Cyclic Difference Sets

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich

182 Leonard D. Baumert California Institute of Technology Pasadena, CA / USA

Cyclic Difference Sets

Springer-Verlag Berlin. Heidelberg- New York 1971

I S B N 3-540-05368-9 S p r i n g e r - V e r l a g B e r l i n • H e i d e l b e r g • N e w Y o r k I S B N 0-387-05368-9 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data hanks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. @ by Springer-Verlag Berlin - Heidelberg 1971. Library of Congress Catalog Card Number 73-153466 Printed in Germany. Offsetdruck: Julius Beltz, Weinheim/Bergstr,

CYCLIC DIFFERENCE SETS

A fairly comprehensive survey of the general theory of cyclic difference sets is given below°

The aim of ~his survey is to provide a cohesive presentation of

the known facts as well as an introduction to some of the outstanding problems. The more general topics of block designs and difference sets in finite groups are introduced but only those aspects of these subjects which shed some light on problems arising for cyclic difference sets are developed. It is not expected that many will wish to read this survey sequentially from the beginning.

For this reason the chapters and to a lesser degree the sections

within them are largely independent of each other, having been written that way in order to encourage the reader to skip around and follow his own interests.

However

a certain familiarity with the contents of Chapter I is presupposed elsewhere. Beyond this, interconnections between the various sections and chapters are indicated when they seem relevant.

This structure~

coupled with the aim of making the

later material understandable to as many as possible, has led to the anomaly that, in some cases, quite elementary concepts are defined in the later chapters, whereas these

same

concepts,

and a great deal more, were presupposed in earlier sections.

In addition to the specific references inserted in the text, the books of Marshall Hall, Jr., "Combinatorial Theory", Blaisdell Publishing Company, 1967, of H. B. Mann, "Addition Theorems", Interscience Publishers, "Combinatorial Mathematics",

1965, and of Ho J. Ryser,

Carus Mathematical Monograph No. 14, 1963, may be used

as general references for a large part of this material° This survey was compiled in connection with research carried out at the Jet Propulsion Laboratory,

California Institute of Technology, under Contract No.

NAS 7-100, sponsored by the National Aeronautics and Space Administration.

CONTENTS I.

II.

III.

IV.

V.

INTRODUCTION

. . . . . . . . . . . . . . . . . . . . . . . . .

A.

Difference

B.

Shifts~

C.

Block Designs,

D.

The Characteristic

E.

Multipliers

F.

The Hall-polynomial,

G.

Group Difference

EXISTENCE

Sets

i

. . . . . . . . . . . . . . . . . . . . . .

Equivalence

Complements

. . . . . . . . . . . . . . .

Incidence Matrices. Function

i

The Incidence

Equation . . . . .

and its Autocorrelation

Function

....

. . . . . . . . . . . . . . . . . . . . . . . .

QUESTIONS

w-multipliers

4 6 7

. . . . . . . . . . . . . .

Sets . . . . . . . . . . . . . . . . . . . .

8 9

. . . . . . . . . . . . . . . . . . . . . .

ii

A.

The Main Existence

B.

The Bruck-Ryser-Chowla

Co

Integral

D.

The Theorem of Hall and Ryser . . . . . . . . . . . . . . . . .

24

E.

Results of Mann,

. . . . . . . . . . .

26

. . . . . . . . . . . .

54

MULTIPLIERS

Problems

2

Solutions

. . . . . . . . . . . . . . . . .

Theorem

. . . . . . . . . . . . . . . .

to the Incidence

Rankin,

Equation

Turyn and Yamamoto

A N D CONSTRUCTIVE

ii

EXISTENCE

TESTS

12

. . . . . . . . . .

18

A.

Multiplier

Theorems . . . . . . . . . . . . . . . . . . . . .

54

Bo

Difference

Sets Fixed by a Multiplier

. . . . . . . . . . . . .

60

C.

Multipliers

. . . . . . . . . . . . .

62

D.

Polynomial

DIFFERENCE

and Diophantine Congruences

Planar Difference

B.

Hadamard Difference

C.

Barker Sequences.

Ao

Sets

. . . . . . . . . . . . . . . . .

.

N th Power Residue

.

77 77

Sets . . . . . . . . . . . . . . . . . . .

90

SETS

Hadamard Matrices . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

Singer Sets and Their Generalizations. .

65

. . . . . . . . . . . . . . . . . . .

Circulant

OF DIFFERENCE

and Welch . B.

. . . . . . . . . . . . . . . . . . .

SETS OF SPECIAL TYPE

Ao

FAMILIES

Equations

.

.

.

.

Difference

.

.

.

The Results

of Gordon,

99 Mills

. . . . . . . . . . .

Sets and Cyclotomy

96

. . . . . . . . .

99 119

V.

VI.

FAMILIES OF DIFFERENCE C.

More Cyelotomic

D.

Generalized

MI SCELLANY

SETS

(continued)

Difference

Sets

. . . . . . . . . . . . . . .

Cyclotomy and Difference

Sets

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

A.

Multiple

Inequivalent

B.

Searches

C.

Some Examples

D.

A Table of Difference

Difference

Sets

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sets

. . . . . . . . . . . . . . . . .

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . .

127 131 1 43 143 14h 146 148 159

I.

INTRODUCTION

The main purpose of this chapter is to provide the basic definitions and vocabulary of the study of difference sets so that subsequent chapters need not be interrupted at inopportune times b y the introduction of such material.

Thus none

of the concepts are pursued in detail - such development being deferred to the appropriate part of a later chapter.

A.

Difference Sets A

v, k, ~ - difference set

