the 2-flat (*, 0, *)
13=(1,1,1,1,0,0,0,0)
corresponding to the 2-flat (0,*, *) with boolean functions fi
(X3, X2, XI)
= 1 + Xi
thus defines an (m + 1) x 2m generator matrix G. According to Equation (5A.a) this leads to the set of 16 codewords: (5.4.g) (0,0,0,0,0,0,0,0) (1,1, 1, I, 1, 1, 1, 1) (0,0,0,0,1,1,1,1) (1, I, 1, 1, 0, 0, 0, 0) (0,0,1,1.0,0,1,1) (1,1,0,0,1,1,0,0) (0,1,0,1,0,1,0,1) (1,0,1,0,1,0,1.0) (0,0,1,1,1,1,0,0) (1,1, 0, 0, 0, 0,1,1) (0,1,0,1,1,0,1,0) (1,0,1,0,0,1. 0,1) (0,1,1,0,0, I, 1,0) (1,0,0,1,1,0,0,1) (0,1,1,0,1,0,0,1) (1,0,0,1,0,1,1,0).
§5. Codes and designs
901
is converted into the real (row) vector (5A.b)
by replacing a O-entry in I by a I-entry in F and a I-entry in I by a ( -1)entry in F. Modulation is then achieved by transmitting the step function in the interval [0, 2m - 1] defined by 2171_l
(5.4.c)
F(t) =
L FBin(i)I[i.i+IJ(t). ;=0
Note that the sum (5.4.d)
fro =
10
2m
F(t) dt
of all coefficients of F is the number of O-components minus two times the number of I-components in I, thus (5A.e)
fro = 2m
-
2 wgt(f).
We shall illustrate this by the example of the first-order Reed-Muller code RM(1,3). The codewords of the first order Reed-Muller code RM(1, m) with m = 3 of length 8 can be regarded as incidence vectors of special subsets of points of AG(3, 2). The table (5A.f) 1 = (1,1,1,1,1,1,1,1)
corresponding to the whole space
II = (1, 0, 1,0,1,0,1,0) corresponding to the 2-flat (*, *,0)
12 =
(1, 1,0,0,1, 1,0,0)
corresponding to the 2-flat (*, 0, *)
13=(1,1,1,1,0,0,0,0)
corresponding to the 2-flat (0,*, *) with boolean functions fi
(X3, X2, XI)
= 1 + Xi
thus defines an (m + 1) x 2m generator matrix G. According to Equation (5A.a) this leads to the set of 16 codewords: (5.4.g) (0,0,0,0,0,0,0,0) (1,1, 1, I, 1, 1, 1, 1) (0,0,0,0,1,1,1,1) (1, I, 1, 1, 0, 0, 0, 0) (0,0,1,1.0,0,1,1) (1,1,0,0,1,1,0,0) (0,1,0,1,0,1,0,1) (1,0,1,0,1,0,1.0) (0,0,1,1,1,1,0,0) (1,1, 0, 0, 0, 0,1,1) (0,1,0,1,1,0,1,0) (1,0,1,0,0,1. 0,1) (0,1,1,0,0, I, 1,0) (1,0,0,1,1,0,0,1) (0,1,1,0,1,0,0,1) (1,0,0,1,0,1,1,0).
§5. Codes and designs
903
products (5.4.j)
feu) = (Wu 11jJ) =
lT
Wu (t)1jJ(t) dt
where for u E GF(2)'", Wu(t) is given by 2m
(5.4.k)
WuCt) = L(_l)u.Binary(j-J) ·lU-l,jj(t). j=l
The vector (5.4.1) is the row labelled by u of the Hadamard matrix H2m, cf. Equations (4.1.j). As the received signal function corresponds to a sampling vector (5.4.m)
1jJ=F+e,
after a Walsh-Hadamard transform one obtains (5.4.n) Rather than reducing the noise vector, e.g. by the methods of Section 4, we try to ignore the error by estimating a maximum-likelihood signal function as follows. With (5.4.0) given by (5.4.p) by formula (5.4.d) we obtain the identity (S.4.q)
12m - F(u)1
= 2 wgt(ul. (fJf)
since (u . v)v E GF(2)'" is the incidence vector ul. of the hyperspace of AG(2, m) orthogonal to the vector u E GF(2) 111 • As the RM(1, m) codes consist of all incidence vectors of such hyperspaces or their cosets (i.e. the complement), a
§5. Codes and designs
903
products (5.4.j)
feu) = (Wu 11jJ) =
lT
Wu (t)1jJ(t) dt
where for u E GF(2)'", Wu(t) is given by 2m
(5.4.k)
WuCt) = L(_l)u.Binary(j-J) ·lU-l,jj(t). j=l
The vector (5.4.1) is the row labelled by u of the Hadamard matrix H2m, cf. Equations (4.1.j). As the received signal function corresponds to a sampling vector (5.4.m)
1jJ=F+e,
after a Walsh-Hadamard transform one obtains (5.4.n) Rather than reducing the noise vector, e.g. by the methods of Section 4, we try to ignore the error by estimating a maximum-likelihood signal function as follows. With (5.4.0) given by (5.4.p) by formula (5.4.d) we obtain the identity (S.4.q)
12m - F(u)1
= 2 wgt(ul. (fJf)
since (u . v)v E GF(2)'" is the incidence vector ul. of the hyperspace of AG(2, m) orthogonal to the vector u E GF(2) 111 • As the RM(1, m) codes consist of all incidence vectors of such hyperspaces or their cosets (i.e. the complement), a
90S
§5. Codes and designs
check equations which can be used for error control purposes: for any received word U E lFn, the syndrome s given by Equation (S . 1. g) depends only on the error vector e, which changed the transmitted unknown codeword c into U = c + e. For IF = GF(2) the error pattern supp(e) uniquely determines the error. Thus in this case the correction algorithm only has to identify and correct the error positions. A position j is said to be checked by W E C.L, if j E supp(w). Let Wj C C be a set of vectors checking j, such that (S.S.c)
Vw, Wi E Wj: w
i= Wi
=}
Isupp(w) n supp(w')1
= 1.
Wj is called an orthogonal check set on position j. The [7, 3,4] simplex code illustrates this definition. S3 is the dual of 1{3 and has parity check matrix, cf. (S.2.d),
(S.S.d)
H~ (~
Seven of the 27-
1 0 0
0
1
0
0
0 1
~)
= 16 parity checks are S 6
1 0
0
0
0
1
0
0
0
0
0
0
0 0
1
0 0
0
0
0 0
1
0
3 4
0
0
1 0
1 2 1 0
0
(S.5.e)
3
0
1
1 0 0
1
0
1
0
0
0
0 0
Rows one, five and seven form the parity check set Wo (S.S.f)
xo+xJ +X3=O Xo +X4+X5 =0 Xo +X2 +X6=0
which are orthogonal on coordinate O. Of course these correspond to the lines through the point Po in Figure S.2. Similarly the lines through PI give three parity checks orthogonal on coordinate 1, and so on.
90S
§5. Codes and designs
check equations which can be used for error control purposes: for any received word U E lFn , the syndrome s given by Equation (S .l.g) depends only on the error vector e, which changed the transmitted unknown codeword c into U = c + e. For IF = GF(2) the error pattern supp(e) uniquely determines the error. Thus in this case the correction algorithm only has to identify and correct the error positions. A position j is said to be checked by W E C.L, if j E supp(w). Let Wj C C be a set of vectors checking j, such that (S.S.c)
Vw, Wi E Wj: w
i= Wi =}
Isupp(w) n supp(w')1
= 1.
Wj is called an orthogonal check set on position j. The [7, 3,4] simplex code illustrates this definition. S3 is the dual of 1{3 and has parity check matrix, cf. (S.2.d),
(S.S.d)
H~ (~
Seven of the 27-
0 0 (S.5.e)
3
0 1 0
1 0 0 0
1
0 0
1 0
0 1
1
0
~)
= 16 parity checks are
1 2 1 0
3 4 1 0
1
0
0 0
0
0 0 0
1
S 6
0 0 0 0 1 0 0
0 0 0 0
0 0 0 0
1
0 1
0 0 0
Rows one, five and seven form the parity check set Wo (S.S.f)
Xo
+Xl
+X3=O
Xo +X4 +X5 =0 XO+X2 +X6=0 which are orthogonal on coordinate O. Of course these correspond to the lines through the point Po in Figure S.2. Similarly the lines through PI give three parity checks orthogonal on coordinate 1, and so on.
§5. Codes and designs
907
u(x)
~
~tmajOrit~ j Figure 5.6. One-step majority logic decoder for cyclic codes, cf. Lin and Costello (1983). The decoding procedure can be described as follows: Step 1. With gate 1 turned on and gate 2 turned off, the received vector u is read into the buffer register. Step 2. The j parity check sums orthogonal on en-i are formed by summing the appropriate received digits. Step 3. The j orthogonal check sums are fed into a majority logic gate. The first received digit Un-i is read out of the buffer and is corrected by the output of the majority logic gate. Step 4. At the end of step 3, the buffer register has been shifted one place to the right with gate 2 on. Now the second received digit is in the rightmost stage of the buffer register and is corrected in exactly the same manner as the first received digit was. The decoder repeats steps 2 and 3. Step 5. The received vector is decoded digit by digit in the manner above until a total of n shifts.
to t = ~dmin - 1 errors for q = 2r, where dmin = q + 1 (see Theorem II.ll.2l), cf. Lin and Costello (1983), Chapter 8. For the same reason Reed-Muller (cf. Section 5.4) and other geometric codes (cf. Section 6) are threshold decodable in a very natural manner. What is more, the geometric parallelism inherent in the AG(n, q) "affords" the design of parallel decoding algorithms, cf. Beth (1981a). For further details the reader is referred to MacWilliams and Sloane (1977), Lin and Costello (1983) and the original papers cited there.
5.6. Threshold Decoding and the Golay-Witt-Mathieu Machine In this section we demonstrate an application of design theory and coding theory, in building a demonstrator for geometric decoding principles. This machine, the GWM machine, see Figure 5.7, was built by Beth and Fumy and presented to the most senior of the authors of this volume during a Fest-colloquium. Details are contained in the paper by Fumy (1981), on which this section is based.
§5. Codes and designs
907
u(x)
~
~tmajOrit~ j Figure 5.6. One-step majority logic decoder for cyclic codes, cf. Lin and Costello (1983). The decoding procedure can be described as follows: Step 1. With gate 1 turned on and gate 2 turned off, the received vector u is read into the buffer register. Step 2. The j parity check sums orthogonal on en-i are formed by summing the appropriate received digits. Step 3. The j orthogonal check sums are fed into a majority logic gate. The first received digit Un-i is read out of the buffer and is corrected by the output of the majority logic gate. Step 4. At the end of step 3, the buffer register has been shifted one place to the right with gate 2 on. Now the second received digit is in the rightmost stage of the buffer register and is corrected in exactly the same manner as the first received digit was. The decoder repeats steps 2 and 3. Step 5. The received vector is decoded digit by digit in the manner above until a total of n shifts.
to t = ~dmin - 1 errors for q = 2r, where dmin = q + 1 (see Theorem II.ll.2l), cf. Lin and Costello (1983), Chapter 8. For the same reason Reed-Muller (cf. Section 5.4) and other geometric codes (cf. Section 6) are threshold decodable in a very natural manner. What is more, the geometric parallelism inherent in the AG(n, q) "affords" the design of parallel decoding algorithms, cf. Beth (1981a). For further details the reader is referred to MacWilliams and Sloane (1977), Lin and Costello (1983) and the original papers cited there.
5.6. Threshold Decoding and the Golay-Witt-Mathieu Machine In this section we demonstrate an application of design theory and coding theory, in building a demonstrator for geometric decoding principles. This machine, the GWM machine, see Figure 5.7, was built by Beth and Fumy and presented to the most senior of the authors of this volume during a Fest-colloquium. Details are contained in the paper by Fumy (1981), on which this section is based.
909
§5. Codes and desigl1s
The close connections between the Witt designs S(5, 8; 24), S(5, 6; 12), the Mathieu groups M24 and MI2 and the Golay codes have been investigated thoroughly and are widely known, cf. Witt (1938a, b), Goethals (1971), Beth and Jungnickel (1981). An interesting construction which gives the S(5, 8; 24) and the S(5, 6; 12) simultaneously and the representations of their automorphism groups M24 and M 12 , respectively, is described in Chapter IV. The incidence matrix of the large Witt design S(5, 8; 24) generates the extended Golay code G24 of length n = 24, k = 24 and w = 8 over GF(2). The problem of finding the unique block containing five given points amongst 24 can be considered as part of the decoding problem of this code. There are a great number of wellknown decoding algorithms for the (24, 12) Golay code, cf. Fumy (1980). Two of these decoding procedures make use of the properties of the large Witt design. Gibson and Blake (1978) base their method on the miracle octad generator of Curtis (1976), cf. Beth, Fumy and RieB (1984). The more elegant method of Goethals (1971), which is a clever threshold decoding scheme, will be presented in this section. For this we recapitulate the following facts from Section IVA. LetD = (V, B) be an S(5, 8; 24) and let hi denote the number of blocks B E B containing a chosen i-subset of V. Then the equations (5.6.a)
bo = 759, b4
=5
bl
and
= 253, b2 = 77, b5 = ).. = 1
b3
= 21,
hold. Let U be a subset of a given block BinD. The number I1(B, U) of blocks C in D with B n C = U depends only on u = IU I and may thus be denoted by l1 u , having the following values: (5.6.b)
118
= 1,
117
112
= 16
and
= n6 = n5 = 113 = 111 = 0, 110 = 30.
114
= 4,
For given blocks A, B of D with IA + B I = 12, the symmetric difference A + B of A and B is called a dodecad, cf. Lemma IVA.3. There exist at most 132 pairs of blocks (Y, Z) forming the same dodecad. The row space G 24 of the incidence matrix D is self-dual, i.e. G 24 = G~. This can be seen from equations (5.6.b), as two blocks of D intersect in an even number of points, implying G24 £: Gf4 and the numerical fact 1 + 759+2576+ 759 + 1 = 4096 = 2 12 , as its weight distribution is (5.6.c)
weight number of vectors
909
§5. Codes and desigl1s
The close connections between the Witt designs S(5, 8; 24), S(5, 6; 12), the Mathieu groups M24 and MI2 and the Golay codes have been investigated thoroughly and are widely known, cf. Witt (1938a, b), Goethals (1971), Beth and Jungnickel (1981). An interesting construction which gives the S(5, 8; 24) and the S(5, 6; 12) simultaneously and the representations of their automorphism groups M24 and M 12 , respectively, is described in Chapter IV. The incidence matrix of the large Witt design S(5, 8; 24) generates the extended Golay code G24 of length n = 24, k = 24 and w = 8 over GF(2). The problem of finding the unique block containing five given points amongst 24 can be considered as part of the decoding problem of this code. There are a great number of wellknown decoding algorithms for the (24, 12) Golay code, cf. Fumy (1980). Two of these decoding procedures make use of the properties of the large Witt design. Gibson and Blake (1978) base their method on the miracle octad generator of Curtis (1976), cf. Beth, Fumy and RieB (1984). The more elegant method of Goethals (1971), which is a clever threshold decoding scheme, will be presented in this section. For this we recapitulate the following facts from Section IVA. LetD = (V, B) be an S(5, 8; 24) and let hi denote the number of blocks B E B containing a chosen i-subset of V. Then the equations (5.6.a)
bo = 759, b4
=5
bl
and
= 253, b2 = 77, b5 = ).. = 1
b3
= 21,
hold. Let U be a subset of a given block BinD. The number I1(B, U) of blocks C in D with B n C = U depends only on u = IU I and may thus be denoted by l1 u , having the following values: (5.6.b)
118
= 1,
117
112
= 16
and
= n6 = n5 = 113 = 111 = 0, 110 = 30.
114
= 4,
For given blocks A, B of D with IA + B I = 12, the symmetric difference A + B of A and B is called a dodecad, cf. Lemma IVA.3. There exist at most 132 pairs of blocks (Y, Z) forming the same dodecad. The row space G 24 of the incidence matrix D is self-dual, i.e. G 24 = G~. This can be seen from equations (5.6.b), as two blocks of D intersect in an even number of points, implying G24 £: Gf4 and the numerical fact 1 + 759+2576+ 759 + 1 = 4096 = 2 12 , as its weight distribution is (5.6.c)
weight number of vectors
XIII. Applications of designs
910
Much as in the above Sections 5.1 ff. we consider vectors h .1. c, i.e. h· c = 0 mod 2 for all C E G24 as so-called parity checks. Since G24 is self-dual, any set of codevectors can be chosen to be the parity checks on the code itself. Decoding of a received vector u = c + e with C E G 24 and wgt(e) < 4 is performed step-by-step. To decode a given coordinate Ui, take the 253 vectors of weight 8 associated with the 253 blocks passing through Ui as parity checks. (5.6.d)
If wgt(e) = 0 none of the parity checks fails. If wgt(e) = 1 and supp(e) = {i} all the 253 parity checks fail, while forwgt(e) = 1 and supp(e) = {j}, (i =1= j), 77 parity checks fail, cf. equation (5.6.a)
n.
As in Fumy (1980) we consider first the case wgt(e) = 2, supp(e) = {i, From equation (5.6.a) we deduce that among the 253 parity checks for Ui there are 77 that also check U j and 253 - 77 = 176 that do not check U j. Hence, in this case 176 parity checks will fail. Now assume supp(e) = {j, k}, (k =1= i, j). From equation (5.6.a) we deduce that among the 253 parity checks that check Ui, there are 21 that check Uj and Ub 77 - 21 = 56 that check Uj but not Uk. 56 that check Uk but not U j, and 253 - 21 - 56 - 56 = 120 that check neither U j nor Uk. Hence, 56 + 56 = 112 parity checks fail in this case. Treating the other cases similarly, we obtain Table 5.1. This table shows that decoding can be performed by a simple threshold test: set ei = 1 if at least 141 parity checks fail, set ei = 0 if at most 125 parity checks fail, and four errors have occurred if exactly 128 parity checks fail. Table 5.1. Number of parity checks for G 24 (takenfrom Fumy 1981). wgt(e) ei
ei
=0
=1
U{(lB;,u) = I}
0 I
0 77
2 3 4 4 3 2
112 125 128 128 141
1
253
176
§5. Codes and designs
911
This leads to the decoding algorithm shown in Figure 5.8. FORi:= 1 TO 24 DO
z:= 0 FORB
E
Bi DO
IF wgt(uAND1s) odd THEN z:= z+ 1 IF z > 128 THEN u := u + ei : correcting coordinate Ui EXIT FI FI
OD IF
=
z 128 THEN EXIT
; 4 errors detected
FI
OD
Figure 5 ..8. Simple threshold decoding algorithm (taken from Fumy 1981).
We conclude with a description of the implementation built using 1980s technology. The threshold decoding algorithm was implemented using a standard microprocessor unit (f1P 8085). An essential part of this machine is a table of the 759 blocks of the large Witt design arranged (and programmed on an early EPROM 2764) in a certain order, such that the first 22 blocks restricted on the first 12 points yield a Hadamard design S2(3, 6; 12), and the first 132 blocks restricted on the first 12 points yield a Steiner system S(5, 6; 12), see Section IV.4. By means of a coloured LED-interactive display, the machine is able to
(a) display the blocks of the designs S(5, 8; 24), S(5, 6; 12) and S(3, 6; 12), (b) decode a given input u into a codeword C E G24, a feature which obviously includes (i) finding the unique block of the Steiner system S(5, 8; 24) that contains five given points, (ii) finding the unique block of the Steiner system S(5, 6; 12) that contains five given points out of a dodecad (for this purpose, two of the dodecads are marked with colours).
This machine has been used successfully in courses on codes and designs at several universities for more than a decade.
912
XIII. Applications of designs
5.7. Quantum Codes During the last few years the newly founded areas of quantum cryptography and especially quantum computing have generated considerable interest among computer scientists and physicists. This section will give some new insights into an essential fact underlying these areas, namely that quantum mechanics presents a canonical link between discrete mathematics and Hilbert space operator theory. Therefore it is only natural that methods from design theory such as Hadamard transform techniques (cf. Section §4) and methods from group theory, geometry, and coding theory apply to quantum computing. The prospect of speeding up certain classes of computations by utilising the quantum-mechanical superposition principle and the physics of entanglement has received a great deal of attention lately (e.g., Ekert 1995). The potentially useful quantum algorithms include factorisation of large numbers (Shor 1994), database search (Grover 1996), and simulation of quantum-mechanical systems (Lloyd 1996). Recent theoretical and experimental progress in atomic physics and quantum optics has shown that small-scale quantum circuitry is feasible, see e.g. Monroe, Meekhof et al. (1995), Turchette, Hood et al. (1995), Cirac and Zoller (1995), and Pellizari, Gardiner et al. (1995). However, building a quantum computer seems to be an extremely difficult task. The major obstacle is that of coupling of the quantum computer with the environment, which destroys quantum-mechanical superposition very rapidly. This effect is usually referred to as decoherence (see Landauer 1995). It is thus of crucial importance to find schemes which suppress and undo the effects of decoherence. Schemes to protect static quantum states against decoherence were found independently by Shor (1995) and Steane (1996a, b). Their proposals gave rise to a large number of subsequent publications (see, for example, Knill and Laflamme (1997), Calderbank, Rains et al. (1997b); and references therein). The theory of quantum error-correcting codes has been developed following these first papers. These publications have initiated a worldwide cooperation of researchers from physics, computer science, and mathematics resulting in theorems and experiments not thought possible before 1995. As these results are based on coding and design theory, and are likely to change our understanding of quantum theory, we dedicate a section of this book to this topic. The basic structure underlying this concept is that of a so-called qubit (quantum bit), i.e. a state 1,8) E 7-£1 in the complex two-dimensional space spanned by the base vectors 10) and II) corresponding to the classical bits 0, 1 E GP(2). We
§5. Codes and designs
913
can state that the qubit 1,8) has the form (5.7.a)
1,8) = cos(tp)IO)
+ ei¢} sin(tp) I1)
so that a measurement with respect to the basis 10), 11) gives the outcome 6 10) with probability p = cos 2 (tp) or 11) with probability q = 1 - P = sin 2 (tp). The state space of a quantum computer of n qubits is the 2n -dimensional complex Hilbert space (5.7.b)
spanned by the vectors Ix) of all binary words x E GF(2)n. It is endowed with the inner product (·I·) and the corresponding L2-norm. Thus each 11/1) E 1-[2" has the form (5.7.c)
11/1) =
L
Ax Ix)
xeGF(2)n
of a superposition of n-qubit states, where Ax (5.7.d)
(1/1 11/1) =
L
E
C and the normalisation
IAxl 2 = 1
xEGF(2)"
holds, to give a probability distribution after the measurement. Because of the principles of quantum mechanics, such states cannot be copied (due to the so-called "no-don ing-theo rem") and any measurement destroys the original quantum state by projecting it onto a subspace of the Hilbert space. Therefore the usual rules of infonnation processing cannot be applied; however, a suitable modification of the concept of error-correcting codes leads to surprising results in stabilising quantum states (cf. Pellizari, Beth et al. 1996), as we shall describe now. A quantum error-correcting code encodes original quantum states into a certain protected subspace spanned by so-called code states 11/Iw,) so that errors and decoherence in a small number of individual qubits have little or no effect on the original state. We will concentrate on the most simple case of errors. There are essentially two basic types of errors, generating the group G ofiocaI errors:
6
To visualise this the reader may consider 1.8) as a photon polarised at the angle 'P with respect to the coordinate system given by the basis.
XIII. Applications of designs
914
A bit-flip error combinatorially resembling the addition + I mod 2 on an individual qubit, i. e., a Hilbert space 'H2 in the tensor product, corresponds to applying the so-called Pauli matrix 0"x = (~ ~) to that qubit. As quantum states also carry phase-information, we furthermore account for sign-flip errors generated by the ~J Pauli matrix O"z =
G
The error group E ::: U(2n), (S.7.e) consists of all possible error operators on n qubits. Following Beth and Grassl (1996) we describe a construction of a code, first discovered independently by Calderbank: and Shor (199S) and Steane (1996a, b), which protects against errors of this kind: given a "classical" binary linear [n, k] code C of length n and dimension k the states of the quantum code are given by (S.7.f)
To simplify the notation, we omit the overall normalisation factor in what follows. The vectors Wi in (S.7.f) are appropriately chosen elements of GF(2)n /CJ.. HereCJ. is the binary dual [n, n -k]-code ofC, cf. Section S.1. Thus, in addition to the choice of the classical code C, for the construction of a quantum code a subset W £ GF(2)n /CJ. has to be given to define the vectors Wi. To illustrate this we consider the smallest example. Take C := {(O, 0, 0), (1, 1, I)} ~ GF(2)3
to be the dual of the RM(1, 3) code n as displayed in Figure S.3. Thus W := GF(2)3/CJ. provides an appropriate choice to use Wo
= (0,0,0),
WI
= (1, 1, 1)
for encoding 7
1
(S.7.g)
10)
f-+
IVrwo) = .,fi (10,0,0)
(S.7.h)
11)
f-+
IVrwl) = .,fi (10,0,0)
7
1
The state 10,0,0)
+ 11,1,1) -11, 1, 1).
+ 11, I, I) is a so-caUed GHZ state as already mentioned in Section 4.1.
915
§5. Codes and designs
The protected subspace He <
H23
of the code vectors is then
As C is one-error correcting (classically), the quantum code inherits this property with respect to single bit-flip errors, i. e., the elements (5.7.j)
s3=id0id00"x, el =CTx
s2=id0CTx 0id,
0id0id E E C U(8).
Obviously the subspaces
= He,
Ho
H;
= s;Ho,
i
E
[1, ... ,3]
are mutually orthogonal so that 3
1l?3
= EEl Hi ;=0
is an orthogonal decomposition of the state space H23. Thus any single bit-flip error s; rotates an encoded state 11/1) = a 11/Iwo) + ,B I1/Iw I ) into the unique element s;l1/l) = a(sd1/lwo)) + ,B(s;l1/lw,)) E Hi of the i-th orthogonal subspace H; of H23. The label i can be measured (in the quantum sense) and the adjoint unitary transfom18; reconstructs the original state 11/1). But, if for example, instead of C the code R had been chosen initially, this construction would not have worked for the bit-flip errors 8;, as R has only drrrin = 2. In the quantum scenario, surprisingly more can be achieved however: with R.l = C, representatives of the set GF(2) 3IR.l can be chosen as the four elements {CO, 0, 0), (0,1,1), (1,0,1), (1, 1, O)}. Choosing C2 ::: R to be
(5.7.k)
C2
= {CO, 0, 0), (0,1, I)}
we select the set (5.7.l)
W=
ct IR.l = two := (0,0,0),
WI
:= (1,0, O)}
so that the protected code space 1tR would be spanned by (5.7.m) (5.7.n)
11/Iwo) = 10,0,0) + 10, 1, 1) + 11, 1,0) + 11,0,1) 11/Iw,) = 10, 0, 0) + 10,1,1) -11,1,0) -11,0,1).
§5. Codes and designs
917
form an orthogonal decomposition of 1{8, thus allowing sign-flip error correction in this example.
In general, the states which represent the information of Wi,
after a Hadamard transform become
Beth and Grassl (1996) show that the set W can even be chl2sen more generall y. If with (5.7.s) the following condition (5.7.t)
'v'Wi,Wj : i;j=j
===?
Mon(M o+(Wi -Wj»= C/J
is satisfied, the quantum code can correct terrors, i. e., any error operator
e E E with the total number of positions exposed to bit-flips or sign-flips being
at most t.
By extending the duality principle of sign-flip and bit-flip errors via a Hadama rd transform, a general decoding algorithm can be designed by carrying out the following steps: • projection onto a 1{C+e • bit-flip error correction (subtraction of e) • Hadamard transform • projection onto a 1{w+w' (w' E Mo) • sign-flip error correction (subtraction of w') • re-encoding. The quantum state finally reached by this algorithm does not depend on the error and the original state can be recovered by a unitary operation. The reader may have observed that the conditions (5.7.) are very similar to those given for the definition of orthogonal check sets in Section 5.6. This is a consequence of the decoding principle by orthogonal projection, in this case over the field IF = C.
§5. Codes and designs
917
form an orthogonal decomposition of 1{8, thus allowing sign-flip error correction in this example.
In general, the states which represent the information of Wi,
after a Hadamard transform become
Beth and Grassl (1996) show that the set W can even be chl2sen more generall y. If with (5.7.s) the following condition (5.7.t)
'v'Wi,Wj : i;j=j
===?
Mon(M o+(Wi -Wj»= C/J
is satisfied, the quantum code can correct terrors, i. e., any error operator
e E E with the total number of positions exposed to bit-flips or sign-flips being
at most t.
By extending the duality principle of sign-flip and bit-flip errors via a Hadama rd transform, a general decoding algorithm can be designed by carrying out the following steps: • projection onto a 1{C+e • bit-flip error correction (subtraction of e) • Hadamard transform • projection onto a 1{w+w' (w' E Mo) • sign-flip error correction (subtraction of w') • re-encoding. The quantum state finally reached by this algorithm does not depend on the error and the original state can be recovered by a unitary operation. The reader may have observed that the conditions (5.7.) are very similar to those given for the definition of orthogonal check sets in Section 5.6. This is a consequence of the decoding principle by orthogonal projection, in this case over the field IF = C.
§5. Codes and designs
919
corresponding quantum code treating the cry-errors as weight-1 errors over GF(4) thus providing a better correction capability. For m
= 2 it is defined as the null space (cf. 5.2)
o
1
o (5.7.v)
1i.m =
kerGF(4)
w
of the ":-=-/ x m matrix consisting of the coordinate vectors of all projective points in PG(m, 4). 1i.m has dimension 4'";' - m and minimum distance 3. The dual code C = 1i.t is a linear (5, 24) cod~, yielding a [[5, 1,3]] quantum code which is I-error correcting. A larger class of I-error con'ecting quantum codes has been discovered by Gottesman (1996), which turns out to be a consequence of the properties of underlying designs and geometries (cf. Calderbank, Rains et al. 1996, Theorem 10). Let Sm be the dual of the classical binary Hamming code 1i.m, the so-called simplex codeSlIl oflengthn = 2111 -1, minimum weight 2m - I , and dimension m. Let cr E Aut(Sm) have no fixed one-dimensional subspaces, such as the Singer cycle, cf Section III.6.3. Then the (2l1l, 2,"+1) code 9m generated by the vectors
+ uO', 0)
(5.7.w)
(u
(5.7.x)
(1 ... 1)
(5.7.y)
(w ... w) E GF(4)2
E
GF(4f'" m
yields a [[2m, 2111 - m - 2, 3]] quantum code. • The [[8,3,3]], [[16, 10,3]], [[32,25,3]] I-error correcting quantum codes were discovered by Gottesman (1996) in a totally different fashion, which finds its geometric and design theoretic explanation and generalisation through the approach presented here (see Calderbank, Rains et al. 1996). As a final example it may be mentioned that the [[ 17, 9, 41] quantum code is generated from a (17,2 8 ) linear code C over GF(4) with rninimum distance d = 12. This classical code is a two-weight code (see Calderbank and Kantor 1986).
§5. Codes and designs
919
corresponding quantum code treating the cry-errors as weight-1 errors over GF(4) thus providing a better correction capability. For m
= 2 it is defined as the null space (cf. 5.2)
o
1
o (5.7.v)
1i.m =
kerGF(4)
w
of the ":-=-/ x m matrix consisting of the coordinate vectors of all projective points in PG(m, 4). 1i.m has dimension 4'";' - m and minimum distance 3. The dual code C = 1i.t is a linear (5, 24) cod~, yielding a [[5, 1,3]] quantum code which is I-error correcting. A larger class of I-error con'ecting quantum codes has been discovered by Gottesman (1996), which turns out to be a consequence of the properties of underlying designs and geometries (cf. Calderbank, Rains et al. 1996, Theorem 10). Let Sm be the dual of the classical binary Hamming code 1i.m, the so-called simplex codeSlIl oflengthn = 2111 -1, minimum weight 2m - I , and dimension m. Let cr E Aut(Sm) have no fixed one-dimensional subspaces, such as the Singer cycle, cf Section III.6.3. Then the (2l1l, 2,"+1) code 9m generated by the vectors
+ uO', 0)
(5.7.w)
(u
(5.7.x)
(1 ... 1)
(5.7.y)
(w ... w) E GF(4)2
E
GF(4f'" m
yields a [[2m, 2111 - m - 2, 3]] quantum code. • The [[8,3,3]], [[16, 10,3]], [[32,25,3]] I-error correcting quantum codes were discovered by Gottesman (1996) in a totally different fashion, which finds its geometric and design theoretic explanation and generalisation through the approach presented here (see Calderbank, Rains et al. 1996). As a final example it may be mentioned that the [[ 17, 9, 41] quantum code is generated from a (17,2 8 ) linear code C over GF(4) with rninimum distance d = 12. This classical code is a two-weight code (see Calderbank and Kantor 1986).
920
XIII. Applications of designs
The columns of the generator matrix are the coordinates of the 17 points of an ovoid in PG(3, 4). Like the Hamming codes and the simplex code, C and C.J.. are cyclic. C.J.. is generated by the word (1wlw10 12 ) E GF(4)[Z17]' For further definitions of cyclic quantum codes see Grassl, Beth, and Pellizari (1997), Calderbank, Rains et al. (1996, 1997b) and Beth and Grassl (1998). Calderbank, Rains et al. (1996) show that the quantum codes in our example are best possible. For this reason we repeat their list of the highest achievable minimal distance d in any [[n, k, d]] quantum error-correcting code in Table 5.2.
S.S. MOG, UniMOG and miniMOG We cannot conclude this section on codes and designs without mentioning one of the grand curiosities of design theory, geometry, and coding theory. In Section 5.7 we have shown that quantum state stabilisation can be achieved through an interdisciplinary synergy between quantum physics and discrete mathematics. The relation of these areas via the correspondence between the groups E generated by unitary Pauli matrices and the symplectic GF(4)-geometry of the group If affords the use of (weakly) self-dual codes. A special role in this area is played by the class of self-dual codes. Amongst these the (24, 12, 8) Golay code over GF(2)(cf. Theorem IV.4.4 and Section 5.6) is of exceptional interest. It has been known for some time (cf. Conway and Sloane 1993, Beth, Fumy and RieB 1984) that this gem amongst all codes can be generated from the so-called hexacode. The hexacode is the GF(4)-space (5.8.a)
(00111l,0101ww, 1001ww) < GF(4)6,
thus forming a (6, 26 )-linear code with minimal distance d = 4. This code is a,!; natural foundation to classical coding and design theory by providing alternative ';i J constructions of the Golay code (24, 12, 8) and the large Mathieu-Witt Design il .~ S(5, 8; 24) via the so-called miracle octad generator MOG, cf. Conway and" Sloane (1993), Beth, Fumy and RieB (1984). The construction giving the binary ~~ Golay code (24, 12,8) from the hexacode is worth recalling h e r e . : : ;
:~
For this purpose the hexacodewords are mapped onto 4 x 6-arrays, the so-called ;~ MOG, so that the binary columns of length four correspond to the elements ofl..;;\ GF(4). We define four mappings fo, 10 (the even correspondences) and /J, II ;~ (the odd correspondences) according to Table 5.3. ~ "~~
;
·,",,3
,11
Table 5.2. Achievable minimal distance d in any [[n, k, d]] quantum error-correcting code (taken from Calderbank, Rains et al. 1996) n\k
0
3 4 5 6 7
2 2 3 4 3 4 4 4 5 6
8
9 10 11
12 13
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
2 1 2 3 3 3
1 2 2 2 2
3
3
3 4 5 5 5 5 6 5 6 5 6 6 7 7 8 7 7-8 7 8 7 8 7 8 7-8 8-9 7-9 8-10 8-9 8-9 9 8-10 9 9-10 9 10 10 11
12
11 11
3 4 4 4 4 4-5 5 6 6 6 6 6-7 6-7 6-8 6-8 6-8 7-8 8-9 9 10 10 10
3
4
1 1 1 1 1 2 2 2 2 2 3 3 2 3 3 3 3 4 4 4 4 4 4-5 4 5 4-5 5 5-6 4-5 5-6 5-6 5-6 5-6 5-7 5-6 5-7 5-7 5-7 5-7 5-8 5-7 6-8 6-8 7-8 7-8 8-9 8 9 8-9 9 8-9 9-10 8-9 9-10 8-10
5
6
1 1 1 2 2 2 3 3 3-4 4 4 4-5 4-5 5 5-6 5-6 5-6 5-7 5-7 6-7 7-8 7-8 7-8 7-9 7-9 7-9
1 1 2 2 2 2 3 3 4 4 4 4-5 5 5 5-6 5-6 5-6 5-7 6-7 6-7 6-8 6-8 6-8 6-9 6-9
7
1 1 1 2 2 2 3
3 3 3-4 4 4 4-5 4-5 4-6 4-6 5-6 5-7 5-7 5-8 5-8 6-8 6-8 6-9
8
9
10
11
12
1 1 2 2 2 2 3 3 3 4 4 4 4-5 4-5 4-6 4-6 4-6 4-7 4-7 4-8 6-8 6-8 6-8
1 1 1 2 2 2 3 3 4 4 4 4 4-5 4-5 4-6 4-6 4-6 4-7 5-7 5-8 5-8 5-8
1 1 2 2 2 2 3 3 3 3-4 4 4 4-5 4-5 4-6 4-6 4-6 4-7 5-7 5-7 5-8
1 1 1 2 2 2 3 3 3 3-4 4 4 4-5 4-5 4-6 4-6 4-6 5-7 5-7 5-7
1 1 2 2 2 2 2 3 3 3-4 4 4 4-5 4-5 4-6 4-6 5-6 5-6 5-7
13
1 1
14
1 2 2
1 1 2 2
2
2
2 3 3 3-4 4 4 4-5 4-5 4-5 5-6 5-6 5-6
2 2 3 3 3-4 4 4 4-5 4-5 5-6 5-6 5-6
15
1 1 1 2 2 2 3 3 3
3-4 4 4 4-5 4-5 4-5 4-6
16
17
18
19
20
21
1 1 1 2 1 1 2 1 1 1 1 2 2 2 1 2 1 1 2 2 1 2 2 2 1 2 2 3 2 2 1 2 2 3 2 2 2 3 2 3-4 3 3 2 2 2 4 3-4 3 3 2 2 4 4 3-4 3 3 2 3 4 3-4 3 4 4 4-5 4 4 3-4 3-4 3 4 4-5 4-5 4 4 3-4
22 23
1 1
1
2 2 2 2 2
1 1 2 2 2 2 3
3
3
923
§5. Codes and designs
described in Section 5.7: the [[6,0, 4]]-quantumcode, cf. Table 5.2, is generated by the hexacode. There is a (5,2 5 ) additive code with d = 3 obtained from it which generates one of the indecomposable [[5, 1, 3]]-quantum codes, from which larger codes are built. As the related decoder can, cf. Section 5.7, correct one-quantum-error by implementing the MOG decoder (cf. Beth, Fumy and RieB 1984) as a unitary transform regenerating the unknown initial quantum state, this code is called the UniMOG. Recently J. H. Conway (1997) proposed another game, the so-called rniniMOG, to be played on PG(2, 3) much like the famous puzzle of 15 square tiles and one hole on a 4 x 4 array, cf. Remark IV.2.13 and Egner and Beth (1998). 00
00
o
20
10
Figure 5.9. Lines with "affine slope"
00,
respectively O.
As PG(2, 3) is an S(2, 4; 13), say J, this 13-puzzle can be visualised as follows: in the beginning let the hole be at point 00 of the projective plane, which can be represented as shown in Figure 10.3 below. From this we draw Figure 5.9 showing the lines of "affine slope" 00, respectively O. Taking the point residue Joo we obtain a transversal design TD[4; 3] in which the nine blocks (corresponding to the lines of slopes 0, I or 2), can be coded by the so-called tetracode. This is shown in Figure 5.10, where in tetracode coordinates the lines y = mx + c mod 3 of slope m correspond to the words of length four given by t = (to, t1, t2, t3) = (c + Om, c + 1m, c + 2m, m). The blocks are thus given 00
1=
2
2
1=1 1=0
i=O
0
o
0
2
3
Figure 5.10. In tetracode coordinates the lines y = mx +c mod 3 of slope m correspond to the words oflength four given by t = (to, tI, t2, t3) = (c + Om, c + 1m, c + 2m, m).
923
§5. Codes and designs
described in Section 5.7: the [[6,0, 4]]-quantumcode, cf. Table 5.2, is generated by the hexacode. There is a (5,2 5 ) additive code with d = 3 obtained from it which generates one of the indecomposable [[5, 1, 3]]-quantum codes, from which larger codes are built. As the related decoder can, cf. Section 5.7, correct one-quantum-error by implementing the MOG decoder (cf. Beth, Fumy and RieB 1984) as a unitary transform regenerating the unknown initial quantum state, this code is called the UniMOG. Recently J. H. Conway (1997) proposed another game, the so-called rniniMOG, to be played on PG(2, 3) much like the famous puzzle of 15 square tiles and one hole on a 4 x 4 array, cf. Remark IV.2.13 and Egner and Beth (1998). 00
00
o
20
10
Figure 5.9. Lines with "affine slope"
00,
respectively O.
As PG(2, 3) is an S(2, 4; 13), say J, this 13-puzzle can be visualised as follows: in the beginning let the hole be at point 00 of the projective plane, which can be represented as shown in Figure 10.3 below. From this we draw Figure 5.9 showing the lines of "affine slope" 00, respectively O. Taking the point residue Joo we obtain a transversal design TD[4; 3] in which the nine blocks (corresponding to the lines of slopes 0, I or 2), can be coded by the so-called tetracode. This is shown in Figure 5.10, where in tetracode coordinates the lines y = mx + c mod 3 of slope m correspond to the words of length four given by t = (to, t1, t2, t3) = (c + Om, c + 1m, c + 2m, m). The blocks are thus given 00
1=
2
2
1=1 1=0
i=O
0
o
0
2
3
Figure 5.10. In tetracode coordinates the lines y = mx +c mod 3 of slope m correspond to the words oflength four given by t = (to, tI, t2, t3) = (c + Om, c + 1m, c + 2m, m).
XIII. Applications of designs
924
by the following tetracode matrix (fm,c):
0 (5.S.b)
i= 1 2 3 0
m=O 0 0 0 0 m= 1 0 1 2 1 m=2 0 2 1 2
i= 2 3
1
0
i= 1 2 3
1 1 1 0 2 2 2 0 1 2 0 1 2 0 1 1 1 0 2 2 2 1 0 2
c=O
c=l
c=2
Labelling each node vertex (i, t) of F JO by l(i, t) = 3 . i + c E [0, ... , 11] we obtain the pattern given in Figure 5.11, associated with the corresponding (3 x 4) array, the so-called rniniMOG. The columns represent the point classes of Joo, filled with the twelve labels in the order shown. On this rniniMOG we now describe Conway's 13-puzzle: for this let the hole be at point x of J. The transversal design obtained above can be described using the words of the tetracode. These are the lines along which this game is played. Moving the hole from position x to position y can only be done by moving the label at y to position x and simultaneously exchanging the labels v of a and w of b, if {y, a, b} have formed the point class of y in the transversal design ]X. This substitution is denoted by u I vw and its action can be seen in Figures 5.11 and 5.12. From these visualisations it can rather easily be shown that the stabiliser of the point 00, i.e. the set of sequences generated from exchanges of type u I vw, which bring the hole back to the initial position, is the five-fold transitive group MI2, cf. Chapter IV.
10
o
3
6
9
Figure 5.11. Labelled Joo in tetracode coordinates and the corresponding miniMOG.
9
6
3
o
Figure 5.12. Labelled Jco after applying the sequence (0'1 = 911011,0'2 = 0136, = 9112) generating the pennutation a = (12)(3 6)(9 0)(1011) and the corresponding miniMOG.
0'3
§5. Codes and designs
925
After this coordinate-based design-theoretic exposition of the 13-puzzle, we approach it also from the point of view of group theory. Consider Figure 5.13 for a highly symmetrical visual representation of the incidence structure. Following
Figure 5.13. The incidence structure PG(2, 3) shown as a bipartite graph (. = point. line) within a highly symmetric layout. (The circle represents a Singer-cycle.)
I=
J. H. Conway, the .-ticks denote the points of PG(2, 3) and the I-ticks stand for the lines. A • is connected to a I iff the point is an element of the line. Now M13 can be described as a puzzle (this is what 1. H. Conway did): place twelve counters labelled "1" ... "12" onto twelve of the points. The remaining point is free (it is labeled by a "hole"). An elementary move consists of three actions: (i) move a counter along an edge to a line; (ii) move that counter into the "hole"; (iii) exchange the two counters which occupy the other points of the same line. Hence, an elementary move takes a counter into the hole and simultaneously exchanges two olher counters, specified by the incidence structure of the projective geometry PG(2, 3). The structure Ml3 is the set of all configurations of the puzzle which are reachable by elementary moves from a fixed initial configuration. By this construction MI3 is a certain set of permutations on twelve counters and a hole. Taken as a puzzle, the aim is to restore the initial state from a given one with elementary moves. In this form the puzzle is much a like the famous IS-puzzle of Sam Loyd. For a computer algebraic visualisation, cf. Egner and Beth (1998). Much as the hexacode generates the five-fold transitive large Mathieu group M24 giving one of the decisive steps towards the classification of the sporadic simple groups (Conway, Curtis et al. 1985), the tetracode can surprisingly be used to generate and to extend the fivefold transitive group Ml2 into a sixfold transitive groupoid M 13 , cf. Conway (1997). An essential tool in the proof of this is that the Schur multiplier of order 2 of M\2 can be realised as the group
926
XIII. Applications of designs
of monomial permutations of the signed 24 letters made from the 12 letters of the miniMOG. As in the case ofUniMOG, there seems to be an overwhelming analogy with the action of the error group E for quantum codes. We conclude this section with a mathematician's apology, that, at last, design theory has found an application within mathematics itself.
§6. Discrete Tomography Tomography has been one of those mathematical techniques that have revolutionised science and engineering since the 1970s. Best known for its unique impact on medical diagnostics (cf. Morneburg (ed.) 1995) through the so-called CT and NMR-visualisation (cf. Herman 1980, Natterer 1986, Louis 1989) of deep tissue layers, it is an utterly geometric technique. Originally it was invented for the purpose of cut-line-integral reconstructions of smooth real functions with compact carrier on the plane by Radon (1917). The award of the 1979 Nobel Prize in Physiology or Medicine for the practical application of this theorem to A. M. Cormack (1963) and Sir Godfrey N. Hounsfield, came after the triumphant progress of this technique had begun to revolutionise medicine. Cormack, who had first designed the apparatus almost two decades earlier, actually thought of applications in mining or in the detection of bodies overrun by avalanches (cf. Cormack 1980). Nowadays, the applicability of tomographic techniques has grown far beyond the known applications in medicine to other fields, such as material testing, cf. Engl, Louis and Rundell (1997), and in areas like quantum optics, as we will describe now.
6.1. Phase Space Tomography The so-called phase-space tomography (cf. Leonhardt 1997; Breitenbach, Schiller and Mlynek 1997) is a very recent invention, which can be used for new quantum state measuring techniques in atomic clocks, ion traps (cf. Blatt et al. 1996), etc. This example taken from applied physics shows the intrinsic role which finite geometries, in particular, finite affine planes, can play. The phase space to be measured consists of the so-called Wigner domain (cf. Meyer 1992; Wolf 1979; Kurtsiefer, Pfau and Mlynek 1997) of the conjugate observables of position x and momentum Px of an ion in an ion-trap. Owing to the Heisenberg uncertainty principle, both variables cannot be measured simultaneously with precision. The phase space can be quantised to a discrete N x N-dimensional (X, f)-grid of position-momentum state components in the Wigner domain. Following the rules of quantum mechanics
927
§6. Discrete tomography
(cf. Leonhardt 1997; Cohen-Tannoudji, Diu and Laloe 1997) the Hamiltonians governing this physical system let the initial state vectors 1,!f(0)) being a superposition of position-momentum states Ix)ly) with amplitude a~~ E C undergo a time-evolution according to the transform (6.l.a)
11fr(0))
L
=
a;~:lx)IY)
L
W 11fr(t)) =
(x,y)eZ~
a.~~lx)IY),
(x,y)eZ~
with (6.l.b)
U(t): Ix)ly)
1-+
Ix) Iyt mod N).
From this time evolution, which sweeps over the different "slopes" of onedimensional subspaces in the module Z~, it can be deduced that the unknown phase-space distributions can be measured (with optimal possible precision) by correlation with line-states sweeping through the ZN x ZN grid. This is called phase space tomography (cf. Wootters 1987; Leonhardt 1996). These measurements give a complete set of measurements to provide phase space reconstruction, if and only if N is a prime. This theorem, which looks surprising in physics (Wootters 1987), should look natural to readers of this book. Indeed, the IF-code generated by the incidence vectors of the lines [m,b
= {(x, y) E Z~: y = mx + b mod N}
has full dimension N 2 if and only if (char IF, N) = I and N is a prime. As in the present case IF = C, we obtain an obvious application of affine geometry to physics. In the case char IF = p and N = p, an interesting application arises from the IF p rank-defect of the point-line incidence matrix of an affine plane. This application to discrete fixed point arithmetic precision tomography with error correction was an early proposal (cf. Beth 1980) to overcome the numerical instabilities of inverse Radon transforms in classical computer tomography. Later these methods were developed to treat Euclidean geometry codes (cf. Beth 1993) in applications to quantum optics and quantum computation already mentioned.
6.2. Finite Field Tomography In this section we follow Beth (1980). The discrete Radon transform can be interpreted as a problem of linear algebra as follows: the plane section of the tissue to be tomographed is an array X of p x p cells called points (x, y) with coordinates x, y E [0, ... , p - 1] carrying a density distribution (6.2.a)
d
= «dX,Y, )x,ye[O, ) . .... p-l]
E lFP2
XIII. Applications of designs
928
over a field IF. We shall illustrate this for the case p = 5 in the following example. The discrete Radon transform (DRT) of the density distribution d is the measurement vectorf = CIz)IEL, where for each line I E L (6.2.b)
Iz =
L
dx . y •
(x,y)El
As in classical computer tomography the line integral over the density distribution dx,y corresponds to the sum of those segments of an affine line I, which coincide with straight lines in the real plane, as indicated in Figure 6.1. In order to be able to study the properties of this DRT, some conditions have to be fulfilled by p, X and L. 4 3 y 2 I
0
. .•
·/: . .. •
· . . /.
·
0
2 3 4 x
Figure 6.1. The 5 x 5 plane with the affine line I : y = x
Let N be the (p2 (6.2.c)
+ p)
X
+ 2.
p2 incidence matrix of AG(2, p): with
nl,v
={
I
if v E /,
o
else.
With this notation the DRT of the density distribution dbased on the affine plane AG(2, p) is the ideal measurement-vector f E lF P2 +p fulfilling the equation f=Nd. Thus the problem of computing the tomogram is reduced to the problem of "inverting" the matrix N under simultaneous erasure of systematic and random errors superimposed on f The tomographic scanner can be considered as a noisy channel delivering realistic measurement vectors f' as outputs, where the input was the unknown distribution d. The encoder is the matrix producing ideal measurement vectors, the codewords. Neglecting systematic errors e = f - f' for the moment, the channel can be considered as a device adding noise patterns - the error vectors. In this setting the problem of image reconstruction is equivalent to the decoding problem of error-correcting (n, k, d) codes (cf. Section 5).
929
§6. Discrete tomography
Here, however, the linear encoder G cannot be a matrix of size k x n of full row rank k, as in our example of DRT, the encoder is the matrix N. Thus it is important to determine the IF-rank of N, since the code in consideration is the IF-column space of N. It is obvious that Zp x Zp is a group of automorphisms ofAG(2, p). Moreover, the code C generated by the columns of N over IF is an ideal in the group algebra
(6.2.d) From this observation it is easy to obtain the formulas for the IF-rank of N, the minimum weight and a decoding procedure. It is known that
(6.2.e)
if char f =f. p, if char IF
= p,
cf. Theorem II. 11.13. This theorem shows that either - in order to avoid ilJposedness - N has a maximal rank implying that the code does not provide any fault tolerance, or - accepting quite a loss of rank - the code can correct errors, as its minimum weight is p, cf. Beth (1993).
9
In the first case, char IF =f. p, the group algebra is semisimple. Therefore - the image reconstruction problem being well-posed - it is possible to apply discrete Fourier transforms over IF in order to reconstruct the picture from the DRT by simultaneously treating systematic errors by filtering techniques. In the more interesting case where char IF = p, the group algebra has a large errors. Fortunately, the radical and the code has rate ~, and can correct virtual rank deficiency of this code is in fact quite feasible in a tomographic application, as the actual picture of, e.g., a skull is physically realised by suspending the skull in the middle of the picture plane supported by an annular water pillow. Therefore approximately only half of the points of the plane carry information of interest, and since the density of water is known, the position of the other points can actually be used for redundancy purposes.
9
The decoding is done in two steps: (i) Maximwn-likelihood estimation: observing that over the field IF with char IF = p it is not possible to find p-th roots of unity, in order to apply discrete Fourier transform techniques, the signals for an intermediate state are considered as Z-vectors. Then over a different field apply a Fourier
XIII. Applications of designs
930
transformation on the signal and make a maximum-likelihood estimation by a best-fit test in the function space L2, comparable to the decoding procedure of Reed-Muller codes by the Walsh transformation given in MacWilliams and Sloane (1977), cf. Section 5.4. (ii) Determining the density distribution: again considering the signal as a "pure" codeword over lEi', the actual computation of the (Pi!) information positions of d is a matter of Mobius inversion over IF', which again is an analogue to Fourier-Walsh-Hadamard transforms over lEi'. The detailed description of this for the case p = 2 can be found in MacWilliams and Sloane (1977), whereas the interesting case for large primes p is treated in Beth (1993). The DRT presented here is based on interpreting the image reconstruction problem as a maximum-likelihood decision problem naturally occurring in the setting of linear algebra over finite fields GF(q). Computations in GF(q) have many practical advantages: the number p2 of points in the discrete picture plane determines the characteristic of the field lEi'. According to the desired density resolution, the field lEi' = GF(q) can be chosen large enough (q = pS) to represent all possible values. The arithmetic is considerably faster than over Q, lR or C and extremely accurate, since the underlying operations are integral and therefore produce no round-off errors (cf. McClellan 1979).
§7. Designs in Data Structures and Computer Algorithms Occidit miseros crambre repetita magistros Juvenal, Satire VI1.154
7.1. Filing Schemes Even though today modem database technology seems to be a fully developed and mature technology, cf. Lockemann and Schmidt (1987), which feeds a still growing industry, we shall not hesitate to mention a historical breakthrough, cf. Abraham, Ghosh and Ray-Chaudhuri (1968), in data storage technology which may still have some application in future technologies. This application of design theory dates back to the time when computer mass memory was very slow (such as magnetic tape) or expensive, while magnetic core memory (cf. Section 8.1) was small, even more expensive and thus used only for the computation itself.
§7. Data structures and compUler algorithms
931
An excellent description of the need and applications of efficient filing systems is given in the introduction of Ray-Chaudhuri (1968). In what follows we rely on a paper by Beth and Vijayan (1978):
In any filing system, first of all, there is a more or less large set of individual records which, for obvious reasons, can be stored in some "slow" memory. Secondly, to allow retrieval from this set of records, there is an inverted list of addresses called accession numbers of the individual records. This inverted list is somehow arranged with respect to a set of queries out of a set of attributes. This list of accession numbers is to be stored in some faster memory. The memory locations where these accession numbers will be stored will be called addresses. Finally, there is a storing rule and a retrieval rule. The storing rule is an algorithm which determines the addresses where the accession numbers of a given record nave to be stored. The retrieval rule, somewhat similar to the "inverse" of the storing rule, will give all addresses where the accession numbers of those individuals are stored (see Figure. 7.1). The reader will see that it may happen that some accession numbers are stored in more than one address. But this can be good to make the filing scheme more efficient.
Input query
"
I
AddJ;~sse~ of (,query)
/
~
\~
"Faster" memory
with inverted list Acces
0
Slow memory with all records Output: Records of all individuals fulfilling the query (!)
~'-~~~~U~se~r~~~---'
(?) ' - - - - - - - - - - - - '
Figure 7.1. Retrieval technique for a filing scheme (taken from Beth and Vijayan 1978). In such a retrieval scheme the speed obviously depends on the "retrieval rule".
=----
§7. Data structures and computer algorithms
,",!!!'-
933
Obvious ly a t-balanc ed system requires a balance d incidenc e structure which is at-cover , cf. Colboum and van Oorscho t (1989). In order to make the algorith ms for the retrieval rule efficient, Ray-Cha udhuri (1968) propose d the use of linear algebra over finite geometr ies. Beth and Vijayan (1978) propose in their paper the so-calle d group filing systems , which we now describe formally . Let G a finite group and (Q, B) be a partial t-design S = SA (t, k; IQ/) with B = SJ such that there exists aBE B with the property that for Q E (~), there exists at least one element y E G such that Q ~ By. Let c: (~) -+ G be a choice function which compute s exactly one y such that By ;2 Q. Let y = c(Q). Choose s: I -+ 2A to satisfy the followin g property. Let (AQ.y)Q,y be a family of disjoint subsets of the set A of addresse s which are chosen to be so large that
Vi
E
I :
[ls(i)
n AQ,yl = 1 {} wei) n By = Q].
Then (F, A, s) is a storage rule. For r: (~) -+ 2A being the mapping given by r(Q) AQ,y, ((~), r) is a t-balanc ed retrieval rule.
=
The filing system (F, S, G, s, r) introduc ed above with storage and retrieval rule is called a group filing system. To illustrate this we repeat a simple example from Beth and Vijayan (1978). The design S2(2, 5; 11) is given by (GF(Il) , B) with B=dev Co, i.e. the followin g blocks C i in B, :::s i :::s 10.
°
Co={1 ,3,4,5,9 }
C, = {2, 4, 5, 6, lO} C2 = {3, 4,6,7, o}
C3={4 ,5,7,8,1 } C4 = {5, 7, 8,9, 2}
C5 = {6, 8, 9,10, 3} C6 = {7, 9,10,0, 4} C7 = {8, 10,0, 1, 5} Cs={9 ,0,1,2, 6}
C9 = {l0, 1,2,3, 7} CIO = {O, 2, 3,4, 8}
...,.-
§7. Data structures and computer algorithms
-
-
-
'""'!!
933
Obvious ly a t-balanc ed system requires a balanced incidenc e structure which is at-cover , cf. Colbour n and van Oorscho t (1989). In order to make the algorith ms for the retrieval rule efficient, Ray-Cha udhuri (1968) propose d the use of linear algebra over finite geometr ies. Beth and Vijayan (1978) propose in their paper the so-calle d group filing systems , which we now describe formally .
=
Let G a finite group and (Q, B) be a partial t-design S SA (t, k; IQ/) with 13 = 13G such that there exists aBE 13 with the property that for Q E (~), there exists at least one element y E G such that Q ~ By. Let c: (~) -+ G be a choice function which compute s exactly one y such that By ;2 Q. Let y = c(Q). Choose s: I -+ 2A to satisfy the followin g property. Let (AQ.y)Q,y be a family of disjoint subsets of the set A of addresse s which are chosen to be so large that Vi
E
I:
[Is(i)
n AQ,yl
= 1 {} wei)
n By
= Q].
Then (F, A, s) is a storage rule. For r: (~) -+ 2A being the mapping given by r(Q) = AQ,y, ((~), r) is a t-balanc ed retrieval rule. The filing system (F, S, G, s, r) introduc ed above with storage and retrieval rule is called a group filing system. To illustrate this we repeat a simple example from Beth and Vijayan (1978). The design S2(2, 5; 11) is given by (GF(ll) , 13) with 13=dev Co, i.e. the followin g blocks C i in 13, :::s i :::s 10.
°
Co={1 ,3,4,5,9 } C, = {2, 4, 5, 6, 1O} C2 = {3, 4,6,7, o} C3={4 ,5,7,8, 1}
C4 = {5, 7, 8,9, 2} C5 = {6, 8, 9,10, 3} C6 = {7, 9,10,0, 4} C7 = {8, 10,0, 1, 5} Cs = {9, 0, 1,2, 6}
C9 = {1O, 1,2,3, 7} CIO = {O, 2, 3,4, 8}
'-
§7. Data structures and computer algorithms
935
in the class Q (nJn). This example is discussed in detail by Lakshman and Agrawala (1986) for designing a message passing protocol on a special network topology, cf. also Section 8.2 and Colboum and van Oorschot (1989).
7.3. Sorting in Rounds Consider a set of 11 elements which should be sorted in k rounds by r processors. In every round r comparisons are made simultaneously. Obviously, sorting n elements in one round needs (~) comparisons. On the other hand, sorting these n elements with one comparison per round requires 0 (11 log n) rounds. For this reason we can restrict k to values I ::s k ::s 11 log 11, while a lower bound is r = Q(ni), cf. Haggkvist and Hell (1982), Colboum and van Oorschot (1989). Alon (1985, 1986) applies the point-hyperplane design obtained from PG(4, q) to construct a sorting algorithm which sorts 11 = q4 + q3 + q2 + q + 1 elements in two rounds on r = O(ni) processors. This latter application is constructive and gives a near optimal algorithm, cf. also Pippenger (1987).
7.4. Designs in the Design and Analysis of Algorithms It has not been widely appreciated by the community of mathematicians and computer scientists that t-designs have successfully been applied for the design and analysis of algorithms. The preceding example of sorting in rounds is only one instance. We list some further applications.
Averaging Complexity. The possibly earliest example of t-designs being used in complexity theory can be found in the paper by Karp and Wigderson (1985) where they developed parallel algorithms for solving the maximal independent set problem. The randomised parallel version runs in expected time o «log /1)3) with 0(n2) processors, while the deterministic version has a run time of 0((logn)4) using 0(n 3 /logn) processors on a graph G = (V, E) withn = iVl vertices. While the complexity theoretic main result of this paper is that the maximal independence set problem lies in the class Nc, cf. van Leeuwen (1990), the point that is of interest in this section is the step where the use of designs is successfully applied. In the proof of their Lemma 2 they compute an average over a "score" function defined over the set of all (-subsets of a given set R. Rewriting this average formula into a different summation order, it can be seen that the overall average depends only on the fact that every element from the basic set R (respectively every pair of distinct elements in R) appears in the same number of t -subsets respectively. Therefore, the algorithm need not range
§7. Data structures and computer algorithms
935
in the class Q (nJn). This example is discussed in detail by Lakshman and Agrawala (1986) for designing a message passing protocol on a special network topology, cf. also Section 8.2 and Colboum and van Oorschot (1989).
7.3. Sorting in Rounds Consider a set of 11 elements which should be sorted in k rounds by r processors. In every round r comparisons are made simultaneously. Obviously, sorting n elements in one round needs (~) comparisons. On the other hand, sorting these n elements with one comparison per round requires 0 (11 log n) rounds. For this reason we can restrict k to values I ::s k ::s 11 log 11, while a lower bound is r = Q(ni), cf. Haggkvist and Hell (1982), Colboum and van Oorschot (1989). Alon (1985, 1986) applies the point-hyperplane design obtained from PG(4, q) to construct a sorting algorithm which sorts 11 = q4 + q3 + q2 + q + 1 elements in two rounds on r = O(ni) processors. This latter application is constructive and gives a near optimal algorithm, cf. also Pippenger (1987).
7.4. Designs in the Design and Analysis of Algorithms It has not been widely appreciated by the community of mathematicians and computer scientists that t-designs have successfully been applied for the design and analysis of algorithms. The preceding example of sorting in rounds is only one instance. We list some further applications.
Averaging Complexity. The possibly earliest example of t-designs being used in complexity theory can be found in the paper by Karp and Wigderson (1985) where they developed parallel algorithms for solving the maximal independent set problem. The randomised parallel version runs in expected time o «log /1)3) with 0(n2) processors, while the deterministic version has a run time of 0((logn)4) using 0(n 3 /logn) processors on a graph G = (V, E) withn = iVl vertices. While the complexity theoretic main result of this paper is that the maximal independence set problem lies in the class Nc, cf. van Leeuwen (1990), the point that is of interest in this section is the step where the use of designs is successfully applied. In the proof of their Lemma 2 they compute an average over a "score" function defined over the set of all (-subsets of a given set R. Rewriting this average formula into a different summation order, it can be seen that the overall average depends only on the fact that every element from the basic set R (respectively every pair of distinct elements in R) appears in the same number of t -subsets respectively. Therefore, the algorithm need not range
936
XIII. Applications of designs
over the full class of all t-subsets, but rather over a block design 8)..(2, t; JRI). Here, the balancedness of the design obviously is used to replace a procedure of much higher complexity, similar to those we have seen in the question of experimental designs (cf. Section 2). Analysis of Algorithms. Another example for the use of designs in the construction of efficient algorithms appears in the area of algebra and related computer algebra packages: in the problem of factorisation of polynomials over a finite field GF(q) into linear factors the so-called root finding problem (cf. Berlekamp 1970) is the essential problem. The idea of using designs in this context is due to van Oorschot and Vanstone (1989, 1990). The roots of polynomials can be split into two classes, e.g. quadratic residues and non-residues. For the prime q, use the two different sets formed by the quadratic residues R, respectively the quadratic non-residues N as start blocks; then theincidencestructuredev(R, N) is a Steiner system 82:::2 (2, ~; q). With 2 Berlekamp's famous gcd trick we compute (7.4. a)
gcd (I(x), (x
+ c)'S-' - I).
For a random choice of c one can show that a given pair of roots is separated with probability +o(~) by the use of this design. For an example in GF(Il) see van Oorschot and Vanstone (1990). Further algorithms in this area are by now available in the different versions of the computer algebra packages Axiom, GAP, Magma, Maple and Mathematica. 8
!
In order to find roots over extension fields (to be constructed) the gcd trick can be specialised to compute gcd(f(x), t(aix) - fJ), with j E [0, ... , m - 1] and fJ E GF(q), where a is a root of an irreducible polynomial of degree m over GF(q) with j and fJ varying. Again, the design formed by points and hyperplanes of the affine geometry AG(m, q) can be used to determine the probability that roots are not separated by the algorithms to be smaller than ~. Cf. Vanstone and van Oorschot (1989, 1990), and Beutelspacher, Jungnickel et al. (1992) for more information on this topic. Lower Bounds for Algorithms. A problem closely related to the construction of balanced t-designs is the so-called set-covering problem (cf. Colbourn and van Oorschot 1989). This problem being of general interest in the context
8 Axiom®, MapJe® and Mathematica® are registered trademarks.
§8. Designs in hardware
937
of theories of partially ordered sets and lattices has an application in key management, cf. 10.2, but also in the design of broadcast systems when the positioning of repeaters for radio and television or cell compartments for mobile communication have to be optimised. In the analysis of special integer programming algorithms for solving this problem, as described by Colboum and van Oorschot (1989), the branching arising in this class of algorithms leads to lower complexity bounds. Here binary branching trees occur, which are closely related to Steiner triple systems. De Brandes and Rod! (1984) have shown that there exist S(2, 3; v) whose minimum set cover has a size v - O(v!). From these exponential lower bounds are derived. In this application, it is the intersection property of Steiner triple systems of two blocks meeting in one point that leads to this result, which of course is a consequence of balancedness. But also designs where the block size is about! of the set of points available, namely Hadamard designs, can be used to give lower complexity bounds. This was done by Chor and Goldreich (1988), who proved that the worst case probabilistic communication complexity can be bounded from below by a recent combinatorial result in the theory of Hadamard designs, due to Frankl, Rodl, et al. (1988): (7.4.b)
In a Hadamard design on v points, let X be a set of i elements and Y be a set of j blocks. Let c(X, Y) be the total number of occurrences of elements of X in blocksofY. Then
1 2(ij
1 -/iIV) :s c(X, Y) :s 2(ij + /iIV) .•
The last example shows that through applications of the type discussed in this section not only have properties of t-designs been successfully applied, but it can also be expected that in the future more questions wiIl arise in practical areas, that stimulate surprising results in mathematical structures that have been known for a long time.
§8. Designs in Hardware 8.1. Design of Magnetic Core Access Switches One other "historical" example of applications of designs concerns the design of magnetic core access switches, where n cores are given (cf. Figure 8.1 and Minnick and Haynes 1962). Each core can hold one bit of information. To access the different cores m, wires WI, ••. , Wm carrying a unit current are used
938
XIII. Applications of designs
Figure 8.1. Magnetic core memory (courtesy ofD. Zerfowski).
to address the cores by induction. The wires can wind around the core positively or negatively or do not wind the core at all. According to the direction of the winding and the current CYj E {O, I} on the wires, in each core i there is an induced signal Ci which is a superposition of the signals CYj Pi,j generated by the single wires W j. The contribution of wire W j to Ci is given by
(8.I.a)
if wire
Wj
winds core Ci in a positive sense,
if wire
Wj
winds core Ci in a negative sense,
if wire
Wj
does not wire core Ci.
To select a core i, the corresponding signal Ci has to exceed a given activation threshold A. So a flow ai := (ai,I, ... , ai,lIl) of current has to be found with 2:7=1 ai.j Pk,j 2: A if and only if k = i. Obviously there are many variations for the above situation. For instance the set of windings can be restricted to positive and non-winding (unipolar) (Pi.j E {O, I}) or only negative and positive (bipolar) windings (Pi,j E {-I, I}).
§8. Designs in hardware
939
Also the restriction to unit currents can be weakened allowing several different currents. The overall goal in this problem is • to minimise the number 111 of wires, and • to optimise the signal-to-noise-ratio • under a simultaneous maximization of the so-called load sharing factor on each wire.
1: }
This can obviously be solved using a balanced incomplete block design (V, B, I), as these goals correspond exactly to those described in previous sections. Let X = {I, ... , n} be the set of cores and B= {B I , ..• , Bm} the set of blocks. Then the incidence I is given by the matrix p. For unipolar windings p is the 0-1 incidence matrix of a block design and the theory of Section 2 shows that the optimisation requirements are solved by choosing a BlED for (p). In the case of bipolar windings (Pi.j E +1, -1), the design is formed by the blocks B j = {i I Pi,j = I}. From Section 2.1 it is clear that the optimisation goals are achieved by a symmetric block design SJ..(2,k; v), as its (1, -I) incidence matrix guarantees a signal of strength Ci = r on wire i and a signal of strength 2A - r to all other cores. With a bias wire (corresponding to the extended design) carrying a predetermined bias r - lA, the activation threshold is set off to 2r - 2)" with zero mean noise amplitudes. Symmetric designs with maximal r - )., are therefore to be used, preferably Hadamard designs, cf. Constantine (1958, 1960) and Singleton (1962).
8.2. Interconnection Networks Designs are also used for the construction of interconnection networks which are necessary for establishing communication paths between v nodes (vertices), i.e. input and output nodes, in an efficient way. Systems of buses can be modelled by k-subsets (blocks) of nodes with restricted connection between the nodes themselves. Often switch nodes for bus interconnection are inserted to get a hierarchical, logical, or physical communication structure. This has to be done under the conflicting constraints: • to use a small number of buses; • each node should connect to a small number of buses (fan out/in problem); • only a small number of switch nodes is allowed;
941
§S. Designs in hardware for perfonning the desired arrangements.
In contrast to the ideas of Hagelbarger an alternative approach has been given by Beth and Hatz (1991, 1992) and Hatz (1992). In this concept of "design machines" the cross-bar switches are eliminated by the use of routing chips that are based on Steiner systems. The processors are connected via routing chips which can realise the complete connection graph between all inputs. There is no longer an asymmetry between inputs and outputs. Therefore a time division method based on I-factorisation of the complete graph K 2m is applied, cf. Example 1.5.7. An ASIC-(VLSI) chip performs the switching between these sub graphs in a time multiplexed manner. One step of the time multiplex can be regarded as an operation of an automorphism on the set of blocks
U{{i -
B; = {{oo, in
(8.2.a)
j, i
+ j} : j
=
I, ... ,i-I}, with
i = 0, ... , n - 2.
The automorphism ifJ operating on the set of blocks is defined by
ifJ: B; -+ B(i+I)
..
ifJ: fl, J} ~
mod(n-2)
{U+ 1,j+1}
if j E
Zn-],
+ 1, oo}
if j =
00.
{i
During the repeated application of the automorphism ifJ, all I-factors of S(2, 2; n) are "visited". Figure 8.3 visualises a mechanical or geometrical in-
terpretation of the subgraphs Kg. In the situation shown on the left there exist connections between points 1 and 6, 2andS, 3 and 4 as well as oand 00. The automorphism ifJ is represented by a rotation of the slice containing the connection lines by A VLSI implementation of the above concept of switching between separate factors is realised as a chip (see Figure 8.4). During one complete "rotation" of the connection structure each node has temporary connection to
2; .
o
o
C9
1
5
~
4
2
~
5~2
3
Figure 8.3. Geometrical interpretation of I-factorisation.
941
§S. Designs in hardware for perfonning the desired arrangements.
In contrast to the ideas of Hagelbarger an alternative approach has been given by Beth and Hatz (1991, 1992) and Hatz (1992). In this concept of "design machines" the cross-bar switches are eliminated by the use of routing chips that are based on Steiner systems. The processors are connected via routing chips which can realise the complete connection graph between all inputs. There is no longer an asymmetry between inputs and outputs. Therefore a time division method based on I-factorisation of the complete graph K 2m is applied, cf. Example 1.5.7. An ASIC-(VLSI) chip performs the switching between these sub graphs in a time multiplexed manner. One step of the time multiplex can be regarded as an operation of an automorphism on the set of blocks
U{{i -
B; = {{oo, in
(8.2.a)
j, i
+ j} : j
=
I, ... ,i-I}, with
i = 0, ... , n - 2.
The automorphism ifJ operating on the set of blocks is defined by
ifJ: B; -+ B(i+I)
..
ifJ: fl, J} ~
mod(n-2)
{U+ 1,j+1}
if j E
Zn-],
+ 1, oo}
if j =
00.
{i
During the repeated application of the automorphism ifJ, all I-factors of S(2, 2; n) are "visited". Figure 8.3 visualises a mechanical or geometrical in-
terpretation of the subgraphs Kg. In the situation shown on the left there exist connections between points 1 and 6, 2andS, 3 and 4 as well as oand 00. The automorphism ifJ is represented by a rotation of the slice containing the connection lines by A VLSI implementation of the above concept of switching between separate factors is realised as a chip (see Figure 8.4). During one complete "rotation" of the connection structure each node has temporary connection to
2; .
o
o
C9
1
5
~
4
2
~
5~2
3
Figure 8.3. Geometrical interpretation of I-factorisation.
§8. Designs in hardware
943
munication paths are disjoint. Each such barrel shifter is a set of multiplexed, hardware coded shifters. To minimise the needed chip area, not all shifts by I, ... , n are realised in hardware, but the missing shifts can be performed by concatenated executions of a suitable number of hardware realised shifts. Owing to the underlying structure of ZI7 these can naturally be chosen through the use of appropriate difference sets. For further details see Kilian, Kipnis and Leiserson (1987) and the references quoted in Colboum and van Oorschot (1989).
Figure 8.5. 12 bit barrel shifter for shifts 0, ... ,7 (445 /Lm x 100 /Lm), cf. Beth and
Hatz (1990).
8.4. VLSI Testing Reduces to High-Table-Design A special class of tournament designs, cf. Section 1.2, is used in a paper that addresses a serious challenge in modem VLSI-hardware design. To localise point defects occurring between several interconnection Jayers of modem multilayer VLSI chips Hess, Weiland and Bornefeld (1997) propose a multilevel checkerboard test structure (Figures 8.6-8.10). This checkerboard
Figure 8.6. Transformation of the lowest layer of a vector matrix to a checkerboard test structure (taken from Hess, Weiland and Bomefeld 1997).
§8. Designs in hardware
943
munication paths are disjoint. Each such barrel shifter is a set of multiplexed, hardware coded shifters. To minimise the needed chip area, not all shifts by I, ... , n are realised in hardware, but the missing shifts can be performed by concatenated executions of a suitable number of hardware realised shifts. Owing to the underlying structure of ZI7 these can naturally be chosen through the use of appropriate difference sets. For further details see Kilian, Kipnis and Leiserson (1987) and the references quoted in Colboum and van Oorschot (1989).
Figure 8.5. 12 bit barrel shifter for shifts 0, ... ,7 (445 /Lm x 100 /Lm), cf. Beth and
Hatz (1990).
8.4. VLSI Testing Reduces to High-Table-Design A special class of tournament designs, cf. Section 1.2, is used in a paper that addresses a serious challenge in modem VLSI-hardware design. To localise point defects occurring between several interconnection Jayers of modem multilayer VLSI chips Hess, Weiland and Bornefeld (1997) propose a multilevel checkerboard test structure (Figures 8.6-8.10). This checkerboard
Figure 8.6. Transformation of the lowest layer of a vector matrix to a checkerboard test structure (taken from Hess, Weiland and Bomefeld 1997).
L
944
XIII. Applications of designs
Figure 8.7. Frame of a checkerboard test structure with 23 x 12 chips, taken from Hess, Weiland and Bomefeld (1997).
6 7 8 5
7 3 84725-+ 5
Figure 8.8. Example of the permutation procedure for n = 8 values. On the left the complete neighbourhood graph is shown. Each node represents a line connected to one pad. 1\vo nodes connected to these pads are adjacently placed anywhere inside a test chip with only non-conducting material between them. The lines in the right matrix mark pairs to line" 1". Taken from Hess, Weiland and Bornefeld (I997).
n_:~!1_0 1,~ '.
:
~.~ Fellow
0-1- (n-1)-2-(n-2)"'(#-I) -(#+1)-# 3 4 5 ••• 2n - 2 2n - 1 2n
# _ 1 in seat no. 1 2
•
#+1
Figure 8.9. Geometrical interpretation of cycles Co
= 1.
945
§8. Designs in hardware N
ci
ci 0::
«)
V
'" .::
ci
ci ci
2
3 4
\0
r- oo
ci
ci ci
0::
0::
'" ''""' '1;l"' '~"' '1;l"' ""'" '"'"' '"'"' '"'""' 1 2 3 4 5 6 7 8 0::
fellow #
0
seating no. I ~ seating no. 2 seating no. 3
0::
0::
5 6 7
XXX
"lXXY VXXY X X\
seatingno.4/X fellow #
4
5
3 627
1 0
Figure S.lO. Our solution for n = 8 reproducing the solution by Hess, Weiland and Bomefeld (1997).
structure divides the large defect sensitive area surrounded by boundary pads into m x T1 many subchips, well distinguishable by electrical measurements via the boundary pads in a special wiring pattern of a routing channel, described in the form of a ~ x n-subpermutation matrix with n = m + 1. The following combinatorially equivalent reformulation shows that this reduces to a design theoretic problem.
r
A high-table design (HTD) is a tournament of n = 8 fellows dining on a long linear table under the auspices of the sovereign's portrait hanging from the wall: the task is to arrange a suite of different seatings, so that after ndinners any two fellows have been placed next to each other exactly once. As for each dinner on a linear table of n seats there are n -1 neighbourships the necessary condition is (8.4. a)
G)
n
n= n -1 = 2'
By this reformulation we show that solutions to the high-table design problem can be constructed by factoring the complete graph Kn+l into edge-disjoint cycles of length n + 1. Indeed, if a fixed (n + l)st point plays the role of the sovereign, the remaining fellow elements of each cycle form a linear table seating as requested. Take the complete graph on n + 1 = disjoint cycles of length n + I:
2n + I points and partion it into nedge
(i) Solution I: Label the points with the elements of Zn U {ex)}.
Construction HTDl: The cycles Ci = (Co + i mod n) for i = 0, ... , partion the complete graph Kn+l (with cycle Co given in Figure 8.9).
n- 1
946
XIII. Applications of designs
Proof: Differences except +rt = -~ occur twice in each cycle. 00 plays the role of the sovereign. (ii) Solution II: Label the points with the elements of Zn+l and let S Units(Z,,+I) I{±l}'
Construction HDT2: With C = {O, 1,2, ... , n} the cycles a· C = (0, a, 2a, ... , na)
for a
E
S
partition the complete graph Kn+l if and only if lSI = is prime. 0 plays the rOle of the exceptional element.
rt if and only if n + 1
§9. Difference Sets Rule Matter and Waves It is well known that phase shifting devices are important tools in controlling wave-like phenomena, due to the effect of interference. Sound will be extinguished if the two speakers of a high-quality stereo set have been wrongly wired, producing a phase shift of JT:. Parabolic antennas, so-called satellite dishes, are built to collect radio waves in phase at the focal point. Lenses and mirrors can be generalised in diffractive optics to phase elements, e.g. for shaping beams of light. In optical astronomy so-called adaptive mirrors have been developed to compensate continuously the atmospheric aberration of light. From this a potential new mass product, the mirror array (see Figure 9.1) has been developed, providing an application as optical multiplexers such as cross-bar switches (cf. Beth and Hatz 1991) or for the projection of high-resolution TV pictures. This is a visible example of an active phased array, a technology which is also applied in motion free antennae, flat satellite TV antennae, ultrasound transducers and loudspeaker columns. In order to show the principle, in this section we shall present two examples of the theory of difference sets applied to the design of passive phase arrays.
controlling electrode mechanical support
substrate of Si
Figure 9.1. Divertible micro mirrors forming an adaptive mirror array.
§9. Difference sets rule matter and waves
947
9.1. Concert Hall Acoustics Brought Home Such an application for acoustic waves is given by Schroeder (1986), from which we quote: Recent research, based on a subjective evaluation of the acoustics of 20 major European concert halls, has shown that many modem halls have poor acoustics because their ceilings are low relative to their widths. Such halls do not provide the listener with enough laterally travelling sound waves - as opposed to sound travelling in the frontlback direction and arriving at the listener's head in his "median" plane (the symmetry plane through his head). Such median-plane sound, of course, gives rise to two very similar acoustic signals at the listener's two ears, and it is thought that the resulting excessive "binaural similarity" is responsible for the poor acoustical quality. How can one correct this shortcoming? One component of the median-plane sound, the direct sound from stage, cannot be suppressed because it is needed to localise properly the different instruments on the stage. In principle, the ceiling could be raised so that the lateral sound would dominate again, as in the preferred (but unprofitable) old-style high and narrow halls. But modern air-conditioning has made the low ceiling possible - and that is where mounting building costs will keep it.
For related material see also Schroeder (1979), and Schroeder, Gottlob and Siebrasse (1974). Alternatively, the reflected sound from the ceiling (found in abundance in almost any living room) could be dispersed into a lateral pattern by diffraction which is well understood, e.g. in optics, cf. Born and Wolf (1970). Such diffraction patterns are generated by interference of waves scattered from a so-called phase-reflex grating mounted on the ceiling (Figure 9.2). The localisation of the higher frequencies of audible sound is described by the far field (subwoofers can be placed anywhere). This is the Fraunhofer diffraction pattern which in first order can be approximated by the Fourier transform (cf. Beth 1984; Schroeder 1986) of the complex amplitude of the wave signal as a
Figure 9.2. 17 x 19 phase-reflex grating based on quadratic residues, see (9.1.j).
948
XIII. Applications of designs
spatial function of the grating surface. In the dual space, i.e. the Fourier domain, the properties of the desired grating can be expressed very elegantly:
(9. 1. a)
A broad diffraction pattern is characterised by a wide spectrum of the grating. For good "equalised" audibility the power spectrum should also be flat.
These requirements can be modelled and implemented rather directly, if the grating is realised by a step-function of troughs of width wand depth an, see Figure 9.3, generated by a periodic sequence (an)nE[O: L-l]. The phase of a vertically incident wave of wavelength), is changed through reflection in the n-th trough of depth an E [0: N - 1] by (9.l.b)
CPn
4nan
= -).-.
troughs cross-dimension of hall - .
-
Figure 9.3. Proposed concert hall ceiling design, a reflection phase-grating based on a quadratic residue sequence for the prime 23.
With these elementary preparations from physics we now proceed to characterise the spectrum generated by the complex phase factors (e irpn )nEZL' For this we recall, cf. Babovsky, Beth et al. (1987), that the DFT-spectrum (Rm)mEZL of a complex sequence (rn)nEZL is given by (9.l.c)
with the power spectrum (9.l.d)
where (9. I.e)
Ck
=L nEZL
rnr;+k' k E ZL
949
§9. Difference sets rule matter and waves
is the k-th autocorrelation coefficient (cf. also Section lOA, property R3) of (rn)n E 'llL.
It was Schroeder's idea (1980) to propose phase-reflex gratings with the sequence an, rather than chosen randomly, to be defined algebraically by
(9.U)
._ bnA an ·- 2L
where bn := n 2 mod L for an odd prime L. It has the following spectral properties. Let (9.l.g)
be the n-th reflection coefficient. Then by elementary number theory (9.1.h)
sums to (9.l.i)
Ck
= {O L
for k ¢ 0 k=O.
mod L
Applying the DFT we obtain IRml2 = p for all m residue ceiling is shown in Figure 9.3.
E 'llL'
Such a quadratic
Because of formula (9 .1. b) the phase shift generated by the troughs is proportional to the frequency and the reflection coefficient for a frequency I f is rl for the frequency factor l, so that Equation (9 .l.h) holds also for this sequence. Thus a quadratic residue ceiling is highly diffusing for log2 LL - 1J octaves, e.g. more than four octaves in the case L = 23 of Figure 9.3. The real problem lies in the design of two-dimensional arrays, which scatter equal intensities into all diffraction directions over the (lower) hemisphere. This can be achieved by defining the two-dimensional step function array (an,m)neZp,mezq with primes p, q such that the reflection factors are given by (9.l.j)
rnm := e
2 m2) 2Jr.(np+q . I
This phase array being the direct product of two one-dimensional arrays is separable (Figure 904), and thus has a large symmetry in the x- and in
XIII. Applications of designs
950
y-directions. This symmetry could result in unwanted nodes and antinodes along the "median" plane and parallel to the rows of the seats.
(a)
(b)
(c)
Figure 9.4. Separable phase plate (a) given by Equation (9.l.j) with p = 19, q = 17 in grey value representation, showing the x-y symmetry, d. Figure 9.2. While the ideal far
field of this distribution is as flat as predicted by (9.1.i) a simulation of the far field (b) of the phase plate with an absorbing neighbourhood shows spatial modulation. (c) shows the same as (b) with reflecting neighbourhood. These simulation have been performed with the system DigiOpt®. To overcome this disadvantage the construction can be modified by applying some design theory in addition to the number theory already exercised. If the array, for example is built from a Hadamard difference set as a twodimensional q x (q + 2) array according to the Stanton-Sprott theorem (cf. Theorem VI.8.2), an even better array can be achieved, cf. Fenimore and Cannon (1978). From Equations (9.l.d) and (9.l.e) it is natural that twodimensional difference sets with suitable automorphism groups form the building pattern for diffusers fulfilling the flatness condition (9.l.a) equally well over a large interval of frequency factors, which indeed correspond to numerical multipliers of the difference set. For further examples of sequences and arrays fulfilling these properties see Pott (1995), Fenimore and Cannon (1978) and Gottesman and Fenimore (1989). Such arrays have independently been developed for applications in astronomic~., . imaging techniques. In the subsequent examples the importance of difference set theory will become evident.
9.2. Astronomical Imaging In radio astronomy antenna arrays are used for a wide range offrequencies, cially because of their ability to restore "images" which are observed 'h.'n .. ~h. a turbulent atmosphere. This technique is based on interferometric ovrlth"oio generated by non-redundant masks to be used within telescopes. An H.1~,aU'''u
------~
§9. Difference sets rule matter and waves
951
antenna array can be considered as a two-dimensional grid, where each pair of grid points serves as an interferometer selecting the signal component of specific wavelength analogously to the description of Equation (9.l.b). To avoid interference of the components of the same spatial frequency the arrays should be non-redundant. Such antenna arrays are usually large spatially spread out installations, sometimes several kilometres apart, e.g. the Australian Radio Telescop e. But the same principle can be applied to telescopes for much shorter wavelengths, namely in X-ray and y-ray astronomy. Here the antenna array can be thought of as being replaced by a two-dimensional mask, put into the aperture of the telescope. The mask implements the coded aperture principle, cf. Gottesman and Fenimore (1989), which in the context of diffractive optics is applied in binary masks, cf. Aagedal, Beth et al. (1996). A single opening of a simple pinhole camera is replaced by many pinholes randomly. In this wayan imaging system can be constructed, which has the high angular resolution of a single pinhole with the signal-to-noise ratio improved by a factor of .IN along the lines of SNR improvement due to Hadamard transfon n optics, cf. Section 4. As N can be larger than 106 in these frequency ranges, the results are spectacu lar, cf. Fenimore and Cannon (1978). In a series of papers Kopilovich (1984, 1988a, 1992) and Kopilov ich and Sodin (1991, 1994) proposed algorithms for the construction of such masks using two-dimensional difference sets (see Chapter VI, also cf. 9.1). One method starts with relative difference sets (RDS, see §VI. 9) with paramet ers (m, n, k, A) on a rectangular m x n grid. This is a set R = {(ai, bi)} of k points of the grid, such that any point (/-L, v) not belonging to a grid side has exactly J.. representations of the fonn (/-L, v) = (ai - aj mod m, b - b mod /1). One i j example of an RDS with A = 1 is the set lei, j2 mod p)}, i = 0, ., ., p - 1, on the p x p integer square with p prime. Another construction method uses cyclic difference sets (see Definition VI. 1.2). Since (u, k, J..) cyclic difference sets for A = 1 exist for all k = q + 1 and u = , q2 + q + 1 with q a power of a prime number, these difference sets give linear masks of k + 1 elements on the interval [0, u]. By folding and applying shears these linear masks can be transformed into two-dimensional masks. , An example (Kopilovich 1984) for /1 = 20 based on the (381,20 , 1) cyclic difference set (i.e. to, 1, 19,28,9 6, 118, 151, 153, 176,20 2,240,2 54,290 , 296,300 ,307,33 7,361,3 66, 369}) is shown in Figure 9.5. Masks made from such difference sets and RDS give so-called uniformly redundant arrays, i.e. as two-dimensional difference sets TDS, cf. Kopilovich and Sodin (1994). They
XIII. Applications of designs
952
have a two-dimensional autocorrelation function (cn,m)n,m which is two-valued (9.2.a)
crs ,
=
k { )..
ifr = 0
modm ors =0
modn,
elsewhere.
As in the example in Figure 9.5, this combines the high angular resolution with a side-lobe-free point spread function of the telescope with small aperture. What is more, the ranges of frequencies covered can be made extremely large, if the difference set has many numerical multipliers, cf. Section VI.2. Another family of masks with three-valued correlation functions has been proposed by Kumar (1988), with applications ranging beyond coded aperture imaging into spread spectrum technology, where pseudorandom patterns, cf. Section lOA, are required for frequency allocations or so-called hopping patterns, cf. Chambers (1985), Schroeder (1986).
o
o 20 40 60 80 100
120 140 160 180 200 220 240 260 280 300
320 340 360
380
1 2 3 4 5 6 7 8 9
••
10 11 12 13 14 15 16 17 18 19
•
•
•• •• • • • • • • •• • •• • •
Figure 9.5. Mask on a 20 x 20 grid based on a (381,20, 1) cyclic difference set.
9.3. An Application in Neutron Spectroscopy An interesting application of designs to neutron spectroscopy has been given by Gompf (1970), Wilhelmi and Gompf (1970) (see also Pickert 1974). As shown in Figure 9.6, a beam of neutrons is sent through a test object in which these
§9. Difference sets rule matter and waves
Chopper
953
Analyser
Figure 9.6. Measurement of speed distribution of neutrons.
neutrons are slowed down resulting in a corresponding speed distribution. After passing the object, the beam reaches a rotating disc which is subdivided into v sectors. Some of these sectors contain slots; the others are closed for neutrons, which leads to corresponding neutron pulses (containing a collection of neutrons of different speed) behind the chopper. A synchronously rotating, sector segmented analyser measures the number of arriving particles. Since neutrons with different speeds need different times to cover the distance between the chopper and the analyser, the beam is distributed over several analyser segments. One way to determine the speed distribution of the neutrons would be to take a chopper with only one slot and to make measurements over several turns of the chopper and analyser. But in this case most of the neutrons in the beam would not be used and the time of the measurement would be unnecessarily high. For this reason an intelligent pattern of slots in the chopper has to be found to exploit the neutron source in a more efficient way to distinguish the different neutron speeds. Therefore we have to consider the traversal time of neutrons between the chopper and the analyser. We now tum to modelling the chopper. Let V = {i I v > i, i E No} be the set of the v chopper sectors and Bo C Va subset to be found, which determines the transparent sectors. The corresponding incidence is (9.3.a)
ifi
E
Bo
ifi E V\Bo.
Suppose T is the duration of one rotation of the system. Then x j neutrons with traversal time between j ~ and (j + 1) ~ will be detected in the i -th analyser 1 and l + j i mod v. sector, if and only if there exists lEV with al
=
=
XIII. Applications of designs
954
Summing over these x j yields the numbers (9.3.b)
Yi = I>i-jXj,
for all i E V
jeV
of neutrons detected in the i-th analyser sector. Based on the observed Yi the E V, have to be determined using the matrix equation
x j, j
(9.3.c)
Now the ai should be chosen so that the x j can be determined by equation (9.3.c), the dispersions of the Yi should lead to small dispersions of the x j (i.e. small errors in the measured data should lead to small errors in the calculated x j) and the proportion ~ of used neutrons, with k = I{ i : ai = I} I, should be maximised for reasons of efficiency. Since the measurements Yi are superpositions of several events, a simple decoupling can be performed. In the special case Yz = 1, Yi = 0, i E V \ {I}, obviously there exist bj, j E V, with
(9.3.d)
which is a solution of equation (9.3.c). Therefore the problem can be reformulated: for given integers v and k with k < v find a v-tuple (a; );eV with {a;: i E V} ~ {O, I}, k = I{i: ai = Ill. Furthermore (ai)ieV should minimise the square sum Ljev bJ, where the v-tuple (bj)jev is uniquely determined by equation (9.3.d) for I = O. Solving this minimisation problem leads to the following. Given a Bo C V, with IBol =k, IVI = v and the subsets Bi = {j for each v-tuple (b j ) jeV E R" satisfying
(9.3.e)
""
~b·jeB, J -
{I 0
ifi=O if i E V \ {OJ
+i
I j E Bo},
§9. Difference sets rule matter and waves
955
the inequality (9.3.f)
1 (I 1)2 I>2> k_+ kk jeV
] -
v -
holds, with equality if and only if for all i (9.3.g)
IBo
E
V \ {OJ
k-l
n Bd = k - - =: A. v-I
In the case of equality, one has I
(9.3.h)
bj
={
if j E Bo,
k I-k k(u-k)
otherwise.
Note that IB j n Bd = A for i =1= j and again we have an application of a S)..(2, k; v)-design, where A = ki~=g and the blocks of the design are Bj := {i + j mod v: i E Bo} with j E V. Since the design is cyclic according to the set-up of the experiment, we have derived a situation where cyclic difference sets are naturally required. The material covered in this section provides some examples for the applications of difference sets and their generalisations in a variety of engineering problems. Usually such applications require translating the difference set into a periodic (in most cases binary) sequence or array and exploiting their correlation properties. Note that the autocorrelation function of a sequence (or array) is a measure for how much the given sequence (or array) differs from its translates. In particular, periodic binary sequences with good correlation properties have important applications in various areas of engineering. For instance, one needs sequences with a two-level autocorrelation function, that is, all non-trivial autocorrelation coefficients equal some constant y. The case where y is as small as theoretically possible in absolute value has turned out to be especially useful; such sequences are called perfect. Unfortunately, in many cases perfect sequences cannot exist, and so one has also considered "almost perfect" sequences, where one allows one non-trivial autocorrelation coefficient to be different from y. For this topic, the interested reader should consult the recent survey by Jungnicke1 and Pott (1999) and the references given there; these authors also describe the application of the twin prime difference sets (see Theorem VI.8.2) to astronomical imaging first proposed by Fenimore and Cannon (1978) in some detail. For an extensive treatment of the connections between difference sets, sequences and their correlation properties (and their applications), we refer to the recent collection of papers edited by Helleseth, Jungnickel et al. (1999).
956
XlII. Applications of designs
§10. No Waves, No Rules, but Security
Und zehn ist keins? Das ist das Hexen-Einmal-Eins! Secure information transfer and processing has become an important topic of computer and system science. Many methods from discrete mathematics are applied here or have been developed for this purpose. The area of security is considered to consist of the areas confidentiality, integrity and authenticity (CIA). The former notion of security and confidentiality, usually being associated with the classical (not: classified!) field of cryptology, has attracted many mathematicians to apply their knowledge. Except for the last Section lOA this book does not touch upon this topic. The subsequent examples of designs are therefore concerned with the two latter areas, integrity and authenticity.
10.1. Authentication Codes A very important application for designs has been found in the area of secure and authenticated message transfer or communication, see Simmons (1992a). While methods from coding theory (cf. Section 5) are commonly used to protect messages against errors occurring on the communication channel (e.g. against physical disturbance like thunder during radio transmission), in several cases of valuable applications, additional authentication of the sender is desired. Combining these two features leads to a special class of codes called authentication codes. Authentication has become the predominant topic of security, especially in networks and computers. This development has been foreseen by G. J. Simmons (1985b), who at that time had been responsible for designing command-controlsystems with very high security level (cf. Beth and Diffie (eds.) 1996). For an introduction to this topic and the duality between coding theory and authentication theory, we quote G. J. Simmons (1985a), who by now is generally considered the father of authentication theory. We consider a communications scenario in which a transmitter attempts to inform a remote receiver of the state of a source by sending messages through an imperfect communications channel. There are two fundamentally different ways in which the receiver can end up being misinformed. The channel may be noisy so that symbols in the transmitted message can be received in error, or the channel may be under the control of an opponent who can either deliberately modify legitimate messages or else introduce fraudulent ones
§10. No waves, no rules, but security
957
to deceive the receiver. The device by which the receiver improves his chances of detecting error (deception) is the same in either case: the deliberate introduction of redundant information into the transmitted message. The way in which this redundant information is introduced and ·used, though, is diametrically opposite in the two cases. For a statistically described noisy channel, coding theory is concerned with schemes (codes) that introduce redundancy in such a way that the most likely alterations to the encoded messages are in some sense close to the code they derive from. The receiver can then use a maximum likelihood detector to decide which (acceptable) message he should infer as having been transmitted from the (possibly altered) code that was received. In other words, the object in coding theory is to cluster the most likely alterations of an acceptable code as closely as possible (in an appropriate metric) to the code itself, and disjoint from the corresponding clusters about other acceptable codes. In Wyner (1975) and Simmons (1981) the present author [i.e. Simmons1showed that the problem of detecting either the deliberate modification of legitimate messages or the introduction offraudulent messages; i.e., of transmitter and digital message authentication, could be modeled in complete generality by replacing the classical noisy communications channel of coding theory with a game-theoretic noiseless channel in which an intelligent opponent, who knows the system and can observe the channel, plays so as to optimise his chances of deceiving the receiver. To provide some degree of immunity to deception (of the receiver), the transmitter also introduces redundancy in this case, but does so in such a way that, for any message the transmitter may send, the altered messages that the opponent would introduce using his optimal strategy are spread randomly, i.e., as uniformly as possible (again with respect to an appropriate metric) over the set of possible messages, M. Authentication theory is concerned with devising and analyzing schemes (codes) to achieve this "spreading". It is in this sense that coding theory and authentication theory are dual theories: one is concerned with clustering the most likely alterations as closely about the originalciJde as possible and the other with spreading the optimal (to the opponent) alterations as uniformly as possibly over M. The probability that the receiver will be deceived by the opponent, Pd, can be bounded below by any of several expressions involving the entropy of the source H(S), of the channel H (M), of the encoding rules used by the transmitter to assign messages to states of the source H(E), etc. For example: (lO.l.a)
10g(Pd)
~
H(MES) - H(E) - H(M).
The authentication system is said to be perfect if equality holds in (1 O.l.a), since in this case all of the information capacity of a transmitted message is used to either inform the receiver as to the state of the source or else to confound the-opponent. In a sense, inequality (1 O.l.a) defines an authentication channel bound similar to the communication channel bounds of coding theory. Constructions for perfect authentication systems are consequently of great interest since they fully realise the capacity of the authentication channel.
This scenario can be reduced to a set theoretical model as done by Simmons in his early keynote paper, ct. Simmons (l985a), which is here reproduced with the author's consent.
958
XIII. Applications of designs
There is a source (set) S with a probability distribution S on its elements for which the binary entropy is H(S). H(S) is the average amount of information about the source communicated to the receiver by the transmitter in each message. There is also a message space M consisting of all of the possible messages that the transmitter can send to the receiver. Since an unstated assumption is that the transmitter can communicate to the receiver any observation he makes of the source, IMI 2:: lSI where lSI is interpreted to be the cardinality of stales of S that have nonzero probability of occurrence. It should be obvious that authentication depends on the set of messages that the receiver may receive being partitioned into two nonempty parts: a collection of messages that the receiver will accept as authentic and another collection that he will reject as inauthentic. If 1M I = lSI, all messages would have to be acceptable to the receiver, hence no authentication would be possible in this case. Therefore, IMI > lSI and as we shall see later the even stronger inequality H(M) > H(S) holds as well. Figure 10.1 schematically shows the essential features of what has been described thus far.
Figure 10.1. Schematic description of the source space S and the message space M (Figure 1 from Simmons 1985a). Any message in the shaded region of M would be rejected by the receiver, while any message in the set MJ would be accepted as authentic. Figure 10.1 also illustrates that it is possible for the opponent to fail to deceive the receiver, even though he succeeds in getting him to accept a message that was not sent by the transmitter. Assume that the state of the source is S2 and that the transmitter chooses to encode this information by sending message 11!2 to the receiver. If the opponent - not knowing the information shown in Figure 10.1 of course - intercepts the message 11!2 and replaces it with 11!3, the receiver would accept m 3 as being authentic since it is one of the messages that the transmitter might have sent, even though it was not the message actually sent in this case. However the receiver would interpret 11!3 to mean that the source state was s2 - as observed by the transmitter. The opponent would lose in this case, in spite of the fact that he succeeded in having the receiver accept a fraudulent message, since the receiver is not misinformed as to the state of the source. There is a well-known precept in cryptography, known as Kerckhoff's principle, that the opponent knows the system, i.e, the information contained in Figure 10.1. It is equally reasonable to assume the same for authentication. Consequently there would be no authentication possible for the receiver using the scheme shown in Figure 10.1 alone. What is done instead is to have many such encoding rules in an authentication system all of which are known to the opponent - with the choice of the particular encoding rule in use being known only to the transmitter and receiver, similar in many respects
959
§lO. No waves, no rules, but security
M
Figure 10.2. Suggestion of a general scheme (Figure 2 from Simmons 1985a).
to the "key" known only to the transmitter and receiver in a cryptosystem. Figure 10.2 suggests the general scheme. Each encoding rule, ei, determines a proper subset Mi of M, IMi I 2: 151, and a mapping - perhaps one to many - of 5 onto Mi. The inverse mapping D is a well-defined function, i.e., for any e E £ and m E Mi the function D(e, m) defines a unique state in 5 U 1/), where I/) is the null set. Even this very intuitive description of authentication should make clear the reason for describing authentication as a problem in "spreading" messages in M. If m] is an acceptable message only in set MI, then the opponent, knowing the system, would be able to conclude that el was the coding rule being used if he saw m 1 in the channel and would then be able to substitute another message with certainty of deceiving the receiver. To avoid this it is necessary that each message occur in sufficiently many authenticating sets to (ideally) leave the opponent no more able to "guess" at an acceptable message after he has observed what the transmitter sent than he could have before the observation. This ideal can be achieved in infinitely many perfect authentication systems.
We now transform this early description into the language of design theory as follows. The information to be transferred represents one of k states from a state set S. One uses a set M of v ~ k possible messages together with an encoding rule for mapping the states into messages. Each encoding rule e E E, of b = IE I possible ones, is a one-to-one mapping of the elements from some subset of S to those of some subset of M. Now an authentication code is defined by the set of states, the set of messages, the encoding rules, represented by the (b x k) encoding matrix over M. For authenticated information transfer two parties use an encoding rule, which is only known by these parties. The enemy may know the complete set of all possible encoding rules but not the e E E actually used. This will be illustrated by possibly the earliest example of an authentication code given by Gilbert, MacWilliams and Sloane (1974) after Simmons had
XIII. Applications of designs
960
• 2
Figure 10.3. Projective plane PG(2, 3). acquainted them with this problem. For this consider the projective plane PG(2, q), for instance with q = 3, given in Figure 10.3. Encode the state s E GF(q) U too} (considered as the line at infinity) under the encoding rule
(10.l.b)
e: s
1-+
(s, c) E (GF(q) U {oo})2,
where e is a point of PG(2, q) and (s, c) is the affine line of slope s and affine offset c of the unique block containing the point e in the derived design (PG(2, q»s of the point s, cf. Figure 5.10. For q = 3 Table 10.1 shows the principle for this authentication code without secrecy. There are two interesting probabilities which have to be considered. First the probability Pi that an enemy can insert a message without detection at the receiver's side and second the probability Pa that an enemy can alter a delivered message without detection at the receiver's side. In this situation (lO.l.c)
k v
Pi::: -
and
k-l Pa::: - - . v-I
For our example Pa = ~:::i is optimal. In general, designs are associated with authentication schemes almost in a one-to-one manner, as was shown by Stinson (1988, I992a). These results are summarised in the following theorem. First it should be noted that special classes of designs are connected to special types of authentication codes. For example Stinson (1988) shows the following result. Cartesian authentication systems (i.e. with H(S 1M) = 0, thus without secrecy) with N = kn and b = n 2 are uniquely determined as transversal de-
§10. No waves, no rules, but security
961
Table 10.1. Principle of authentication code for PG(2, 3) State
Encoding rule
Message
00
00
01
10
11
20 0 00 01 02 0 00 02 01 2 00 01 02 1
21 1 12 10
000 001 002 0000 20 21 22 200
2
0
11
00 11 10
12 0 10 11
12 2
02 12 22 2 21 22 20 1 22 21 20 00 20 21 22 00
10 11
12 100 00 01 02 000
signs TD[k; n] with (lO.I.d)
H(S) - H(M) = H(E
I M)
= log2(n)
and thus Pi = Pa = ~ are optimal. • Last but not least we mention a result which represents the culmination of the theory of authentication as conceived by Simmons (l985b). For this we introduce the notion of an authentication system with peifect secrecy, cf. Simmons (1992a), p. 15, which is defined by requiring that the probability Pd = max (Pi> Pa) of deception fulfils the equality (lO.I.e)
log Pd = H(M
I ES) + H(S)
- H(M).
Then the fundamental theorem, cf. Stinson (1992a) (Theorem 4.2), shows the importance and exclusive rOle of designs in authentication theory.
Suppose we have an authentication code which provides peifect secrecy in which Pi = ~ and Pa = ~::: Then b :::: ~~:::~, and equality occurs if and only
J.
if
962
XlII. Applications of designs
• v-I == 0 mod k(k - 1); • there exists a S(2, k; v); and • both the source states and encoding rules are equiprobable. •
10.2. Shared Control by Designs
Threshold schemes were independently discovered by Blakley (1979) and Shamir (1979) when they devised shared secret schemes for robust key management for cryptosystems. While originally devised for solving the problem of recovery from a loss of keys, this technique has become an important topic in modem system management and control. Not only can the access and authorisation structure in large organisations, e.g. multisignature systems used in banks, be implemented in this technology, there is also an almost unlimited use of such techniques with the recently growing demand for distributed computing and secure outsourcing of computational tasks in large networks, cf. Otten (1992). Last but not least we shall mention the immense demand for such shared control systems to be expected from legally conforming democratically controlled key escrow systems, which are presently discussed worldwide,1O cf. Beth (1995a). The reader is referred to Simmons (1992a), Chapter 9, where shared control systems are surveyed extensively. The main idea which relates the shared control systems via threshold schemes to design theory is the following. Given a cryptographic key or an access privilege, one wants to divide the key into several, say w, parts of information and (securely) distribute these pieces of information among a set of people in such a way that any t of these w pieces of information gain access but fewer cannot. That means that a reconstruction of the secret can be obtained by the knowledge of at least t parts of the secret key. Obviously the secret key even can be reconstructed if at most w - t partial secrets are lost. Such shared secret schemes are called (t, w)-threshold schemes. Before we describe design theoretic examples of such structures which can be provided by finite geometries or Latin squares, we refer to an algebraic solution which had been invented almost as early as that by Blakley (1979).
10 Beth, Th.: "Zur Sicherheit in der Infonnationstechnik", FinalReport at the Deutscher Bundestag,
Bonn 1990, see also Infonnatik-Spektrnm, 1990, no. 13, pp. 204-215. Beth, Th.: Report at public hearing "Zukunft der Medien in Wirtschaft und GesellschaftDeutschlands Weg in die Infonnationsgesellschaft", Deutscher Bundestag, Bonn 1997.
§lO. No waves, no rules, but security
963
Shamir's (1979) construction is based on interpolation of polynomials over GF(q). Using Lagrange interpolation and the fact that any polynomial of de-
gree t - 1 is uniquely determined by t sample values at different positions, one randomly chooses such a polynomial p(x) and considers the constant part p(O) as secret. Mter the evaluation of p(x) at I different positions the corresponding values are distributed among I people. Any t-subset of these I so-called shadows determines the original polynomial and, in particular, its constant term and therefore the secret. There is an obvious way to implement different levels of privileges, because in a hierarchy more important persons can have higher privileges than others, which are nicely represented by this application of classical mathematics. Note that this solution has the disadvantage of lacking perfect secrecy; it can however be used for a large range of applications, e.g. in outsourced anonymous computing, cf. Otten (1992). Here we describe, however, the appealing solution of the problem that is provided by applying design theory.
10.3. Blakley's and Simmons' geometric construction Originally Blakley described shared secret schemes in a geometrical setting in which he prepared a large technological step, invented soon after by Simmons. He used intersecting lines in a finite projective space PG(n, q) the points of which are the one-dimensional subspaces of GF(q)n+l. As there exist (10.3.a)
n7~Lr+2(qi - 1)
n;=, (qi -
1)
r-spaces in PG(n, q), it consists of ["i l ] = qn+~~l points and [,,;1] = (q(:~'-N(~"~/) lines, where the number of li~es thr~ugh a point in PG(n,;) is ~ q_1 ,see
Proposltlon .. 1216 .. .
[n
Blakley exemplifies this in the special case PG(3, q) in which k = = q+1 q points are lying on a line and v = [~]q = q3 + q2 + q + 1 points exist in all. Furthermore, in PG(3, q), each pair of different lines intersects in at most one point. On the other hand, each point lies on q2 + q + 1 lines. For this reason the secret, being encoded by the coordinates of a point p, can be divided into private information pieces, for instance, the different lines through p. Now, it is obvious that any two out of w :::: q2 + q + 1 pieces uniquely determine the point p. Hence we have a (2, q2 + q + I)-threshold scheme. As in the case of polynomials this secret sharing system is not perfectly secure, even if generalised to higher dimensions.
964
XIII. Applications of designs lineg
Figure 10.4. Geometric representation of a shared secret P (taken from Simmons 1992a).
In Figure 1004 a situation to overcome problems of this type is visualised, namely a (3, q + I)-threshold scheme. This was invented by Simmons (1990) after he pointed out that the uncertainty about the secret is reduced by every degree of freedom lost through coalition of partners. In the example of a so-called perfect secret sharing scheme described in Figure lOA, this disadvantage is overcome. Here all possible secrets lie on the given line g, the indicator variety. The concrete secret, the coordinates of P, has to be determined by three parties PI, P2 and P3 in the sense that the point P lies in the unique intersection from line g and the plane n spanned by PI, P2 and P3 in the domain variety. Obviously, all these three points are necessary to identify P. Note that the points have to be chosen such that no three lie on a line, so that we have q + 1 participants in this scheme if we use a conic in n, cf. Remark VIII.5 .34. With this geometrisation (and algebraisation) the problem is solved (and implemented) very efficiently. The main requirement of shared control is to divide access privilege among w people, such that any t of these w gain access but fewer cannot. Defining a set V of v shadows (partial privileges), a set !C of m keys and a set B of b disjoint w-subsets of V, a (t, w, v)-threshold scheme is defined by the pair (B, !C) together with a mapping f : B -+ !C such that for every t -subset T of V and for all blocks B E B which contain T, the equation f(B) = K holds for some fixed K E !C. With this definition the subset T determines a unique key. Now, this secret key can be shared among w people by selecting a block B with f (B) = K and distributing the shadows in B among w people. As a result of this construction, any group of t of these w people uniquely determine K by f. Obviously, Steiner systems S(t, k; v) directly lead to (t, k, v)-threshold schemes. Symmetric designs with index A. lead to (t, k, v)-threshold schemes with t = A. + 1. A (t, w, v)-threshold scheme is perfect if no s shadows, s < t, give any information about the block containing these shadows. The upper bound on the number m of keys in a perfect (t, w, v)-threshold scheme is due to Stinson and
§lO. No waves, no rules, but security
965
Vanstone (1998): (1O.3.b)
v - (t - 1) , - w - (t -1)
m<
with equality if and only if there exists a (t - I)-resolvable S(t, w; v) that is partitionable into S(t - 1, w; v) designs, cf. Definition I.5.4. The preceding protocol has the shortcoming that each principal involved has to divulge his own partial secret which might breach the security. To overcome this problem each secret can be used only once, resulting in an overhead of secret distribution which can, e.g. be realised in some special military command and control systems (cf. Simmons I992a), but in wider applications alternative methods are needed. One way has been given by Otten (1992), Beth, Klein, Otten (1993) and Beth, Knobloch, Otten (1993) who proposed multiparty computations of the secret, without the need to disclose the partial secrets. This scheme, too, has the disadvanta&e of spreading the secret among the participants of the computation. Alternatively, Beth (1988) proposed using a zero-knowledge authentication scheme which results in a shared control instead of a shared secret scheme. In contrast to the previously described methods, the principals involved only disclose the image of their partial secret under a specific homomorphic oneway function. The personal secrets are kept secret from all other instances, especially from the decision instance. Thus, the distributed secrets can be used several times without restrictions on security (see also Beth 1995a, b). For more details the reader may want to consult the survey by Stinson (l992a) and forthcoming proceedings of the CRYPTO, EUROCRYPT and ASIACRYPT series.
10.4. Pseudorandom sequences While deterministic digital computers play an increasingly dominant role in practically all aspects oflife, there is an "almost unquenchable thirst for random numbers", cf. Schroeder (1986), Knuth (1969), Niederreiter (1992). Monte Carlo methods have become decisive tools in numerics, simulations and planning. Algorithm development and complexity theory have gained considerably from non-deterministic and probabilistic concepts, e.g. Las Vegas methods. For many purposes proper random sequences derived from physical processes even though difficult to obtain are useful and desirable; they are however
XIII. Applications of designs
966
expensive. Therefore most software packages contain deterministic algorithms, so-called pseudorandom generators (PRO), which have of course deficiencies with respect to their generating procedure. Some elementary generators are described by Schroeder (1986). Almost all of these PROs have been designed for special purposes, e.g. in statistics or approximation theory. Reusing them for instance for generation of noise (e.g. for simulation of communication links) usually turns out to be a bad choice, as, e.g., undesired spectral properties may inevitably be associated with the algorithm of the generator. Especially cryptographic applications require the availability of very long high-quality binary random sequences, cf. Sloane (1973), by which any communication can be made secure, cf. Beth (1982) and LUke (1992). The idea for this goes back to the AT&T engineer o. S. Vernam, who in 1917 (published as late as 1926) proposed the use of simple Vigenere ciphers with a random key, cf. Beth (1982), Beth and Hess and Wid (1983). This so-called Vernam cipher had been developed for use in telegraph systems where the plaintext, being a binary sequence in this case, is superimposed on a binary key sequence of length at least that of the plaintext length via mod 2 addition. This key sequence was historically realised as a sequence on a paper tape being used by both transmitter and receiver, who synchronously apply each portion of this tape only once (one time pad); see Figure 10.5. receiver
sender adder
adder
r-----,
plain
~-+~--~--+~*++-~
Figure 10.5. Vemam cipher with one-time pad (taken from Beth 1983a).
Thus emphasis has to be placed upon the question of how to generate the key sequence in order to make the Vemam system secure. Since coin-tossing - aside from all objections - is not quite what one would caIl a real-time random process which can be used in modem computer systems, one has to refer back to "approximate randomisation" by PROs. Under the assumption that such systems are available, an electronic stream cipher system basically has the form shown in Figure 10.6.
§10. No waves, no rules, but security sender
967
receiver adder adder , - - - - , plain 1-----4+1--+---+->+++--->1 decoder
Figure 10.6. A sequential cipher system (taken from Beth 1983a).
The tempting idea to replace coin-tossing by so-called feedback-shift registers is easily implemented, cf. Gollmann (1994). For this and deeper reasons PRGs are still produced and sold worldwide, even though they are known to be totally insecure, cf. Herlestam (1982). Their replacement by so-called non-linear-feedback-shift registers has been a major topic in cryptographic research since the late 1970s, cf. Herlestam (1982), Beker and Piper (1982), when new non-linear PRGs have been proposed, cf. Jennings (1980), Beth and Piper (1984), Gollmann (1985), Chan and Games (1990), Wolfman (1992), Bradley and Pott (1995). They are usually composed from linear-feedback-shift registers (LFSRs), whose feedback polynomial f(x) E GF(2)[x] is primitive of degree n (Figure 10.7). Thus such an LFSR produces a maximal (0, 1)-sequence of period 2" - 1, which has linear complexity n, however; cf. Jungnickel (1993a). The art and science of stream cipher technology relies on the combination of these sequences to
Figure 10.7. A linear feedback-shift register with feedback polynomial j(x) L7=1 Ci xi (taken from Beth 1983a).
=I+
XIII. Applications of designs
968
I
J
initialisation S
---- r--
. . . . . .
I
-.
register j I 1x ,-d x ,-21 • 1 • 1 • 1 • 1 • 1 • 1 • I x 1 I xo 1
...
~\
. . . . . . . //II ··· - r--
~
g
fi x ,_" ... , x o)
II 1
Figure 10.S. A non-linear feedback-shift register (taken from Beth 1983a).
fonn a good non-linear pseudonoise generator (PN generator), the behaviour of which can be described wholly in tenns of number theory, polynomial algebra over finite fields, discrete Fourier transfonns, autocorrelation functions and automata theory, cf. Liineburg (1979), Blakley and Purdy (1981), Jansen (1989), Pott (1995), Vielhaber (1997); see Figure 10.8. To protect against the intrusion of an outsider who uses cryptanalysis to determine the predictability of the sequence, cf. Jansen (1989), it is usually assumed that the statistical properties of such sequences can be guaranteed. Quoting Beker and Piper (1982) p. 169, these can be described by the following termi-
nology: If (St) is a binary sequence then a run is a string of consecutive identical sequence elements which is neither preceded nor succeeded by the same symbol. Thus, for example, 011100 I begins with a run of one 0, contains a run of three I s and a run of two Os, and then ends with a run of one 1. A run of Os is called a gap while a run of Is is a block. Suppose that (sr) is a binary sequence of period p (i.e., sm+p = Sm for every m). For any fixed r, we compare the first p terms of (St) and its translate (SHT). If A is the number of positions in which these two sequences agree and D (= p - A) is the number of positions in which they disagree, then the autocorrelation function e(r) is defined by: (IO.4.a)
A-D
C("r)= - - .
p
These of course can be written as a convolution in the spectral domain, cf. Section 9.1, Equation (9. Le). We take some more notions from Beker and Piper (1982) as follows:
§10. No waves, no rules, but security
969
Clearly C (t + p) = C (t) for all t, so it suffices to consider only those t satisfying < p. When t = 0 we have in-phase autocorrelation. In this case, clearly, A = P and D = 0, so that C(O) = 1. For t =I 0 we have out-oJ-phase autocorrelation.
o :s t
The following three randomness postulates for binary sequence of period p were proposed by Golomb (1982). (IOA.b)
Rl If p is even then the cycle of length p shall contain an equal number of zeros and ones. If p is odd then the number of zeros shall be one more or less than the number of ones. R2 In the cycle of length p, half the runs have length 1, a quarter have length 2, an eighth have length 3 and, in general, for each i for which there are at least i+ 1 runs, of the runs have length i. Moreover, for each of these lengths, there are equally many gaps and blocks. R3 The out-of-phase autocorrelation is constant.
t.
Sequences fulfilling these "Golomb axioms" are called PN-sequences (pseudo noise sequences). While these axioms represent a most natural mathematisation of properties of a fair coin tossing sequence, i.e. a minimal set of properties, which cryptographically good sequences should bear, they imply the disappointing consequence for a cryptographer, in as far as from a design theoretic point of view they can be classified completely. This result is due to Piper and Walker (1984). For a binary sequence S So
(lO.4.c)
= (St) of period p the circulant (O-l)-matrix SI
Sp_1
A(s) =
SI
Sp_1
So
is the p x p-incidence matrix of a square I-design D(s) with regular cyclic automorphism group. The Golomb axiom R3 then implies that D(s) is a symmetric 2-design with a cyclic Singer group. Then from axiom Rl and R3 it especially follows that D(s) is a cyclic Hadamard design or the complement of a cyclic Hadamard design, having parameters SA (2, 2.:1. + 1; 4.:1. + 3) or SA (2, 2.:1.; 4.:1.-1) where .:I. is the weight of the intersection of any two row vectors of A (s). For instance, the PN-sequences of period 2n - I connected with the Hamming code Hn are generated via a primitive irreducible polynomial as described above. For detailed study of these cf. Jungnickel (1993a), and Chapter VI. A
XIII. Applications of designs
970
class of binary sequences of much higher linear complexity has been proposed by Gordon, Mills and Welch (1962) using the algebraic geometry of towers of extension fields, cf. §VI. 17. By completely different methods and criteria, other sequences of provably high complexity have been proposed, cf. Blum, Blum and Shub [1986]. The class of deterministic random sequences leaves room for many such constructions, cf. Beth and Dai (1989). Further developments can be found in Chapter 2 of Simmons (1992a). For recent results the reader is referred to the LNCS suite of Proceedings of the conferences CRYPTO, ASIACRYPT and EUROCRYPT, starting with Beth (1983a). Er hatte seine Kapazitat zu denken iiberschritten oder vielleicht konnte dort kein Mensch weiterdenken, wo er gewesen war. Oben, im Kopf, an seiner Schadeldecke, k1ickte etwas, es klickte beangstigend und harte nicht auf, einige Sekunden lang. Er meinte, irrsinnig geworden zu sein, und umkrallte sein Buch mit den Handen. Er lieB den Kopf vorniiber sinken und schloB die Augen, ohnmachtig bei vollem BewuBtsein. Er war am Ende. Ingeborg Bachmann, Das dreij3igste lahr ll
EPILOGUE Ach, es war noch allerlei; aber eigentlich haben Sie nichts versaumt. (Fontane)
II
Bachmann (1993).
Appendix. Tables Questo non piccioi libro e tutto pieno ...
(da Ponte)
In what follows we give a collection of tables of designs which may be used as a quick reference to the state of knowledge on certain parameter sets within a feasible range. In our experience such listings have proved helpful for both theoretical and practical purposes (e.g. in statistics). Naturally such a rather small collection always implies a selection; we think we have made a useful choice, and we hope that the reader will have access to the quoted main sources. In particular, a vast collection of tables is given in the recent CRC handbook of combinatorial designs edited by Colboum and Dinitz (l996a). We urge the reader to consult this handbook and its electronic update Colboum and Dinitz (1998) whenever he or she cannot find the desired information in the small set of tables presented here. No serious design theorist should be without this collection. The compilation we present would not have been possible without the support of several friends and colleagues, among whom we are indebted mainly to Andries Brouwer, Charles Colboum and Alexander Rosa as well as the late Haim Hanani.
§1. Block Designs We here present a table of block designs S)..(2, k; v) with k ~ vl2 and replication number r in the range from 3 to 17. This table is the most recent update of Hall's renowned Table 1 in his book Combinatorial theory originally published for r ~ 15 in 1967, with an extended range up to r ~ 20 in the 1986 edition, see Hall (1986). In our presentation, we rely heavily on the Tables ofparameters of BIBD's with r ~ 41 including existence, enumeration and resolvability results due to R. Mathon and A. Rosa which first appeared in 1985, see Mathon and Rosa (1985b, 1990); the most recent version can be found in Section 1.1.3 of Colboum and Dinitz (1996a). In Table A 1.1 below, we have listed all admissible parameter quintuples (v, b, r, k, A) with r ~ 17,
971
L
972
Appendix. Tables
i.e. those satisfying the necessary existence conditions (I.2.ll.a, b) and Fisher's inequality II.2.6. The parameter quintuples are ordered lexicographically with respect to r, k and J... (in this order); therefore, our enumeration coincides with that of Mathon and Rosa. We note that the tables of Hall go beyond the other tables mentioned by giving specific constructions, but warn the reader that the numberings do not coincide, since Hall does not include parameters of multiples of known designs. The column ND gives the number of non-isomorphic designs with the parameters in question or a lower bound for this number; similarly, the column NR contains information on the number of non-isomorphic resolutions of designs S),.(2, k; v). Note that this number is, in general, larger than the number of non-isomorphic resolvable designs with the given parameters. Here the entry "-" indicates that no resolution is possible, since k does not divide v. If the existence of a design (or a resolution) is - to our knowledge - still in doubt, we indicate this by writing a question mark "?" in column ND or NR, respectively. If the non-existence of a symmetric design is proved by Schutzenberger's theorem or by the Bruck-Chowla-Ryser theorem, we indicate this by "tI II.3.9" or by "tlIIA.6", respectively, in the column Comments and References. Similarly, if the existence of a quasiresidual design (cf. II.8.13) is excluded by Proposition 1.2.7 or by the Hall-Connor theorem II.lO.6, since the corresponding "master design" does not exist, we indicate this fact by "tlRes(#x)". Finally, if the design under consideration cannot be resolvable since it satisfies r = k +J... and would thus be affine by Bose's theorem whereas its parameters are not of the form (II.8.7 .a), we refer to this fact by the entry NR II.8.7. In a similar way, the entry Res(#x) or Ind(#x) indicates that the design in question may be obtained as a residual or an induced subdesign from the symmetric design No. x; cf. II.8.13, II.8.l4. By Ext(#x) we denote the (unique) extension of design No. x, which is a Hadamard design, cf. Theorems.1.9.9 and II.8.lO. An s-fold (quasi-)multiple of design No. x is denoted by s#x. We also indicate special cases of infinite series (like projective or affine geometries, Hadamard designs belonging to a Paley or a twin prime power difference set, etc.) by appropriate symbols (such as PG, AG, Paley, TPP, ... ).
In contrast to Hall's table, we usually refrain from giving exact constructions in general, there are many solutions. Nevertheless, we indicate at least one method or source of construction in the column Comments and References, preferably from the present book. If a design may be obtained as a multiple of a smaller one, this is indicated by the appropriate entry s#x; however, we will include an alternative construction for a simple design, if such a construction is known (to us) and reasonably easy. We also give references for further details in particular concerning the numbers ND and NR. Once again, we refer the reader to Section 1.1.3 of Colbourn and Dinitz (1996a) for corresponding information on designs in the range 18 :s r :s 41.
(
Table A1.I. Existence of designs with r::: 17 No.
v
b
r
k
A
1 2 3 4 5 6 7 8 9 10 11 12
7 9
7 12
13
13
14 15 16 17 18 19 20 21 22
6 16 21 11 13 7 10 25 31 16 15 8 15 36 43 22 15 9 25
10 20 21 11 26 14 15 30 31 16 35 14 21 42 43 22 15 24 50
3 4 4 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8
3 3 4 3 4 5 5 3 3 4 5 6 6 3 4 5 6 7 7 7 3 4
1 1 1 2 1 1 2 1 2 2 1 1 2 1 3 2 1 1 2 3 2 1
1 1 1 1 1 1 1 2 4 3 1 1 3 80 4 0 0 0 0 5 36 18
23 24 25
13 9 21
26 18 28
8 8 8
4 4 6
2 3 2
2461
13
ND
11 0
NR
Comments and references PG(2, 2); cf. 1.4.4 AG(2, 3) Res(#3); cf. 1.2.9 PG(2, 3); cf. 1.2.3 Res(#7), NR II.8.7; cf. 1.5.14 AG(2, 4) Res(#6); cf. IV.5.19, VIII.4.12 PG(2, 4); cf. IV.5.19, Vill.4.12 Paley; cf. III.9.6, IV.7.1O, VI. 1.12 ill.9.6; cf. Mathon, Phelps and Rosa (1983) 2#1 or dev({O, 1, 3), {O, 2, 3}) mod 7; cf. Nandi (1946a) Res(#13), (Vill.1.6.a); cf. Nandi (1946a) AG(2, 5) Res(#12); cf. MacInnes (1907) PG(2, 5); cf. MacInnes (1907) I.9.7, VI. 1.3; cf. Hussain (1945) I.5.8, ill.9.3; cf. Mathon, Phelps and Rosa (1983) AG2(3, 2), Res(#20), Ext(#l); cf. Gibbons (1976) 'i!Res(#19) 'i!Res(#18) 'i! II.4.lO 'i! II.3.9 PG2(3, 2); cf. Nandi (1946a) Ind(#40); cf. Mathon and Rosa (1977) VII.3.2; cf. Kramer, Magliveras and Mathon (1989) and Spence (1996) 2#3 ordev({O, I, 3, 9), (0,2,5, 6l) mod 13; cf. Pietsch (l994) Ind(#41), VIL1.6, VII.5.2; cf. Gibbons (1976) Connor (1952)
=
0 1
=
=
7 1 0 0
9
(cant.)
Table A1.I. (cant.) v
b
r
k
;.,.
ND
26 27 28 29 30 31 32 33 34 35
49 57 29 19 10 7 28 10 46 16
56 57 29 57 30 21 63 18 69 24
8 8 8 9 9 9 9 9 9 9
7 8 8 3 3 3 4 5 6 6
1 1 2 1 2 3 1 4 1 3
1 1 0 ;:::1.1 x 109 960 10 ;:::145 21
36 37 38 39 40 41 42 43 44 45 46
28 64 73 37 25 19 21 6 16 41 21
36 72 73 37 25 19 70 20 40 82 42
9 9 9 9 9 9 10 10 10 10 10
7 8 9 9 9 9 3 3 4 5 5
2 1 1 2 3 4 1 4 2 1 2
7 1 1 4 78 6 ;:::2 x 106 4 22859
47 48 49 50
11 51 21 36
22 85 30 45
10 10 10 10
5 6 7 8
4 1 3 2
4393
No.
NR
Comments and references AG(2, 7) = Res(#27); cf. Hall (1953, 1954a) PG(2, 7); cf. Hall (1953, 1954a)
;:::7 0
? 18920 0 1
~192
1 ~1O
~5
;:::22998
11 II.4.6 ill.9.5, VII.2.2; cf. Colbourn and Dinitz (1996a) Ind(#54), VII.4.9; cf. Colbourn, Colbourn, Harms and Rosa (1983) 3#1 or dev({O, I, 2), {O, I, 4). {O, 2, 4)) mod 7; cf. Morgan (1977b) VII.3.2; cf. Brouwer (1989) Res(#41), NR II.8.7; cf. van Lint, van Tilborg and Wiekema (1977) would have trivial group, see Siftar and Shita (1991) Res(#40), II. 10.8, VII.5.2; cf. Colbourn and Dinitz (1996a) and van Rees (1989) Res(#39), NR II.8.7; cf. Assmus, Mezzaroba and Salwach (1977) AG(2, 8)=Res(#38); cf. Hall, Swift and Walker (1956) PG(2, 8); cf. Hall, Swift and Walker (1956) VI.2.13; cf. Assmus, Mezzaroba and Salwach (1977) Hall (1986); cf. Denniston (1982) Paley, see VI. 1.12; cf. Gibbons (1976) 1.6.8; cf. Wilson (1974d), Dejter, Franek and Rosa (1996) 2#4; cf. Kageyama (1972), Gronau and Prestin (1982) 2#5, VII.5.2; cf. Pietsch (1994) III.9.6; cf. Mathon(1995) 2#6 or dev({3, 6, 7, 12, 14}. {7, 9, 14, 15, 19}); cf. Topalova (1998a) Ind(#63), VII.5.2; cf. Colbourn and Dinitz (1996a)
? 3809 0
0
Res(#54), NR II.8.7; cf. Spence (1992) l1Res(#53)
51 52 53 54 55
81 91 46 31 12
90 91 46 31 44
10 10 10 10 11
9 10 10 10 3
1 1 2 3 2
7 4 0 151
7
~106
~2
56
12
33
11
4
3
~17172470
:::1
57 58 59 60 61 62 63 64 65 66 67
45 12 45 100 111 56 23 25 13 9 7
99 22 55 110 111 56 23 100 52 36 28
11 11 II 11 11 11 11 12 12 12 12
5 6 9 10 11 11 11 3 3 3 3
1 5 2 1 1 2 5 1 2 3 4
~16
?
68 69 70 71
37 19 13 10
111 57 39 30
12 12 12 12
4 4 4 4
1 2 3 4
72 73 74
25 61 31
60 122 62
12 12 12
5 6 6
2 1 2
~118884
75
21
42
12
6
3
:::236
~II.3.9
11603 ~16
0 0
0
~5
1102 ~1014 ~92714
22521 35
AG(2, 9) = Res(#52); cf. Lam, Kolesova and Thiel (1991) PG(2, 9); cf. Lam, Kolesova and Thiel (1991)
~395
~3 ~423 ~3702
13769944 ~748
Vill.l.14; cf. Spence (1992) Ind(#84); cf. Curran and Vanstone (1988/89), Royle (1989) IX.5.8; cf. Kageyama (1972), Colbourn and Dinitz (1996a) VII.7.2; cf. Mathon and Rosa (1985a) Res(#63), Ext(#7); cf. IY.7.l2, Pietsch (1994) Res(#62), NR 11.8.7; cf. Denniston (1980a) ~Res(#61)
Lam, Thiel and Swiercz (1989) II.7.15; cf. Janko and van Trung (1986) Paley, see V1.1.l2; cf. Spence (l995a) VII.4.6; cf. Wilson (1974d) 2#8, VII.5.2; cf. Mathon and Rosa (1985a) 3#2, VII.5.2; cf. Mathon and Lomas (1992) 4#1 or dev({O, 1,2), {O, 1,3), (O, 1, 4), (O, 2, 4}) mod 7; cf. Gronau and Prestin (1982) VIL3.2; cf. Colboum (1980) VII.5.2; cf. Mathon (1995) 3#3, Ind(#97), VII.5.2; cf. Spence (1991a) 2#10 or dev({O, 1,2,4), (O, 1,3, 6), {O, 1, 4, 6}) mod 10; cf. Denny (1998) 2#11, VII.5.2; cf. Topalova (1998b)
? ~72
2#12 ordev({J,5, 11,24,25,27), {4, 6, 7, 20, 26,30)) mod31; cf. lungnickel and Vedder (1987) dev((0,2, 10, 15, 19,20), {O, 3, 7, 9, 10, 16}) mod 21; cf. Kapralov and Topalova (1998) (cant.)
Table A1.I. (cant.) No.
v
b
r
k
A
ND
12 12 12 12 12 12 12 12 12 12 13
6 6 8 9 10 11 12 12 12 12 3
4 5 4 3 2 1 1 2 3 4
:::111 :::2572156 ? :::3375 0 :::1 :::1 0 :::3752 0
76
16
77
13
78 79 80 81 82 83 84 85 86
22 33 55 121 133 67 45 34 27
32 26 33 44 66 132 133 67 45 34 117
87
40
130
13
4
88 89 90
66 14 27
143 26 39
13 13 13
6 7 9
91 92 93 94 95 96 97 98 99 100
40 66 144 157 79 53 40 27 15 22
52 78 156 157 79 53 40 27 70 77
13 13 13 13 13 13 13 13 14 14
10 11 '12 13 13 13 13 13 3 4
NR
:::1 :::1
:::10 11
:::909
:::107
:::2
1 6 4
:::1 :::17896 :::8071
? 0 68
3 2 1 1 2 3 4 6 2 2
? :::2 ? ? ?;2 0 ?;389 208310 :::685521 ?;7921
0 0 ?
:::21
Comments and references 2#13; cf. Jungnicke1 (1985) Ind(#98), VII.5.2; cf. Pietsch (1994) cf. van Rees (1996), McKay and Radziszowski (1996) Res(#84); cf. Mathon and Spence (1996) llRes(#83) AG(2, 11) = Res(#82) PG(2,11) II II.4.6 VL9.2; cf. Mathon and Spence (1996) II II.3.9 AGI(3, 3); cf. Mathon, Phelps and Rosa (1983), Colbourn, Magliveras and Mathon (1992) PGI (3,3), VII.7.5; cf. A.E.Brouwer (personal cornm.), Colbourn and Dinitz (1998) VIll.1.l0 Res(#98), NR II.8.7; cf. Pietsch (1994) AG2(3, 3), Res(#97); cf. Spence (1991a), Lam and Tonchev (1996) NRII.8.7 Res(#95), NR II.8.7
Aschbacher(1971) II IIA.6 PG2(3, 3); cf. Spence (1991a) Paley, see VI. 1.12; cf. Spence (1995a) 2#14, Ind(#140); cf. Mathon and Rosa (1985a) IX.5.7; cf. Franek, Mathon et aI. (1990)
101 102 103
8 15 36
28 42 84
14 14 14
4 5 6
6 4 2
2310 831 :::::5
104 105 106 107 108
15 85 43 29 22
35 170 86 58
6 7 7 7 7
5 1 2 3 4
:::::117
44
14 14 14 14 14
109 110 111 112 113 114 115 116 117 118 119 120 121 122 123
15 78 169 183 92 31 16 11 7 6 16 61 31 21 16
30 91 182 183 92 155 80 55 35 30 60 183 93 63 48
14 14 14 14 14 15 15 15 15 15 15 15 15 15 15
7 12 13 14 14 3 3 3 3 3 4 5 5 5 5
6 2 1 1 2 2 3 5 6 3 1 2 3 4
124 125
13 11
39 33
15 15
5 5
5 6
4
? :::::2
2#15; cf. Mathon and Rosa (1985a) IX.8.5, Ind(#141); cf. Tiessen and van Rees (1997) Vrn.3.13; cf. Jungnickel (1990d), Wertheimer (1991) Ind(#142); cf. Mathon (1995)
? :::::4 :::::1 :::::3393 :::::57810 0 :::::1 :::::1 0 :::::6 x 10 16 :::::10 13 :::::436800 109 6 :::::6 x 105 :::::10 >1
:::::i09 :::::11
:::::30 :::::127
:::::1
0 :::::6 x 105
VII.1.6, Vill.3.13; cf. Jungnickel (1990d), Abel (1994) VII.5.2 dev({O, 2, 6, 8, 9, 10, 13}, to, 3, 5, 6, 12, 13, 17}) mod 22; cf. Tiessen and van Rees (1997) 2#20, Ind(#143); cf. Denny (1998) jJRes(#I13) AG(2, 13) = Res(#112) PG(2.13) jJ II.3.9 PGI (4,2); cf. Mathon and Rosa (1985a) VII.4.9, VII.5.2; cf. Mathon and Rosa (1985a) VII.3.1, VII.5.2; cf. Mathon and Rosa (1985a) #9 + #31; cf. Colboum and Dinitz (1996a) 3#4; cf. Mathon and Rosa (1985a) 3#5, Ind(#170); cf. Mathon and Rosa (1985a) VII.3.5, VII.3.6; cf. Colboum (1980) VII.5.2 3#6; cf. Colboum and Dinitz (1996a) VII.5.2, Ind(#171); cf. Brouwer and Wilbrink (1984) VII.5.2; cf. Tonchev and Raev (1982a) 3#7, VII.5.2; cf. Jungnickel (1985) (cont.)
Table AI.I. (cont.)
!
!
No.
v
b
r
k
).
ND
126 127 128 129
76 26 16 91
190 65 40 195
15 15 15 15
6 6 6 7
j
3 5 1
::::1 ::::1 ::::15 ::::2
130 131
16 32
30 35
15 15
8 9
7 6
::::92436200 ::::104
132 133 134 135 136 137 138 139 140 141 142 143 144
136 46 28 56 91 196 211 106 71 43 36 31 33
204 69 42 70 105 210 211 106 71 43 36 31 176
15 15 15 15 15 15 15 15 15 15 15 15 16
10 10 10 12 13 14 15 15 15 15 15 15 3
1 3 5 3 2 1 1 2 3 5 6 7
? ? ::::3 ::::4 0 0 0 0 ::::8 0 ::::25634 ::::1266891 ::::10 13
145 146 147 148 149
9 49 25 17
48 196 100 68 52
16 16 16 16 16
3 4 4 4 4
4 1 2 3 4
13
16585031 ::::769 ::::19 ::::1 ::::2462
NR
?
::::5
0 0
::::529 ::::9
Comments and references VIII. 1.5 VIII.1.18 VII.5.2, Ind(#176); cf. Mathon (1995) dev({O, 10,27,28,31,43,50), {O, 11,20,25,49,55, 57}, (O, 13,26,39,52,65, 78}) mod 91; cf. Bhat-Nayak, Kane et al. (1983) AG3(4, 2), VII.5.2; cf. Kocay and van Rees (1991) Res(#142); cf. Bussemaker, Mathon and Seidel (1981)
van Lint, Tonchev and Landgev (1990) Res(#140); cf. Haemers (1980) :ilRes(#139) :ilRes(#138) :il II.4.lO :il II.3.9 Haemers (1980) :il II.4.6 II.3.19, V1.9.3; cf. Spence (1995c) PG3(4, 2); cf. Norman (1976) 1.6.11, Vil.3'!, Vil.4.I; cf. Mathon and Rosa (I985b), Tonchev and Vanstone (1992), Buratti and Zuanni (1998) 4#2, VII.5.2; cf. Colboum and Dinitz (1996a) VII.3.5; cf. Colboum and Dinitz (1998) 2#22, VII.5.2 (simple) Ind(#185) 4#3, VII.5.2; cf. Colboum and Dinitz (1996a)
150 151 152 153 154 155 156
9 65 81 21 49 113 57
36 208 216 56 112 226 114
16 16 16 16 16 16 16
4 5 6 6 7 8 8
6 1 1 4 2 1 2
:::::1.25 x 108 :::::2 ? :::::1 :::::73 ? :::::1362
157 158 159 160 161
29 17 145 25 33
58 34 232 40 48
16 16 16 16 16
8 8 10 10 11
4 7 1 6 5
:::::2 :::::11 ? :::::43 :::::19
162 163 164 165 166 167 168 169 l70 172
177 45 65 105 225 241 121 81 61 49 41
236 60 80 120 240 241 121 81 61 49 41
16 16 16 16 16 16 16 16 16 16 16
12 12 13 14 15 16 16 16 16 16 16
1 4 3 2 1 1 2 3 4 5 6
? :::::1 ? ? ? ? ? ? :::::6 :::::12146 :::::115307
173
18
102
l7
3
2
:::::4 x 10 14
:::::1
174
52
221
17
4
:::::206
:::::30
171
:::::1 :::::73
2#24, VII.5.2; cf. Colboum and Dinitz (1996a) VII.7.2; cf. Co1boum (1980) Example VII.1.8 2#26; cf. Jungnickel (1986) 2#27 or dev(±{O, 1,7, 19,23,44,47, 49}) mod 57; cf. Jungnickel and Vedder (1987) VII.5.2; cf. Colboum and Dinitz (1996a) VII.5.2, Ind(#186); cf. Tonchev and Raev (1982b)
0
0
VII.5.2, Ind(#I72); cf. Colboum and Dinitz (1996a) Ind(#171), NR II.8.7; cf. Brouwer and Wilbrink (1984) Ind(#170) NR II.8.7
?
Landjev and Topalova (1997) Brouwer and Wilbrink (1984); cf. KOlmel (1994) Bridges, Hall and Hayden (1981); cf. Colboum and Dinitz (1996a) VIlA.9, Hanani (1974a); cf. Mathon and Rosa (1985a) VII.7.5; cf. Colboum and Dinitz (1998) (cont.)
Table At.t. (cont.) No.
v
b
r
k
A
ND
NR
175 176
35 18
119 51
17 17
5 6
2 5
2:1 2:3
2:1 2:2
177 178 179
35 120 18
85 255 34
17 17 17
7 8 9
3 1 8
2:2 2:94 2:103
2:1 0
180 181 182
52 120 256
68 136 272
17 17 17
13 5 16
4 2 1
2:1 0 2:189
0 0 2:189
183
273
273
17
17
184 185
137 69
137 69
17 17
17 17
2 4
0 2:4
186
35
35
17
17
8
2:108131
2:22
?
Comments and references VII.7.3 dev({oo, 0, 2, 8, 9, 12}, {O, 1, 2, 4, 7, 15J, {O, 1,3,7,8, 12}) mod 17; cf. Kageyama (1983) Abel (1994) Penttila, Royle and Simpson (1996), Seiden (1963) NR II.S.7; cf. Bussemaker, Mathon and Seidel (1981) Ind(#185), NR II.8.7 :1Res(#184) AG(2, 16) = Res(#183); cf. Colboum and Dinitz (1996a) PG(2, 16); cf. Colboum and Dinitz (1996a) :1 II.4.6 Shrikhande and Singhi (1975), Whiteman (1988), Colboum and Dinitz (1998) TPP, see VI.8.2; cf. Spence (J995c)
(
§2. Symmetric designs
981
§2. Symmetric Designs
In this section we provide further information on the existence problem for symmetric designs, going beyond the range covered in § 1. Let us begin by listing the known families of symmetric designs; we remark that Series 3, 4 and 5 below were first obtained in an unpublished manuscript of R. M. Wilson with the title Symmetric group divisible designs. Moreover, some of the following series belong to families of difference sets; these will be discussed in §3.
. qd+l_l k qd-l_l d-l h . NaturaI Senes. v = -q1-' =~ n = q , were q IS a q- 1 ' A = --1-' qprime power and d an integer 2:2. These parameters are realised by the classical designs PGd -1 (d, q), see Proposition 1.2.16. There are many other examples having the same parameters; see, for instance, Theorem XII.2.23 and Corollary X.9.20.
Hadamard Series. v = 4n - 1, k = 2n - 1, A= n - 1. Existence is conjectured for all positive integers n; some constructions were presented in sections 1.9 and V1.8. There is an extensive literature on Hadamard designs and Hadamard matrices; we refer the reader to the books by Wallis, Street and Wallis (1972), Geramita and Seberry (1979) and Agaian (1985) and to the recent survey by Seberry and Yamada (1992). Extensive tables regarding the existence of (various special types of) Hadamard matrices may be found in §IV.24 of Colboum and Dinitz (1996a). In what follows, we only point out a few results. Some particularly important early constructions are due to Williamson (1944, 1947), Baumert and Hall (1965) and Turyn (1972) who constructed the first infinite series of "Williamson matrices". For along time, the smallest open case was n = 67; Hadamard designs of this order as well as examples for n = 103, 134 and 268 are now known, see Sawade (1985). Thus the smallest open c,ase is at present n = 107, and the next ones are n = 167 and 179. Given any odd positive integer t , thereexistsaHadamarddesignofordern = 2S tforalls 2: L210g2 (t-3)J by an impOltant result of Seberry (1976); see also Street and Street (1987) for a proof of her result. Her bound was recently improved to s 2: 4r ~ log2«t - 1)/21 + 2 by Craigen (1995) who also gave several other constructions and a table of best known exponents s for various values of t. A further improvement to s 2: 4L log2i~-I) J + 6 is due to Craigen, Holzmann and Kharaghani (1997). A similar result for regular Hadamard matrices is due to Craigen and Kharaghani (1994). Some powerful new constructions for Hadamard designs were given by Yamada (1989) and Miyamoto (1991). In particular, Yamada constructed such designs for n = 127,151,233 and 466, and Miyamoto obtained the cases n = 219,249, 267, 269, 373 and 445. Dokovic (1992a, b, 1994) settled three
Appendix. Tables
982
further small cases, namely n = 163,239 and 463. According to Table IV.24.33 of Colboum and Dinitz (1996a), there remain 48 odd integers n :::: 1000 for which the existence of a Hadamard matrix of order 4n is still in doubt. In Table A2.1 below we exhibit these open cases; for each value n, the integer in brackets gives the smallest t for which a Hadamard matIix of order 2t n is known.
Table A2.1. Values of n ::: 1000 for which the existence of a Hadamard matrix of order 4n is undecided 107 (3) 347 (3) 523 (3) 653 (3) 787 (3) 907 (3)
167 (3) 359 (4) 537 (3) 659 (4) 789 (3) 917 (3)
179 (3) 419 (4) 571 (3) 669 (3) 823 (3) 919 (3)
191 (3) 443 (3) 573 (3) 719 (4) 839 (4) 933 (4)
223 (3) 479 (4) 599 (5) 721 (3) 853 (3) 947 (3)
251 (3) 487 (3) 631 (3) 739 (5) 859 (3) 955 (3)
283 (3) 491 (5) 643 (3) 749 (4) 863 (3) 971 (5)
311 515 647 751 883 991
(3) (3) (3) (3) (3) (3)
Series 1. Let q be a prime power and d a positive integer. Then there exists a symmetric design with parameters v = qd+1 1 + q d+1 (
q-l
1)
,
d qd+! - 1 k =q . " ' - - - q-l
and
qd _ 1 A=qd. _ _
q-l
(and hence n = q2d). This series is due to Wallis (1971) and can also be realised by the McFarland difference sets; see Corollary II.8.17 and Theorem VI.9.2. Series 2. v = 4t 2 , k = 2t 2 - t, A = t 2 - t and hence n = t 2 ; existence is conjectured for all positive integers t. SymmetIic designs with parameters of this type were first studied by Menon (1962). Examples are known to exist whenever 2t is the order of a Hadamard matrix (see Corollary II.8.l7) and if both 2t - 1 and 2t + 1 are prime powers, cf. Wallis, Street and Wallis (1972), pp. 341-346 and Brouwer and van Lint (1984), where also an example with t = 7 is constructed. Finally, a large class of examples is provided by Hadamard difference sets, see Theorem VI.12.1S. Series 3. Let d be a positive integer. Then there exists a symmetric design with parameters
(and hence n = 32d ). There are corresponding difference sets due to Spence (1977), see Theorem VI.9.3.
983
§2. Symmetric designs
Series 4. Let q and n be prime powers satisfying n = q2 exists a symmetric design with order n and parameters
+ q + 1. Then there
(so that v = kJ... + 1). This series is due to Shrikhande and Singhi (1975); see also Whiteman (1988) or Theorem 17.5.5 of Hall (1986).
Series 5. Let q be a prime power and d a positive integer. Then there exists a symmetric design with parameters . 2q(qd - 1) V=
q-l
+ 1,
k = qd
A = qd-l
and
(and hence n = qd (q + 1)/2). This series is due to Brouwer (1983); see also de Launey (1986) or Theorem 17.5.7 of Hall (1986).
Series 6. Let q be a prime power and d a positive integer, and assume that q - 1 is the order of a projective plane. Then there exists a symmetric design with parameters
v = qd+1 _ q
+ 1,
k = qd
and
A = qd-I
(and hence n = qd _qd-I). This series is due to Mitchell (1979) and Rajkundlia (1978, 1983); generalising the work of Beker and Piper (1977).
Series 7. Let p and q be prime powers satisfying q = (pd+1 - l)/(p - 1) for some positive integer d. Then there exists a symmetric design with parameters k =q 2111-1 p d
an
d
1\.= q 1
2m-2 d( d 1) P pp-l
(and hence n = q2111-2 p2d) for every positive integerm. The smallest interesting case is p = 2,d = 1, q = 3, m = 2; this gives a symmetric (160, 54, 18)-design which is due to Spence, Tonchev and van Trung (1993). This construction was generalised to the family with p = 2 above by Spence (1993a) and to primes p by Jungnickel and Pou (1994); the final generalisation is due to lonin (1998a, 1999a). It can be obtained by a construction combining certain generalised balanced weighing matrices and McFarland difference sets.
Appendix. Tables
984
Series 8. Let p and q be prime powers satisfying q = pd+l + P - 1 for some positive integer d. Then there exists a symmetric design with parameters pd(q2m _ 1) V=
(p - 1)(pd
+ 1) ,
and
(and hence n = q2m-2 p2d) for every positive integer m. The smallest interesting case is p = 2, d = 1, q = 5, m = 2, giving a symmetric (416,166, 66)-design; this construction is also due to louin (1998a, 1999a). It combines generalised balanced weighing matrices and the complements of McFarland difference sets.
Series 9. Let q be a prime power satisfying q = (3 d +1 + 1) /2 for some positive integer d. Then there exists a symmetric design with parameters
(and hence n = 32d q2m-2) for every positive integer m. Again, this construction is due to lonin (1998a, 1999a); this times one combines generalised balanced weighing matrices and Spence difference sets.
Series 10. Let q be a prime power satisfying q = 3d +1 - 2 for some positive integer d. Then there exists a symmetric design with parameters
and 3d (3 2d q 2m-2 - 1)
= --=2-:':(3=-:-d---:1-:)--'(and hence n = 32d q2m-2) for every positive integer m. Again, this construction A
is due to lonin (1998a, 1999a); this time one combines generalised balanced weighing matrices and the complements of Spence difference sets.
Series 11. Let d be a positive integer. Then there exists a symmetric design with parameters
v=
73 d +1 - 64
9
(and hence n = 64· 73 d -
1
).
k = 73 d
and
A = 9· 73 d -
1
This series is due tolonin (1999b).
§2. Symmetric designs
985
Series 12. Let q = 2P - 1 be a Mersenne prime, and let d be a positive integer. Then there exists a symmetric design with parameters
2(q
+ 1)«q + 1)2d -
v=l+
q+
1)
2'
k = q(q
+ 1)2d -
q
q+2
and
(and hence n = q(q + 1)2d~1 /2). For q = 3, this series is due to Flanning (1995); the general case was given by lonin and Shrikhande (1998) in terms of the complementary designs.
Series 13.
v = 22d+4(22d +2 - 1)/3,
k = 22d +I(22d+3 + 1)/3,
A = 22d+I(22d+1 + 1)/3,
n = 22d+2 ,
where d is a non-negative integer. This series is realised by the difference sets due to Davis and Jedwab (1997a); see Theorem VI.9.4.
Series 14. v
= 4q 2d q2d -
1
q2 -1 '
A=q2d~l(q_l)q
k
= 2d~1 (2(q2d -
2d~1
q +1
q
and
+1
1)
+ 1)
'
n=q4d~2,
q+l where q is the square of an odd prime or a power of either 2 or 3. This series is realised by the difference sets of Chen (1997, 1998); see Theorem VI.9.5.
Series 15.
v=
h«2h - 1)2m - 1)
, k = h(2h _ 1)2m-l, h-l A = h(h - 1)(2h - Ipm-2 and n = h2(2h _ 1)2m-2, where m and d are positive integers, h = ±3 ·2d and 12h - 11 is a prime power. This series is due to lonin (1998b, 1999a), who obtained it by using regular Hadamard matrices (resp. Hadamard difference sets). The next two series are due to lonin (1999a); they are constructed by using the Davis-Iedwab difference sets (Series 13) and their complements.
986
Appendix. Tables
Series 16.
where In and d are positive integers and q = 22d~+1 is a prime power. Series 17.
where In and d are positive integers and q = 22d+3
-
3 is a prime power.
The following two series belong to cyclotomic difference sets, see §3. Series 18. v = 4t 2 + 1, k = t 2 , J... = (t2 - 1)/4, n = (3t 2 + 1)/4, where tis an odd positive integer and v a prime power. These designs may be obtained by using the biquadratic residues modulo v as a difference set; see Theorem VI.8.11 or Hall (1986). Series 19. v = 4t 2 + 9, k = t 2 + 3, A = (t 2 + 3)/4, n = 3(t 2 + 3)/4, where t is an odd positive integer and v a prime power. These designs may be obtained by using the biquadratic residues modulo v together with 0 as a difference set; see Theorem VI.8.11 or Hall (1986). Unfortunately, for several of the preceding series one only knows a few examples; indeed, it is an open problem whether or not the series 4, 18 and 19 yield infinitely many symmetric designs. Similarly, it is undecided whether or not infinitely many values of q arise in the series 6, 7, 8, 9, 10 and 12. In what follows, we present two tables of parameters for symmetric designs SJ..(2, k; v), that is, we list parameter triples (v, k, J...) subject to the following three conditions: (a) J...(v - 1) = k(k - 1) (arithmetic condition); (b) the number-theoretic conditions provided by Corollary II.3.9 (Schiitzenberger) and Theorem 11.4.6 (Bruck, Chowla, Ryser), respectively;
------------------.,~ §2. Symmetric designs
987
(c) k 2 - k + 1 > v > 2k + 1 (thus we exclude projectiv e planes and Hadama rd designs, see Remark II.3.12).
In Table A2.2, we list all such triples with n = k - A :::: 30, includin g those for which existenc e is still undecid ed; in Table A2.3, we list all such triples for which a symmet ric design is known to exist in the range 31 :::: n :::: 100. In the column headed ND, we give a lower bound for the number of isomorp hism classes of designs with the respecti ve paramet ers. Explicit descript ions of symmet ric designs withn :::: 25 may be found in TableI.5 .25 ofColbo um and Dinitz (l996a).
Table A2.2. Admissible parameter triples for symmetric designs with n :::: 30 n=k-J. ..
v
k
,\.
4 6 7
16 25 37
6 9 9
2 3 2
3 78 4
7 9
31 56
10 11
3 2
151 2:5
9
45
12
3
2:3752
9 9 10
40 36 41
13 15 16
4 6 6
2:389 2:25634 2:115307
11 11
79 49
13 16
2 5
2:2 2: 12146
12
71
15
3
2:8
12 13 13 14 15 15 16 16 16 16 16
61 81 69 121 71 61 154 115 96 85 78
16 16 17 16 21 25 18 19 20 21 22
4 3 4 2 6 10 2 3 4 5 6
2:6
ND
References and comments
?
2:4 ? 2:2 2:1 ? ? 2:2 2:213964 ::-::3
Series 1, Series 2; Hussain (1945) Series 5, Series 6; Denniston (1982) Series 12, VI.2.13; Assrnus, Mezzaroba and Salwach (1977) Series 4, VIII.1.14; Spence (1992) Theorem II.7.15; Janko and van Trung (1986) Series 1, Series 3; Mathon and Spence (1996) PG2(3, 3); Spence (1991a) Series 2, Series 3; Spence (1995c) Bridges, Hall and Hayden (1981); Spence (1993b), Colbourn and Dinitz (1996a) Aschbacher (1971) Brouwer and Wilbrink (1984); KOlmeI (1994) Haemers (1980), Beker and Haemers (1980) Series 6; Landjev and Topalova (1997) Series 4; Colbourn and Dinitz (1998) Janko and van Trung (l985b) Series 5
Series I; Sane (1982) PGz(3, 4); JungnickeI (1984b)
Janko and van Trung (1985a), Tonchev (1987a) (cant.)
988
Appendix. Tables
Table A2.2. (cant.) n =k-J...
v
k
16
70
24
8
::::12
16
66
26
10
::::19
16
64
28
12
::::8784
18 18 19 19 19 19 20 20 21
191 79 211 155
20 27 21 22 25 28 24 25 26
2 9 2 3
21 21 22 22 22
109 85 201 127 97
23 23 25 25 25
101 85 139 121 131
J...
References and comments
ND
Janko and van Trung (1984, 1992), Golemac (1993) van Trung (1982), Bridges (1983); Matulic-Bedenic, Horvatic-Baldasar and Kramer (1995) Series 2; Ding, Houghten, Lam, Thiel and Tonchev (1998)
? ::::1463
Series 6; Held and Pavcevic (1997)
? ? ::::1
301 103 352 253 204
28 36 25 28 33 25 34 27 28 29
6 9 4 5 5 7 15 3 6 11 2 11 2 3 4
25 25 25 25 25
175 156 133 120 112
30 31 33 35 37
5 6 8 10 12
::::2 ::::10 17 ::::1 ? ?
25 25 26 27 27 27 27 27 27 28 28
105 100 127 407 291 177 141 121 111 311 249
40 45 36 29 30 33 36 40 45 31 32
15 20 10 2 3 6 9 13 18 3 4
::::4 ::::1 ? ? ? ? ? ::::1029
Series 18
? ? ::::1 ?
Series 6
::::1 ?
Series 19
? ?
? ? ? ? ? ? Series 1; Sane (1982) PGz(3, 5); Jungnickel (1984b)
VI.8.21
Janko (1999) Series 2
PG3(4, 3); Jungnickel (1984b)
? ? ?
(cant.)
989
§2. Symmetric designs
Table A2.2. (cant.) n=k-J...
28 28 28 28 29 29 30
v
k
J...
171 149 131 113 265 181 239
35 37 40 49 33 36 35
7 9 12 21 4 7 5
ND
References and comments
? ?
?
:::1
Series 5
? ? ?
Table A2.3. Parameters of known symmetric designs with 31 S n S 100 n=k-J...
31 36 36 36 36 37 45 48 49 49 49 54 56 64 64 64 64 64 66 72 73 75 81 81 81 81 81 81
References and comments
v
k
J...
223 176 189 160 144 197 181 253 441 400 196 241 505 640 585 341 320 256 265 721 739 311 891 820 378 364 351 324
37 50 48 54 66 49 81 64 56 57 91 81 64 72 73 85 88 120 121 81 82 125 90 91 117 121 126 153
6 14 12 18 30 12 36 16 7 8 42 27 8 8 9 21 24 56 55 9 9 50 9
Series 13 Series 2 Series 5 Series 6 Series 4 Series 5 Series 1
10
PGz(3,9)
36 40 45 72
Series 1, Series 3
Series 4 IV.8.1O, Higman (1969) Janko (1997) Series 7 Series 2 Series 18 Series 5 Series 6 Series 1 PG2(3,7)
Series 2 Series 6 Series 6 Series 1 PG2(3,8) PG3(4,4)
PG4(5,3)
Series 3 Series 2 (cont.)
Appendix. Tables
990 Table A2.3. (cont.) n =k-A
91 96 100 100 100
v
k
A
References and comments
365 409 621 416 400
169 153 125 166 190
78 57 25 66 90
Series 5 Series 12 Series 6 Series 8 Series 2
§3. Abelian Difference Sets In this section, we provide information on the existence of abelian I difference sets. Let us begin by listing the known series of abelian difference sets. Series 1 (Singer Difference Sets). qd+l
v=
-1
qd-l _
A.=
q -1 '
q-l
1 ,
n=
qd-l,
where q is a prime power and d an integer :::::2; see Theorem VI.l.10. We shall use the notation PG(d, q) for this difference set. Often, there are (many) inequivalent cyclic difference sets with the same parameters, e.g. the GMWdifference sets; see §VI.17. Series 2 (Paley Difference Sets). v = q = 4n - 1,
k
= 2n -
1,
A. = n - 1,
G = (GF(q), +),
where q is a prime power; see Theorem VI.l.12. We will denote this difference set by P(q). Series 3 (Twin Prime Power Difference Sets).
1
Some families of non-abelian difference sets were given in Chapter VI. Kibler (1978) listed non-abelian difference sets with k < 20; see also §IV. 13 of Colboum and Dinitz (1996a).
§3. Abelian difference sets
where both q and q
991
+ 2 are prime powers, and G
= (GF(q), +) EB (GF(q +2), +);
see Theorem VI.8.2. This difference set will be denoted by TPP(q).
Series 4 (Hadamard Difference Sets).
Here G may be any group which is a direct product of an abelian group of order 22d+2 and exponent at most 2d +2 (for some non-negative integer d) with groups of type Z~i where each mi is a power of 3, and groups of type Z!j where the Pj are (not necessarily distinct)odd primes; see Theorem VI.12.1S.
Series 5 (McFarland Difference Sets).
where q is a prime power and d a positive integer. Here G may be any abelian group of order v which contains a subgroup isomorphic to E A (qd+ 1); for q = 4, such difference sets also exist in any abelian group of order v which contains a subgroup of order 2 2d+3 and exponent at most 4. See Theorems VI.9.2 and V1.11.12.
Series 6 (Spence Difference Sets).
where d is a positive integer. Here G may be any abelian group of order v which contains a subgroup isomorphic to E A (3 d + 1); see Theorems VI.9.3 and VI.ll.12. We note that a series of non-abelian difference sets with these parameters which admit -1 as a weak multiplier was constructed by Ma (1989a).
Series 7 (Davis-Jedwab Difference Sets). V = 2 2d +4(22d+2
-1)/3,
k = 22d+l(22d+3
A= 22d+l(22d+l
+ 1)/3,
n = 22d+2 ,
+ 1)/3,
Appendix. Tables
992
where d is a non-negative integer. Here G may be any abelian group of order v which contains a subgroup of order 22d+4 and exponent at most 4, with the exception when the subgroup is Z~ in the case d = 1; see Theorem VI.9.4. Series 8 (Chen Difference Sets).
2d-1 (2(q2d - 1) + 1) q q +1 ' 2d-1 + 1 A=q2d-l(q_1)q and n=q4d-2, q+1
v = 4q2d q2d - 1
k
q2 -1 '
=
where q is the square of an odd prime or a power of either 2 or 3. Here G = K x E, where E is the elementary abelian group of order q2d and where K may be any abelian group of order 4(q2d - 1)/(q2d - 1); see Theorem VI.9.S. Series 9 (Biquadric Residue Difference Sets). Either
v = 4t 2 + 1,
k
= t 2,
A = (t2 -1)/4,
n = (3t 2 + 1)/4
or v = 4t 2 + 9,
k = t 2 + 3,
A = (t2 + 3)/4,
n = 3(t 2 + 3)/4,
where t is an odd positive integer and v a prime power; see Theorem VI.S.l1. Four further series of abelian difference sets constructed by using cyclotomic classes were mentioned in Theorems VI.S.3, VI.S.l1 and VI.S.20; as these constructions seem to yield only very few examples, we will not restate them here. We now give a table of abelian difference sets with 2 ::: n ::: 30
and
k < v /2
which is an updated version of the table given by Jungnickel (1992a). Assmus (19S9) noted that there is much to be said for considering n as the most fundamental parameter for a symmetric design, and we shall follow his approach. The fundamental equation (VI.1.4.a) shows that n(n - 1) is divisible by A; writing n(n - 1) = A/L, the parameters of a symmetric (v, k, A)-design D and its complementary design D' can be written as
(2n
+ A + /L, n + A, A)
and
(2n
+ A + /L, n + /L, /L),
§3. Abelian difference sets
993
respectively. We thus list the possible parameters via the order n, using the divisors A of n(n - 1) in increasing order and stopping with A = n - 1; thus we eliminate complementary designs, begin for each n with the projective plane of this order, and end with the corresponding Hadamard design. We shall list only those parameter quadruples (n, A, v, k) which satisfy (VI.1A.a) and which are not already ruled out by Schiitzenberger's theorem (Corollary II.3.9) or the Bruck-Ryser-Chowla theorem (Theorem IIA.6). For any such set of parameters, we list all abelian groups of the relevant order. Groups are denoted by their type; for instance, (9, 3, 5) stands for Z9 ED Z3 EB Z5. In all cases covered by our table, the existence or non-existence of the respective difference set is decided; the column "Comments and references" contains at least one source for the result in question. Regarding non-existence, many cases can be excluded by the results of Chapter VI. Two frequent arguments are as follows: (a) An entry "t h == -1 mod w" indicates that t would be a multiplier of the difference set in question by Theorem VIA.6; thus the hypothesis of the Mann test (Theorem VI.6.2) would be satisfied for the given values of h and w, leading to a contradiction. (b) An entry "t = m, (x a , yb, zC, ...)" indicates that m would be a multiplier of the difference set in question by Theorem VIA.6 and that the group generated by m has a orbits of size x, b orbits of size y, c orbits of size c, ... on G. The desired contradiction then arises from Theorem VI.2.9, since no possible combination of orbits yields a k-set. In all cases not covered by the results of Chapter VI, the reader is referred to the literature. Quite often a non-existence proof may be found in the book by Lander (1983). This book contains a table covering all cases with k 2': 50; we note that all open cases in this table have been settled by now, see Iiams (1998). An extension of Lander's table for the range 50 < k::: 100 was given by Kopilovich (1989). Most of the open cases in this table have likewise been settled; there remain only three undecided cases with k < 100, see Jungnickel and Pott (1996) and Jungnickel and Schmidt (1997). Recently, a table covering the range 100 < k::: 150 was given by Vera L6pez and Garda Sanchez (1997); ten open cases in their table were settled by Arasu and Sehgal (1998). For explicit listings of small difference sets, the reader should consult Jungnickel and Pott (1996) and Baumert (1971) who listed - up to equivalence - all known cyclic difference sets with k::: 100. If any explicit description is contained in the present book, this will also be indicated.
994
Appendix. Tables
Table A3.I. Admissible parameters for abelian difference sets with n ::: 30 n=k-}"
v
k
}..
Group type
2 3 3
7 13
3
4
4
21
5 5
1 1 2 1
(7) (13) (11) (3,7)
Yes Yes Yes Yes
PG(2, 2), VI.1.3 PG(2, 3), VI. 1.3 P(ll), ill.9.6 PG(2, 4), ill.9.6
4
16
6
2
(16) (8,2) (4,4) (4,2,2) (2,2,2,2)
No Yes Yes Yes Yes
V1.14.20 Series 4 Series 4 Series 4 Series 4, VI. 1.3
4 5 5 6
15 31 19 25
7 6 9 9
3 1 4 3
(3,5) (31) (19) (25) (5,5)
Yes Yes Yes No No
PG(3, 2), TPP(3), VI.2.1O PG(2, 5), VI.2.1O P(19) VI.6.4, VI.5.9 VI.6.4, VI.5.9
6 7 7 7
23 57 37 31
11 8 9 10
5 1 2 3
(23) (3, 19) (37) (31)
Yes Yes Yes No
P(23), VI.2.13 PG(2, 7), VI.2.1O VI.2.13 VI.2.13
7
27
13
6
(27) (9,3)
No No
8 8 9 9
73 31 91 56
9 15 10 11
1 7 1 2
9
45
12
3
(3,3,3) (73) (31) (7, 13) (8,7) (4,2,7) (2,2,2,7) (9,5) (3,3,5)
Yes Yes Yes Yes No No No No Yes
VI.14.26 Bozikov (1985), Arasu (1988b), Wei (1990) P(27) PG(2,8) PG(4, 2), P(31) PG(2,9) VI.6.4 VI.6.4 VI.6.4 VI. 14.20 Series 5 (q = 3, d = 1)
9
40
13
4
9
36
15
6
(8,5) (4,2,5) (2,2,2,5) (4,9) (2,2,9) (4,3,3) (2,2,3,3)
Yes No No No No Yes Yes
PG(3,4) VI.6.4 VI.6.4 VI. 14.20 V1.14.20 Series 6 (d = 1) Series 6 (d = 1), VI.9.S
9 10 10 10 11
35 111 41 39 133
17 11 16 19 12
8 1 6 9 1
(5,7) (3,37) (41) (3, 13) (7, 19)
Yes No No No Yes
11 11
79 49
13 16
2 5
(79) (49) (7,7)
No No No
TPP(5) VI.6.12, VI.7.19 2 1O =-lmod41 VI.5.8 PG(2,11) t = 11, (11, 392)
11
Existence
Comments and references
t = 11, (11,3 2 ,212) VI.14.29
(cont.)
§3. Abelian difference sets
995
Table A3.1. (cont.) v
k
11 12
43 157
21 13
10 (43) 1 (157)
12 12 12 13
71 61 47 183
15 16 23 14
3 4 11 1
13
81
16
3
n=k-J...
13 13 14
69 51 121
17 25 16
14 15
55 241
27 16
15 15 15 16 16 16
71 21 61 25 59 29 273 17 154 18 115 19
)..
Group type
P(43) VI.7.14, VI.7.19
(71) (61) (47) (3,61)
No No Yes Yes
t = 2, (11, 35 2 ) 35 == -1 mod 61 P(47) PG(2,13)
(81) (27, 3) (9,9)
No No No
(9,3,3)
No
(3,3,3, 3)
No
Lander (1983), p.220 Lander (1983), p.220 Bozikov (1985), Arasu (1986) Bozikov (1985), Arasu (1986) Lander (1983), p.183 t = 13, (1 3 ,11 6)
(3,23) (3,17) (121) (11, 11) 13 (5, 11) 1 (241)
16
96
20
4
16 16 16 16
85 78 70 66
21 22 24 26
5 6 8 10
16
64 28
16
63
31
17 17 18
307 67 343
18 33 19
No No No No No No
132 == -1 mod 17 255 == -1 mod 121 2 55 == -1 mod 121 22 == -1 mod 5 VI.6.13, VI.7.19
(71) (61) (59) (3,7, 13) (2,7, 11) (5,23)
No No Yes Yes No No
t = 3, (11, 35 2 )
(32,3) (16,2,3) (8,4,3) (8,2,2,3) (4,4,2,3) (4,2,2,2,3) (2,2, 2, 2, 2, 3) (5,17) (2,3, 13) (2,5,7) (2,3, 11)
No No No No No Yes Yes Yes No No No
VI.I4.20 VI.l4.20 Arasu and Sehgal (l995a) Arasu, Davis et al. (1996) Arasu and Sehgal (1995b) Series 5 (q = 4, d = 1) Series 5 (q = 4, d = 1) PG(3,4) VI.6.4 Lander (1983), p.221 VI.6.4
No No Yes Yes Yes Yes Yes No
VI.l3.7 V1.13.7 VI.l2.2 PG(5,2) TPP(7) PG(2,17) P(67) VI.6.12, VI.7.19
(64) (32,2) all other types 15 (9,7) (3,3,7) 1 (307) 16 (67) 1 all types 12
Comments and references
Yes No
4 12 2
6 10 14 1 2 3
Existence
35 == -1 mod 61 P(59) PG(2,16) VI.6.4 V1.l4.26
(cont.)
Appendix. Tables
996
TableA3.1. (cont.) n=k-A
v
k
A
20 27 35 20 21 22 25 28 37
2 9 17 1 2
19 19 19 19
191 79 71 381 211 155 101 85 75
3 6 9 18
20 20 20
421 139 121
21 24 25
4 5
20
79
39
21 21 21 21 22
131 109 85 83 201
22 22 22 23 23
127 97 87 553 301
Group type
Existence
Comments and references
No No Yes Yes No
t = 3, (11, 95 2) 3 39 == -1 mod 79 P(71) PG(2,19) t = 19, (11,15 14 )
19
(191) (79) (71) (3, 127) (211) (5,31) (101) (5, 17) (3,25) (3,5,5) (421) (139) (121) (11, 11) (79)
No Yes No No No No No No No Yes
26 28 36 41 25
5 7 15 20 3
(131) (109) (5, 17) (83) (3,67)
No Yes No Yes No
19==-lmod5 Series 9 19 == -1 mod 5 19==-lmod5 19==-lmod5 VI.7.14, VI.7.19 269 == -1 mod 139 25 == -1 mod 11 25 == -1 mod 11 P(79) t = 7, (11, 65 2)
28 33 43 24 25
6 11 21 1 2
(127) (97) (3,29) (7,29) (7,43)
No No No Yes No
23 23 24 24 25 25 25 25
103 34 91 45 601 25 95 47 651 26 352 27 253 28 204 29
11 22 1 23 1 2 3 4
(103) (7,13) (601) (5, 19) (3,7,31) all types (11,23) all types
No No No No Yes No No No
25
175
30
5
(25,7) (5,5,7)
No Yes
25
156
31
6
(4,3, 13) (2,2,3, 13)
Yes No
25 25
133 120
33 35
8 10
(7,19) (8,3,5)
Yes No
(4,2,3,5)
No
(2, 2, 2, 3, 5)
No
18 18 18 19 19
Series 9 7 2 == -1 mod 5 P(83) 2==-lmod3 t = 11, (11, 67 2) 224 == -1 mod 97 2==-lmod3 PG(2,23) t = 23, (11, 32, 23 2 , 694 ) Lander (1983), p.221 33 == -1 mod 13 VI.7.14, VI.7.19 32 == -1 mod 5 PG(2,25) VI.2.15 VI.2.14 VI.6.2: s = 6, U = 34 > k, 58 == -1 mod 34 VI.14.16: 53 == -1 mod 7 Series 5 (q = 5, d = 1)
PG(3,5) VI.6.2: p = 5 > s = 3, u = 26, 52 == -1 mod 26 VI.8.21 VI.14.21: s = 4, u* = 30, m=5 VI.14.21: s = 4, u* = 30, m=5 VI.14.16: 5 == -1 mod 6 (cont.)
§4. Small Steiner systems
997
Table A3.1. (cont.) n=k-A
v
k
A Group type Existence Comments and references
112 37 12 all types 10540 15 (3,5,7) 100 45 20 (4,25) (2,2,25) (4,5,5) (2,2,5,5) 99 49 24 (9, 11) (3,3,11) 703 27 1 (19,37) 127 36 10 (127) 103 51 25 (103) 757 28 1 (757) 407 29 2 (11,37) 291 30 3 (3,97) 177 33 6 (3,59)
No No No No No No No Yes No No Yes Yes No No No
141 36 9 (3,47) 121 40 13 (121) (11,11) 111 45 18 (3,37) 107 53 26 (107) 813 29 1 (3, 271) 311 31 3 (311) 249 32 4 (3, 83) 171 35 7 (9,19) (3,3, 19)
No Yes No No Yes No No No No No
28 28 28
149 37 9 (149) 131 40 12 (131) 111 55 27 (3,37)
No No No
29 29 29 29 30 30
871 265 181 115 239 119
1 4 7 28 5 29
Yes No No No No No
25 25 25
25 26 26 26 27 27 27 27
27 27 27 27 28 28 28 28
30 33 36 57 35 59
(13,67) (5,53) (181) (5, 23) (239) (7)(17)
V1.6.4 VI.l4.2I: 53;: -1 mod 21 VI.l4.22 VI.l4.22 VI.l3.12 VI.l3.12 VI.l4.25: qm = 11, rt = 9 TPP(9)
VI.6.12 t = 13, (11,63 2) P(103) PG(2,27)
VI.6.2: 39 ;: -1 mod 37 VI.6.2: 324 ;: -1 mod 97 V1.5.l4: s = t = 3, U = 59, x + 29y + 29z = 33, x 2 + 29y2 + 29z 2 = 45 V1.5.15 PG(4,3)
Lander (1983), p.221 VI.14.16: 39 ;: -1 mod 37 P(107)
VI.7.14, VI.7.19 t = 2, (11, 155 2) VI.6.2: s = 1, 241 ;: -1 mod 249 VI.6.2: s = 1,2 19 ;: -1 mod 171 VI.6.2: s = 1,2 19 ;: -1 mod 171 t = 7;: 2142 mod 149, (11,742) VI.6.2: s = 1,265 ;: -1 mod 131 VI.5.14: s = 3, t = 2, x + 36y = 55, x 2 + 36y2 = 109 PG(2,29)
VI.6.2: 29 == -1 mod 5 t = 29, (11, 15 12 ) VI.6.2: 29 ;: -1 mod 5 t = 2 ;: 362 mod 239, (11, 1192) VI.6.2: 24 ;: -1 mod 17
§4. Small Steiner Systems
In this section we present a list of all triples (t, k, v) satisfying the arithmetic conditions (I.3.2.a) for the existence of an S(t, k; v) in the range 7 :::: v :::: 28. Relying on the original source Doyen and Rosa (1980), we also include the
998
Appendix. Tables
number ND of non-isomorphic systems S(t, k; v) or a lower bound (if such a bound is known to us). The notation Der( ... ) and Ext( ... ) refers to point derivations and extensions, respectively. If the existence of certain systems can be excluded by other results, we give a reference; we also include references with regard to the entries in column "ND". If the existence question is unsettled, this is denoted by a question mark in column "Existence". For a more extensive table, see §I.4.3 of Colboum and Dinitz (l996a). Table A4.1. Steiner systems with n
::s 28
v
S(t,k; v)
Existence
7 8 9 10 12
S(2, 3; 7) S(3, 4; 8) S(2, 3; 9) S(3,4; 10) S(4, 5; 11) S(5, 6; 12)
Yes Yes Yes Yes Yes Yes
1 1 1 1 1 1
13
S(2, 3; 13)
Yes
2
14 15
S(2,4; 13) S(3,4; 14) S(2, 3; 15)
Yes Yes Yes
S(4, 5; 15)
No
111.9.6; cf. Mathon, Phelps and Rosa (1983) 1 PG(2, 3); 1.4.5 4 ill.8.12; cf. Mendelsohn and Hung (1972) 80 1.5.8, VI.9.3; cf. Mathon, Phelps and Rosa (1983) 0 Mendelsohn and Hung (1972)
16
S(2,4; 16)
Yes
AG(2, 4) = Res S(2, 5; 21); IV.5.19,
S(2,6; 16) S(3,4; 16) S(5, 6; 16)
No Yes No
17
S(3, 5; 17) 5(4,5; 17) 5(6,7; 17)
Yes ? No
18
S(4,6; 18) S(5, 6; 18) S(7, 8; 18)
No ? No
19
S(2, 3; 19)
Yes
S(6,7; 19) S(8,9; 19)
? No
20
S(3, 4; 20) S(7, 8; 20) S(8, 9; 19)
Yes ? No
21
S(2, 3; S(2, 5; S(2, 6; S(4, 5; S(8, 9;
Yes Yes No ? ?
11
ND
Comments and references
0 >31300 0
PG(2, 2); 1.404 AG2(3, 2); (1.3A.a), II.8.11 AG(2, 3); 1.4.5
Ext S(2, 3; 9); 111.6.9, IV.2.8 Ext S(3, 4; 10); ill.8.8, IV. 1.2, IV.2.6 Ext S(4, 5; 11); 111.8.8, IY.1.2, IV.2.6
VillA. 12 11.2.7 AG2(4, 2); cf. Lindner and Rosa (1978) Ext S(4, 5; 15) Ext S(2, 4; 16),111.6.9; cf. Witt (1938a)
0
Ext S(5, 6; 16)
0
Witt (1938a)
0
Ext S(6, 7; 17)
::::1.1 x 109 III.9.5, VII.2.2; cf. Colbourn and Dinitz (1996a)
21) 21) 21) 21) 21)
0 >10 17
Ext S(7, 8; 18) 111.6.17; cf. Lindner and Rosa (1976)
0 Ext S(8, 9; 19) ::::2160980 1.6.8; cf. Wilson (1974d) 1 PG(2, 4); IV.5.19, VIllA. 12 0 11.2.7
(cant.)
§5. Infinite series of Steiner systems
999
Table A4.1. (eOlu.) v
S(t,k; v)
Existence
ND
Comments and references
S(3, 4; 22)
Yes
2:119
S(3, 6; 22) S(3, 7; 22) S(5, 6; 22) S(9, 10; 22) S(4, 5; 23) S(4, 7; 23) S(6, 7; 23) S(10, 11; 23) S(5, 6; 24) S(5, 8; 24) S(7, 8; 24) S(l1, 12; 24)
Yes No
1 0
IX.lO.5, Der S(4, 5; 23); cf. Diener (1980), Phelps (1991), Kreher and Frenz (1992) Ext S(2, 5; 21); II1.8.9, ry.J.6, IV.8.1 Ext S(2, 6; 21)
0 1
Der S(5, 6; 24) Ext S(3, 6; 22); III.8.9, IV. 1.6, ry.8.9
S(2, 3; 25) S(2, 4; 25)
Yes Yes
S(2, 5; 25) S(6, 7; 25) S(8, 9; 25)
Yes
26
S(3, S(3, S(3, S(7, S(9,
Yes Yes Yes ?
27
S(2, 3; 27)
22
23
24
25
28
4; 26) 5; 26) 6; 26) 8; 26) 10; 26)
S(4, 5; 27) S(4, 6; 27) S(8, 9; 27) S(10, 11; 27) S(2, 4; 28) S(3, 4; 28) S(5, 6; 28) S(5, 7; 28) S(9, 10; 28) S(ll, 12; 28)
? ? Yes Yes ? ? Yes Yes ? ?
2:3 1
2: 10 14 18
III. 8. 10; cf. Grannell and Griggs (1979) Ext S(4, 7; 23); III.8.9, Iy'1.6, ry.4.18
VII.4.6; cf. Wilson (1974d) VII.3.2; cf. Kramer, Magliveras and Mathon (1989) and Spence (1996) AG(2, 5); cf. MacInnes (1907)
? ? 2:1 2:1 I
IX.lO.l! Der S(4, 6; 27); cf. Hanani (1979) III.6.9, Ext S(2, 5; 25)
2: 1011
AGI (3,3); cf. Mathon, Phelps and Rosa (1983)
? Yes
? Yes
? ? Yes Yes ? Yes ? ?
2:1
>145
2:4~102o 2:1
Der S(5, 7; 28)
VII.3.2; cf. Brouwer (1989) III.6.9; cf. Lindner and Rosa (1978) Denniston (1976)
§S. Infinite Series of Steiner Systems In his classical paper Witt (l938b) gave a list of Steiner systems Set, k; v) which we now reproduce in an updated form; we shall restlict ourselves to the (few) infinite series that are known. The present state of knowledge regarding
Appendix. Tables
1000
Steiner S-designs was discussed in §III.8 (in particular, see Remark ill.8.Il). For more information, including base blocks for the known Steiner S-designs, we refer to §I.4 of Colbourn and Dinitz (1996a).
Table AS.t. Infinite series of Steiner systems Series AG
PG
SG KW
UG AK D
W H STS SQ8 B4 B5 Bk
References
Comments
Parameters S(2, q; qn)
affine geometries AGl(n, q), q a prime power, n::;:2 projective geometries S(2, q + 1; qn+ PGl (n, q), ... + q + 1) q a prime power, n::;:2 spherical geometries, S(3, q + 1; qn + 1) q a prime power, n::;:2 q a prime power, n::;:2, S(3, q + 1; qn + 1) not isomorphic to Series SG S(2, q + 1; q3 + 1) unitals, q a prime power S(3,6; (4m + 2)/3) m::;:3 an integer S(2, 2r; 2r+, + 2r - 2') r < s positive integers S(2, 2e - 1 ; 2e - 1 (2 e - 1» e ::;: 2 an integer Infinitely many examples S(3, 5; v), S(3, 6; v) S(2, 3; v) v == I or 3 (mod 6) 8(3,4; v) v == 2 or 4 (mod 6) 8(2,4; v) v == 1 or 4 (mod 12) 8(2,5; v) t' == 1 or 5 (mod 20) 8(2, k; v) v == 1 (mod k - 1), tI(v - 1) == 0 (mod k(k - 1», t' sufficiently large
1.2.13 1.2.16 III.6.9 Il.7.25 VIII.5.28 Il.7.24 VlII.5.21 VIII.5.17,5.34.(d) Hanani (1979) VIl.4.7, IX.5.1 IX. 10.2 IX.5.6 IX.7.8 XU.S
Tables AS.2 and AS.3list the 33 open cases for the existence of a Steiner system S(2, 6; v) and the 22 open cases for the existence of a Steiner system S(2, 7; v), respectively; see Theorems IX.l2.2 and IX.12.3. These tables are followed by Tables A5.4 and A5.5 for Steiner systems S(2, 8; v) and resolvable quadruple systems RS(3, 4; v); see Theorems IX. 12.4 and IX.12.18, respectively. Table AS.2. Values ofvforwhich the existence of S(2, 6; v) is open 46 291 471 771
51 316 496 796
61 321 501 801
81 346 526
166 351 561
226 376 591
231 406 616
256 411 646
261 436 651
286 441
676
1001
§6. Remark on t-designs with t :::: 3 Table AS.3. Values of v for which the existence of S (2, 7; v) is open
85 799 1645
127 805 1765
133
211
925
1135 1974
1807
253 1387 2479
505 1435 2605
589 1555
715 1638
Table AS.4. Values of v for which the existence of S(2, 8; v) is open 113
169
176
225
281
1065 1457 2185
1121
1128
1177
1464 2241
1513 2577
1520 2913
1233 1569 3305
337 1240 1576 1417
393 1296 1737 3473
624 1345 1793 3753
736 1401 1905
785 1408 1961
Table AS.5. Values of v for which the existence ofRS(3, 4; v) is open
220 1076 2380
235 1100 2540
292 1252 2740
364 1316 2812
460 1820 3620
596 2236 3820
676 2308 6356
724
2324
§6. Remark on (-designs with ( :::::: 3 Many constructions for (not necessarily simple) t-designs with t:::: 3 were presented in Chapter ID. Some other (both recursive and direct) constructions for (farnilies of) both simple and non-simple t-designs were mentioned in §III.8. Teirlinck's general existence theorem for simple t-designs and the recent discovery of simple 7- and 8-designs with small parameters discussed there are particularly important. We will not give any tables of t-designs with t :::: 3 here but refer the reader to the literature. Hanani, Hartman and Kramer (1983) give an excellent survey of 3-designs including extensive tables (for v :::: 32). Kreher and Radziszowski (1987a) give a table of t-designs with t:::: 4 and v :::: 14. An influential detailed study (including tables) of t -designs with t :::: 3 is due to Driessen (1978). Finally, we refer the reader to Sections 1.3.2 and 1.3.3 of Colboum and Dinitz (1996a) for a list of the known infinite families of simple t-designs and a table of simple t-designs with t ::: 3 and v :::: 30, respectively; many further examples were constructed by Bluskov (1997).
Appendix. Tables
1002
§7. Orthogonal Latin Squares
In this final section, we give two tables concerning mutually orthogonal Latin squares: a table of upper bounds on the numbers nk, where nk denotes the largest order for which there exists no set of k mutually orthogonal Latin squares, and a table of the best-known lower bounds on the maximum number N(n) of MOLS of order n, in the range n :s 100. Since one has N(n) = n - I for prime powers n, we only include non-prime power values in the latter table. Most of the bounds quoted here are proved in the present text; we then only give the quotation from the preceding chapters. If a better bound is known, we give a reference and include the bound proved by us in square brackets. For a table of bounds on N (n) up to order 10000, see Section 11.2.6 of Col bourn and Dinitz (1996a); this is also the source for the following table of upper bounds for the numbers nk. Table A7.1. Upper bounds on the numbers nk k
2 7 12
~
k
~
k
~
k
~
k
6 180 7286
3 8 13
10 2774 7288
4 9 14
22 3678 7874
5 10 15
62 5804 8360
6 11 30
75 7222 52502
Table A7.2. Lower bounds jar the number oj MOLS ojorders .:s: 100 n
6 10 12 14
N(n)?:.
References
n
N(n)?:.
References
=1
§X.J3 VIII.4.9, VIII.4.27 VIII. 3. 13 VIII.4.17
36 38 39 40
6 [5] 4 5 [4] 7
VIIl.3.17, Abel (1998) VIII.4.17 VIII.4.17, Abel (1998) VIII.3.17
VIII.3.17 VIII.4.23 VIII.4.17 VIIl.3.!7
42 44 45 46
VIII.4.32 VIlI.3.17 VIII.3.17 VIIl.4.30
48
5 5 6 4 7 [5]
15 18 20 21
2 5 3 4 3 4 5
22 24 26 28
3 5 4 5
VIII.4.23 VIII.3.17 VIII.4.32 VIII.3.17
30 33 34 35
4 5 4 5
VIII.4.32 VIII.3.17 VIII.4.l7 VIII.3.l7
6
50 51 52
5 5 [3]
54
5 [4]
Wojtas (1997) VIIl.3.17 VIII.4.30 VlII.3.17 Colbourn and Dinitz (1996a), 1.7.8 Colbourn and Dinitz (1998), X.1.3 (cant.)
§7. Orthogonal Latin squares
1003
Table A7.2. (cant.) n
N(n)?:.
References
n
55 56 57 58 60 62 63 65 66 68 69 70 72 74 75 76
5 [4] 7 [6] 7 5 4 4 6 7 5 5 6 [5] 6 7 5 5 6 [5]
77 78
6 6
Mills (1977b), 1.7.8 Mills (1977b), 1.7.8 IX.1.19 X.3.l3 1.7.8 X.3.6 I.7.8 VIIIA.27 IXAA IX.4A Zhu (1984), IXA.4 X.1.3 I.7.8 IXAA IXAA Colboum, Yin and Zhu (1995), IXAA 1.7.8 X.1.3
N(n)?:.
References
80
9 [7]
82
8
Abel and Cheng (1994), (X.1.3.g) VIII.4.30
84 85 86 87 88 90 91 92 93 94 95 96 98 99 100
6 6 6 6 7 6 7 6 6 6 6 7 6 8 8
X.3.6 X.3.6 X.3.6 X.3.l3 I.7.8 X.3.9 X.3.l1 X.3.13 X.3.13 X.3.13 X.3.l3 X.3.l3 X.3.l3 I.7.8 VIII.4.30
Notation and Symbols
We begin by explaining the notation used in organising the material presented in this book. Theorems, lemmas, definitions, etc. are numbered consecutively within each section; e.g., II.4.S refers to item no. S in Section 4 of Chapter II (which happens to be a lemma). Within Chapter II, it would be quoted just by giving the number 4.S. Formulae are numbered according to the item where they occur by appending lower case letters; e.g., the formulae in IIA.S are labelled (4.5.a), (4.S.b), ... , (4.S.k). If one of these formulae need be quoted in another chapter, we prefix the number II again: so (1I.4.S.a) would be the result. The end or absence of a proof is always indicated by a •. Finally, in defining a quantity x in terms of a known quantity y, we often write x := y (so the term being defined is the one left of the colon).
General Symbols Here we collect some general notation concerning sets, numbers, algebraic structures, etc., that is more or less standard. The more specific notation concerned with incidence structures, automorphism groups, etc., will be treated separately.
Sets
AUB AnB A\B AxB A+B
the union of A and B the intersection of A and B A without the elements in B the cartesian product of A and B the symmetric difference (A U B)\(A 100S
n B) of A and B
Notation and symbols
1006
2A
No
the power set of A the set of t-subsets of A the set of ordered t-tuples from A the set of ordered t-tuples with distinct entries from A A. is a subset of B A is a proper subset of B the set of positive integers the set of non-negative integers
N~ A~
{xEN:a:sx:sb} A nN~ for A C N
\A\
the cardinality of A the set of primes (resp. prime powers)
(~) N A(t)
AS;B ACB N
P,P*
Lists
See Definition III.9.2 Maps
J:A-+B J:xl-+y xl or J(x)
Sf TI- 1 Xx, Xx
Oij X
XI Xw Tr Trq" /q'
J is a mapping from A to B
J maps x onto y the image of x under y (usually, mappings are written as exponents) {f(x) : XES} for J : A -+ Band S S A {x E A : J(x) E T} for J : A -+ Band T S; B characteristic functions, cf. (II.2.9.b) the Kronecker 0, i.e. oij = I for i = j and 0 otherwise (quadratic) character principal character additive character, cf. (VI.3.17.a) absolute trace, cf. (VI.3.17.b) relative trace, cf. V1.12.9 Numbers
I:7=lai 1 ai [x] or LxJ
D7_ rxl
C)
ala2·· . all the largest integer :sx (for x E JR.) the smallest integer 2:x (for x E JR.) the number of t-subsets of an x-set
Notation and symbols
1007
the number of ordered t-tuples with distinct entries from an x-set, i.e. x(x - 1) ... (x - t + 1) x factorial, i.e. x(x - 1) ... 2· 1 (for x E N) the number of i-dimensional subspaces of the n-dimensional vector space over GF(q) , cf. Lemma 1.2.14 a cyclotomic number, cf. Definition VI.8.8 a divides b / strictly divides b
xl
(i, j)m
alb pi lib
Matrices
AT J Jm,n
In I diag(xj, ... , x n )
the transpose of A a matrix with all entries equal to 1 the (m, n)-matrix with all entries equal to 1 the identity matrix of order n an identity matrix the diagonal matrix of size n with diagonal entries Xl,·· -,Xn
circ (a 11 , AI8lB
••. ,
al n )
a circulant matrix, cf. Definition VIIIA.29 the Kronecker product of A and B, cf. (1.9.6.a)
Rings and fields
Z
Zn Q R C ~m Q(~m) Z(~m)
GF(q) K*
KO V(n, q)
Rf
the ring (or group) of integers the ring (or group) of residues modulo n the field of rational numbers the field of real numbers the field of complex numbers a primitive m-th root of unity in C the m-th cyclotomic field the ring of algebraic integers in Q(~m) the finite field of order q (q a prime power) the multiplicative group of K (K a field) the multiplicative group of non-zero squares in K (K a field) the n-dimensional vector space over GF(q) the group ring of rover R (R a ring, r a group), see Definition VI.3.1
1008
Notation and symbols
Permutation groups the symmetric group on n letters the alternating group on n letters the stabiliser of x, cf. Definition III.3.9 the pointwise stabiliser of B, cf. Definition III.3.9 the setwise stabiliser of B, cf. Definition III.3.9 the image of x under g, cf. Definition III.3.1 (for a given action of G) the orbit of x under G, cf. Definition III.3.9 the Galois group of Q(~m) over Q
Groups in general
H-:sG H<JG gil [g,
hJ
G GxH G$H expG GIN
AB Tg
CG(g) CG(H) Nc(H)
C(G) O(G)
H is a subgroup of G H is a normal subgroup of G h-1ghforg,h E G the commutator of g and h, i.e. ghg-1h- 1 character group of an abelian group G, cf. Remark VI.3.3 the direct product of G and H the direct sum of G and H (for additively written groups) the exponent of G the factor group of G by N (for N <J G) the complex product of A and B (for A, B -:s G) the right translation by g, cf. Exercise V.l.l the centraliser of g in G (for g E G) the centraliser of H in G (for H -:s G) the normaliser of H in G (for H -:s G) the centre of G the maximal normal subgroup of odd order in G, cf. Lemma X.12.1
Particular groups the cyclic group of residues mod n the elementary abelian group of order s (s a prime power) the Mathieu group of order v (v = 11, 12,21,22,23,24), cf. Chapter IV the extension of Mv by a group of order 2 (v = 21, 22), cf. Theorem IV.S.14
Notation and symbols H(n+I,q) e(n + 1, q) T(n,q)
1009
cf. Proposition Ill.S.I2 cf. Notation 11I.6.3 cf. Definition Ill.S.7
For the classical groups GL(n, q), SL(n, q), r L(n, q) and the corresponding affine and projective groups (denoted by prefixing an A or a P, respectively), see Section Ill.5.
Special Symbols
Incidence structures in general (P,B, I)
15 D*
D1+ .. ·+D", mD
D'xD" Dx BD
(D) D(B)
D(B, p) H(D, B) dev D devS D (U)
N(EID) D'(p) Dip D'(B)
incidence structure with point set P, block set Band incidence relation I, cf. Definition I.1.1 complementary structure of D, cf. Definition I. 1. 10 the dual structure of D, cf. Definition 104.9 sum of incidence structures, cf. Definition 1.5.2 multiple of D, cf. Definition 1.5.2 direct product of nets, cf. Definition I. 7 .21 derived structure at x, cf. Definition 11.1.2 residual structure with respect to B, cf. Lemma II.1. 7 and Remark 11.1.8 point residue at x, cf. Lemma II. 1. 7 and Remark H.l.8 reduced structure of D, cf. Definition Ill. 2. 12 structure induced on B, cf. Proposition II.S.14 cf. Definition IV. 7.1 Hussain structure of D at B, cf. Definition IV.7.1 development of D, cf. Definition VI. 1.5 development of S, cf. Definition VII. 1.3 group constructible structure defined by U, cf. Definition X.9.3 net induced by the extension E of D, cf. Theorem X.I1A local structure at p, cf. Lemma XI1.2.7 local structure of a linear space, cf. Definition XII.4.1 reduced structure of D(B), cf. Lemma XII.S.1
Types of incidence structures S).(2, k; v) S).(2, K; v) S).(t, K; v)
2-design, cf. Definition 1.2.9 pairwise balanced design, cf. Definition 1.2.17 cf. Definition 1.3.1
1010 S),.(t,k;v) Set, k; v) STS(v) SQS(v) RS),.(t, k; v) GD)..[K, G; v] TD),.[k, g] RTD),.[k, g] (s, r; JL)-net STD),.(g) S"(2, k; v) AiL(s) LS)..(t, k; v) PS),.(t, k; v) NRS(2, k; v) NRB(v, k)
Notation and symbols t-design, cf. Definition 1.3.1 Steiner system, cf. Definition 1.3.1 Steiner triple system, cf. Definition 1.304 Steiner quadruple system, cf. Definition 1.304 resolvable t-design, cf. Notation I.S.6 divisible design, cf. Definition 1.6.1 transversal design, cf. Notation 1.7.4 resolvable transversal design, cf. Notation 1.704 net, cf. Notation 1.704 symmetric transversal design, cf. Definition 1.7.17 linear space with subspace, cf. Notation 1.8.10 affine 2-design, cf. Theorem 11.8.17 large set of t-designs S)..(t, k; v), cf. Remark VIII.8.S partial t -design, cf. remarks following Theorem 1X.11. 7 nearly resolvable Steiner system S(2, k; v), cf. Remark IX.12.8 nearly resolvable design Sk-I (2, k; v), cf. Remark IX.12.1O
Particular incidence structures AG(n, q) AGd(n, q) PG(n, q) PGd(n, q)
Wi
Ex
affine geometry, cf. Definition 1.2.12 design of points and d-dimensional subspaces of AG(n, q), cf. Definition 1.2.12 projective geometry, cf. Definition 1.2.1S design of points and d-dimensional subspaces of PG(n, q), cf. Definition 1.2.1S Witt design (i = 9, 10, II, 12,21,22,23,24), cf. Notations IV. 1.3 and IY.1.6 design induced on a dodecad X of E (E an S(S, 8; 24», cf. Lemma IVA.3 complete graph, cf. Example II.S.7 complete bipartite graph, cf. Example 11.7.2
Subsets in incidence structures
(B) (p) (Q), (C)
trace of B, cf. Definition II. 1.2 set of blocks on p, cf. Definition 11.1.1 cf. Definition II.l.1
Notation and symbols pq pqr
1011
line through p and q, cf. Definition XII.2.3 plane through p, q and r, cf. Definition XII.2.3 Parameters
v k
number of points, cf. Definition II.2.9 block size, cf. Definition II.2.9 J.... joining number, cf. Definition 11.2.9 r replication number, cf. Theorem II.2.W number of blocks, cf. Theorem II.2.1 0 number of blocks through an s-set, cf. Theorem 1.3.2 and Corollary II.1.4 intersection number of non-parallel blocks, cf. Definition I1.S.1 /h s size of parallel classes, cf. Lemma II.S.2 n order of a symmetric design or a difference set, cf. Lemmas II.3.l1 and VI 1.4 v,k,J.... parameters of difference sets, cf. Definition VI.lO.l m,n,k,J.... parameters of relative difference sets, cf. Definition VI.l 0.1 n,k,J....,/h parameters of strongly regular graphs, cf. Definition II.9.1 b;x;,y number of blocks through an x-set missing a y-yet, cf. Theorem II.6.1 niCB) intersection numbers, cf. Remark II.6.I1 intersection numbers, cf. Remark II.6.11 p-rank of an incidence structure deficiency of net, cf.Definitions X.7.l and X. 7.1 0 point class size, cf. Definition 1.6:1 i cf. Lemma X.7.5 Pjk
Subsets ofN S;..(2, K), B(K, J....), B(k, J....), B(K), B(k) S;..(t, k), Set, k) GD;..(K, G), GD;..(k, g) RS;..(t, k), RB(K, J....), RBCk, J....), RB(k) TD;..(k), TD(k), RTD;..(k), RTDCk) Bu(k) TD*Ck) R'K,;.., Rk,;'" R'K, R'"n, Rk
FK(u)
cf. Notation II.2.19 cf. Notation II.3.1 cf. Definition 1.6.1 cf. Notation II.5.6 cf. Notation II.7A cf. Notation II.S.lO cf. Definition IX.2.10 cf. Definition IX.3.7 cf. Notation IX.3.15
1012
R*k D).(k), D(K)
Notation and symbols cf. Notation IX.6.2 cf. Notation VII.5.3 Numbers
N(g) N*(g)
number of MOLS of order g, cf. Notation VillA.7 number of idempotent M 0 LS of order g, cf. Definition VillA. 10 cf. Remark XA.2
Groups and group rings AutD Aut(E,H)
S
Aa A
== B (mod P)
A(t)
the full automorphism group of D, cf. Definition ill.2.10 automorphism group of a Hussain structure, cf. Definition IV.7.7 sum in group ring, cf. VI.(3.l.b) cf. VI.(3.1.d) cf. Lemma VI.3.7 cf. VI.(3.1.c)
Miscellaneous H",R: b.;jBt
aB 08 (d l , ..• , dk)
GR(g, J....)
OAA(t,k,s),OAA(k,s) (s, r; /-t)-PCP
parallel, cf. Definition 1.204 cf. Notation VII.5.1 cf. Observation Vill.l.2 cf. Definition ill.9.7 cf. Definition ill.9.7 cf. Definition ill.9.7 generalised Hadamard matrix, cf. Definition Vill.3A cf. Definition VillA.1 cf. Definition X.9.3
Bibliography Verachtet mir die Meister nicht, und ehrt rnir ihre Kunst. .. (Wagner)
The following bibliography is not intended to be encyclopedic. In fact, it only contains items referred to in the main text. Aagedal, H., Beth, T., Miiller-Quade, J. and Schmid, M. (1996). Algorithmic design of diffractive optical systems for information processing. Workshop on physics and computation PhysComp96 Boston (eds. T. Toffoli, M. Biafore and J. Leao). New England Complex Systems Institute, pp. 1-6. Abatangelo, V. and Larato, B. (1989). A characterization of Denniston's maximal arcs. Geom. Ded. 30, 197-203. Abatangelo, V., Larato, B. and Rosati, L. A. (1990). Unitals in planes derived from Hughes planes. 1. Comb. In! System Sc. 15, 151-155. Abduhl-Elah, M. S., AI-Dhahir, M. W. and Jungnickel, D. (1987). 83 in PG (2, q). Arch. Math. 49, 141-150. Abe, T. and Yoshiara, S. (1993). On Suzuki's construction of 2-designs over GF(q). Science Rep. Hirosaki Univ. 40, 119-122. Abel, R. J. R. (1991). Four mutually orthogonal Latin squares of order 28 and 52. 1. Comb. Th. (A) 58, 306--309. Abel, R. J. R. (1994). Forty-three balanced incomplete block designs. 1. Comb. Th. (A) 65,252-267. Abel, R. J. R. (1996). Some new BIBDs with A = 1 and 6 ::: k ::: 10.1. Comb. Des. 4, 27-50. Abel, R. J. R. (1998). Personal communication. Abel, R. J. R. and Cheng, Y. W. (1994). Some new MOLS of order 2 n p for p a prime power. Australasian 1. Comb. 10, 175-186. Abel, R. J. R., Ge, G., Greig, M. and Zhu, L. (1998). Resolvable balanced incomplete block designs with a block size of 5. Preprint. Abel, R. J. R. and Greig, M. (1997). Some new RBIBDs with block size 5 and PBDs with block sizes == 1 mod 5. Australasian 1. Comb. 15, 177-202. Abel, R. J. R. and Greig, M. (1998). Balanced incomplete block designs with block size 7. Designs, Codes and Cryptography 13, 5-30. Abel, R. J. R. and Mills, W. H. (1995). Some new BIBDs with k = 6 and A = 1.1. Comb. Des. 3,381-391. Abel, R. J. R. and Todorov, D. T. (1993). Four MOLS of orders 20, 30, 38 and 44. 1. Comb. Th. (A) 64,144--148.
1013
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Index
absolute trace, 398 Adjunction Theorem, 627 accession number, 932 affine difference set, 371 affine incidence space, 807 affine space, 11 algebra, semisimple, 316 antipolarity, 569 arc, 566, 567 complete, 582 arithmetic conditions, 63 Assmus-Mattson Theorem, 161 attribute, 932 authentication code, 959 autocorrelation coefficient, 945 automorphism, 20, 164 automorphism group, 21,166 class regular, 372 averaged treatment effect, 876 axioms of Veblen and Young, 806 axis, 22, 177 Baer subplane, 275, 565 Baranyai's Theorem, 29 Barker sequence, 422 barrel shifter, 943 base block, 24, 175-6,297,465 base point, 24 base vertex, 226 basis, 645 signaling basis, 898 Bays-Lambossy Theorem, 474 biplane, 95, 263 block, I base block, 175-6 good,833 repeated block, 2
block-residue, 64 breaking up blocks, 617 Bruck net, 37 Bruck-Ryser-Chowla Theorem, 93 Bruck-Ryser Theorem, 94 Bruck's Completion Theorem, 714 building block, 374 modulus, 374 building set, 374 covering, 374 extended building set (EBS), 382 Burnside's lemma, 171 capacity, 899 cayley graph, 225 centraliser, 280 centre, 22, 177 centred contrast, 868 character, 313 additive character, 319 principal, 314 quadratic character, 56 trivial,314 chromatic index, 227 circle geometry, 193 class list, 622 class size of incidence structure, 124 class type, 622 claw, 718 maximal, 718 Clebsch graph, 145 clique, 718 grand,720 maximal,718 closure, 819 closure operator, 619 code authentication code, 959
1093
1094 code (COllt.) block code, 153 dual code, 153 equivalent code, 153 extended binary Golay code, 246 first-order Reed-Muller code, 900 Golay code, 234 Hamming code, 894 perfect, 894 point code, 153 self-dual code, 157 simplex code, 919 tetracode, 923 codeword, 893, 918 collinear, 809 collineation, 20 central, 188 colour, 227 companion automorphism, 185 complementary graph, 136 complementary perpendicular array, 563 complementary structure,S complete mapping, 763 completion, 713, 827 component, 732 configuration, 16 Desatgues configuration, 16 Papposconfiguration,18 conic, 581 connected components, 226 Contracted Multiplier Theorem, 334 Conway's 13-puzzle, 242 correlation, 23 counteroval, 238 covering, 687 covering number, 687 covering problem, 687 critical deficiency, 716 cross-ratio, 208 cross-ratio class, 211 cross-ratio distribution, 211 cycle, 148 m-cycle, 221 t-subcycle, 223 cyclic difference family, 220 cyclic difference set, 220 cyclotomic class, 356, 509 order, 357 cyclotomic number, 357 higher cyclotomic number, 519 order, 357 uniform, 360 decoder, 893 decoherence, 912 decomposition, 457 Dedekind Independence Theorem, 315
Index deficiency, 713, 749 degenerate projective plane, 63 degree, 2 deltoid, 601 Dembowski-Wagner Theorem, 8 I 2 derived incidence structure, 53, 63 Desargues configuration, 16 design balanced incotuplete block design, 870 block design, 10, 867 derived, 151 divisible design, 32 embeddable, 129, 149 experimental design, 864 generalised quasiresidual design, 135 generalised residual design, 135 GDD,32 GMW-design, 821 Hadamard design, 50, 53 high-table design, 945 indecomposable, 64 I A-design, 81 type 1,81 large, 213 locally affine, 832 locally projective, 831 locally symmetric, 812, 832 matching design, 588 pairwise balanced design (PBD), 13 Class-uniformly resolvable, 686 partial, 678 uniformly resolvable, 686 quasi derived, 151 quasiresidual, 129 quasismooth,818 quasisymmetric, 88 residual, 129 SDP-design, 848 simple I-design, 5 small, 212 super-simple, 640 symmetric, 22, 77 t -design, 15 TD,37 transversal design, 37 trivial, 20 weighing design, 881, 882 Witt design, 2 I 2, 213, 234 development, 299, 469, 521 DFT-spectrum, 948 difference, 218 difference cycle, 221 difference family, 220, 470 cyclic, 220 multiplier equivalent, 474 radical, 498
Index
:~
.
difference list, 332 order, 332 difference matrix, 532 left (s,r;). )-difference matrix, 766 maximal,766 difference sequence, 221, 598 difference set, 220, 298 abelian, 298 affine difference set, 371 antisymmetric difference set, 356 central, 328 Chen difference set, 366 complementary, 299 cyclic, 220 cyclotomic, 358 Davis-ledwab difference set, 365 decomposable, 457 equivalent, 302 genuinely non-abelian, 410 GMW-difference set, 457, 460 Hadamard difference set, 366 McFarland difference set, 363 Menon difference set, 366 non-abelian, 298 non-trivial, 298 order, 299 Paley difference set, 303 Paley-Hadamard,301 partial difference set (PDS), 230 partial spread difference set, 367 planar, 301 relative difference set (RDS), 369 reversible, 430 Singer difference set, 302 skew difference set, 356 Spence difference set, 365 symmetric, 430 trivial, 298 twin prime power series, 354 difference triple, 482 dilatation, 734 dimension, 829 discrete Radon transfonn, 928 distance, 148 divisible design, 32 symmetric, 370 dodecad, 245 Doyen-Hubaut Theorem, 830 Doyen-Wilson Theorem, 674 duality,23 dual theorem, 23
EBS, 382 edge, 14 edge colouring, 227 elation, 22, 188,818 encoder, 893
1095 epimorphism, 164 error, bit-flip error, 914 sign-pJip error, 914 error-correcting code, 893 error vector, 858, 893 essential elements, 645 experiment, balanced, 878 incomplete, 878 experimental design, 864 extended building set, 382 covering, 382 extension, 53, 63 exterior line, 567 extremal t-design, 109 family, 219 fibre, complete, 782 m-fibre, 781 file, 932 first Kohler graph, 604 First Multiplier Theorem, 307, 322 Fisher's Inequality, 46, 66 flag, 1 flat, 11, 12,835 forbidden subgroup, 370 form, degenerate, 89 equivalent, 89 non-degenerate, 89 quadratic fonn, 89 symmetric bilinear fonn, 89 function, signal function, 898 Fundamental Theorem of Projective Geometry, 188 Gaussian coefficient, 12 Gauss-Markov Theorem, 859 GDD, group divisible design, 32 General GMW construction, 456 generalised Hadamard matrix, 532 generating set, 645 Gewirtz graph, 145 good,833 graph,14 bipartite graph, 149 block graph, 142 Cayley graph, 225 Clebsch graph, 145 complementary, 136 complete bipartite graph, 38 complete graph, 26 conference, 142 connected graph, 226
1096 graph (cont.) cyclic graph, 225 design graph, 144 geometric, 717 Gewirtz graph, 145 L2-graph, 728 net graph, 717 Paley graph, 231 Paley type, 142 parameters, 136 Petersen graph, 13 7 pseudo net graph, 717 r-regular, 16 square lattice graph, 138 strongly regular graph, 136 triangular graph, 137 type I, 142 graph of a PTS, 673 graphical Hadamard matrix, 61 group action, 168 almost flag-transitive, 352 automorphism group, 21, 166 character group, 313 collineation group, 21 decomposition group, 435 Frobenius group, 279 full automorphism group, 21, 166 full collineation group, 21 general affine group, 186 general linear group, 185 generalised dihedral goup, 429 Higman-Sims group, 273 Mathieu group, 234, 243 large Mathieu group, 243 little Mathieu goup, 243 operation, 167 permutation group, 167 pennutation isomorphic, 168 projective general linear group, 187 projective special linear group, 187 quaternian group, 370 regular, 21, 169 semiregular, 169 sharply transitive, 169 sharply t-transitive, 169 Singer group, 297, 298 special linear group, 185 t-homogeneous, 168 transitive, 168 translation group, 731 t-transitive, 168 twisted projective group, 194 group filing system, 933 group ring, 311 GWM machine, 907
Index Hadamard design, 50 Hadamard matrix, 51 excess, 84 graphical, 61 regular, 61 skew,61 symmetric, 61 Hadamard's Inequality, 51 Hall-Connor Theorem, 149 Hall-Paige conjecture, 764 Hall-Paige Theorem, 763 Hall-Ryser Theorem, 333 Hamada's Conjecture, 158 Hamming distance, 894 Hanani's Recursion Lemma, 624 homology, 188 homotheties, 187 hopping pattern, 952 Hussain structure, 262 hyperfactorisation, 586 hypergraph, 13 hyperoval, 464, 567 monomial, 462 regular, 581 hyperplane, 11, 12 ideal reversible, 316 self-orthogonal, 316 imprimitive strongly regular graph, 136 iucidence morphism, 164 incidence preserving, 164 incidence structure, 1 affine, 124 affine resolvable, 124 balanced, 32 class size, 124 cohesive, 809 complementary structure, 5 cyclic, 469 derived, 53, 63 divisible, 32 doubly resolvable, 31 dual structure, 23 equivalent, 473 G-invariant, 469 group constructible, 732 induced substructure, 9 isomorphic, 20, 164 m-fold multiple, 25 multiplier equivalent, 473 non-trivial, 124 normal,96 parameter, 124 r-factorisation,25 r-resolvable, 25
1097
Index reduced structure, 166 repeated,2 regular, 298 resolvable, 25 rigid,479 s-rotational, 487 self-dual, 23 simple, 2 smooth,8lO square, 22 t -balanced, 15 inclusion-exclusion principle, 111 induced,8 integral basis, 436 Integrality Conditions, 140 intersection number, 108,331 intersection parameter, 124 intersection triangle, lO3 Inversion Formula, 314 irregular triangle, 602 isomorphic incidence structure, 20, 164 isosceles triangle, 602 isomorphism, 20, 164 k-choice, 509 consistent, 509 kernel,279 Kiefer ordering, 869 Kirkman system, 28 Kirkman's school-girl problem, 27 kite quadrangle, 601 Kronecker product, 53 A-design, 81 type I, 81 A-design conjecture, 81 Landau symbol, 20 Lander's Conjecture, 429 large set, 597 Latin square, 546 equivalent, 772 self-orthogonal, 547 length,148 line, 11, 12,44,563,809 absolute, 575 linear estimator, 858 linear space, 13, 44 class-uniformly resolvable, 686 generalised Fischer space, 826 locally projective, 828 partial,44 uniformly resolvable, 686 list, 218 evenly distributed, 500 y-invariant, 496 list of differences, 2 I 8, 298
little projective group, 187 logarithmic notation, 489 Lowner ordering, 859 majority logic decoding principle, 906 mandatory representation design, 677 Mann Test, 336 Mariner 9, 900 Maschke's Theorem, 316 Ma's Lemma, 413 matrix adjacency matrix, 139 auxiliary matricix, 36 averaging matrix, 868 centring matrix, 868 circulant, 558 contrast information matrix, 868 generalised information matrix, 865 Hadamard matrix, 51 hermitian, 579 incidence matrix, 3 Lowner optimal, 865 moment matrix, 864 parity check matrix, 894 precision matrix, 864 maximal arc, 566 non-trivial, 566 McFarland's Conjecture, 430 McFarland's Multiplier Theorem, 327 mean parameter vector, 858 measurement, 881 message, 893 Metsch's Completion Theorem, 714 minimal generating set, 645 minimum weight, 160,893 minor, 598 mixed difference, 522 model matrix, 858 MOLS,546 p,-hyperfactorisation, 586 simple, 586 multiset, 218 mUltiplication table, 548 multiplicity, 219 mUltiplier, 304, 334,472 extraneous multiplier, 329 GIN-multiplier, 334 numerical multiplier, 304 right mUltiplier, 304 weak multiplier, 328 Multiplier Conjecture, 323 Multiplier Theorem for Difference Lists, 334 multiset, 218 near-difference matrix, 550 near-field, 279
1098 near-pencil, 69 nearly Kirkman system, 684 nearly resolvable design, 685 neighbour, 603, 606 neighbourhood, 143 nesting, 679 net, 37 Bruck net, 37 closure, 736 completable, 713, 749 completion, 749 coset, 750 embeddable, 713 extendible, 725, 749 A-net, 37 maximal, 725 . non-degenerate form, 89 (s, r, JL)-net, 38 step-t-extendible, 725, 749 translation net, 731 transversal-free, 717 Netto triple system, 196, 843 non-trivial strongly regular graph, 136 nucleus, 567 o-polynomial, 462 object, 881 one time pad, 966 opposite regulus, 738 optimal estimator, 860 optimal linear estimator, 858 orbit, 169 long orbit, 605 short orbit, 469, 605 orbit graph, 606 Orbit Theorem, 173 order, 829 ordered triangle, 809 orthogonalarra~545
degree, 545 index, 545 order, 545 resolvable, 556 strength, 545 orthogonal check set, 905 Orthogonality Relations, 314 orthogonal mate, 726 orthogonal quasi group, 536 orthomorphism, 763 oval,567 p-rank,153 packing, 687 packing number, 687 packing problem, 687 pairwise balanced design (PBD), 13 pairwise disjoint subgroups, 232
Index Palfy's Theorem, 475 Pappos configuration, 18 parallel, 8, 821 parallel class, 25, 556 parameters, 136, 230 parameters of incidence structure, 124 parallelism, 25 parity check, 910 parity check coordinates, 893 part, 25 partial congruence partition (PCP), 232, 732 maximal, 732 partial design, 678 partial difference set, 230 Paley type, 231 partial linear space, 44 partial spread, 734 partial t -spread, 734 maximal, 734 partial transversal, 714 partial triple system (PTS), 673 cyclic (CPTS), 673 regular (RPTS), 673 partition, 25 PBD,13 PCP, 732 perfect secrecy, 961 period, 782 primitive period, 782 permutation group, 167 degree, 168 isomorphic,168 non-trivial, 278 permutation isomorphic, 167 rank,l72 regular, 169 semiregular, 169, 340 sharply transitive, 169 sharply t -transitive, 169 t-homogeneous, 168 t-transitive, 168 transitive, 168 transitive extension, 198 type to, 290 perpendicular array, 563 perpendicular difference array, 542 Petersen graph, 137 phase space tomography, 927 Jr -space, 847 plane, II, 12,810 affine plane, 8 order, 8 affino-projective plane, 846 degenerate projective, 69 Desarguesian, 17 Hall plane, 739 inversive plane, 193
1099
Index Mobius plane, 193 miquelian, 193 near-pencil, 69 non-Desarguesian, 17 order of, 7, 8 Pappi an, 18 projective plane, 6 order, 7 quasidouble, 76 quasimultiple, 75 PN -sequences, 969 point, 1, 11, 12, 167,575 absolute, 143,575 equivalent, 32 point class, 32 point class list, 622 point class type, 622 point residue, 64 polar, 575 polarity, 23, 143,578 orthogonal,581 unitary, 578 pole, 575 position, 905 checked, 905 prime ideal, 4 11 Prime Power Conjecture (PPC), 346 primitive, 170 principle of duality, 23 projective group, 187 twisted, 194 projective incidence space, 806 projective set, 395 projective space, 12 pseudorandom generator, 966 pure difference, 522 quadrangle, 6 irregular, 601 quasi-difference matrix (QDM), 550 quasigroup, 548 diagonal, 763 idempotent, 548 self-orthogonal, 548 qubit, 912 r-factor, 25 randomness postulates, 969 rank,l72 rate, 893 RDS, 369 record,932 rectangle, 601 reduced incidence structure, 166 regression range, 864 regression vector, 858 regular group, 21
regulus, 738 relative difference set, 369 abelian, 370 cyclic, 370 proper, 370 non-abelian, 370 semi regular, 371 splitting, 370 representation, 167 faithful, 167 permutation representation, 167 residue, 357 biquadratic, 357 cubic, 357 e-th power, 357 octie,357 quadratic, 357 quartic, 357 quintic, 357 sextic, 357 resolution, 25 orthogonal, 31 response vector, 858 retrieval rule, 932 l-balanced,932 reversible, 316 right-angled triangle, 602 right translation, 310 Ryser's Conjecture, 429 sample size, 864 Schmidt's Exponent Bound, 449 secant, 567 second Kohler graph, 606 Second Multiplier Theorem, 323 self-conjugate, 411, 438 self-orthogonal,316 semilinear mapping, 185 semilinear space, 44 set, closed,619 eventually periodic, 782 period, 782 set of imprimitivity, 170 shadow, 963 shape, 603 equivalent, 605 side, 603 signal-to-noise ratio, 881 Singer cycle, 190 Singer group, abelian,298 cyclic, 298 non-abelian, 298 Singer subgroup, 190 Singer's Theorem, 190 skew Hadamard matrix, 61
1100
Index
small deficiency, 735 socle,294 SOLS,547 spherical geometry, 193 splitting field, 313 spread, 395 regular, 738 square, 546, 60 I mutually orthogonal, 546 order, 546 orthogonal, 546 SQS,16 strictly cyclic, 601 stabiliser, 169 pointwise stabiliser, 169 setwise stabiliser, 169 Steiner quadruple system, 16 Steiner system, 15 maximal partial, 688 nearly resolvable, 685 Steiner triple system, 16 Stem-Lenz Lemma, 228 storage mapping, 932 storage rule, 932 strictly divide, 327 strongly regular graph (SRG), 136 imprimitive, 136 non-trivial, 136 Paley type, 142 primitive, 136 trivial, 136 type I, 142 type II, 142 subplane, 564 Baer subplane, 564 subspace, 45, 121 proper, 45 sum, 25 support size, 73 Sylvester's Problem, 19 symbol, 167 symmetric design, 22, 77 order of, 77 skew, 61, 362, 818 symmetric difference, 245 symmetric difference property, 848 symmetric Hadamard matrix, 61 syndrome, 893
graphical, 585 hypergraphical, 585 regular, 108 tight, 109 tactical configuration, 16 tactical decomposition, 178 tangent, 567 TD, symmetric, 43 tetracode, 923 3-net,847 threshold algorithm, 904 threshold criterion, 906 threshold scheme, 962 perfect, 964 trace, 2, 166 translate, 297 translation right, 310 translation net, 731, 746 splitting translation net, 746 transversal, 714, 725, 738, 750 transversal design, 37 class regular, 532 symmetric, 43 trapezoid, 601 treatment, 857 triangle, 8, 809 triangular graph, 137 triple system, 481 trivial design, 20 trivial strongly regular graph, 136 Turyn's Exponent Bound, 414, 440 two-way classification model, 866 (t, w)-threshold schemes, 967 (t, w, v)-threshold scheme, 964
,-balanced incidence structure, 15 t-design, 15 bigraphical, 585 extremal, 109
weighing scale, 881 weighting, 622 Wilbrink's Theorem, 341 wreath product, 295
uniformly redundant array, 951 unital, 578 hermitian unital, 579 vector error vector, 858, 893 incidence vector, 3 signal vector, 898 Vemam cipher, 966 vertices, 14 Vizing's Theorem, 227
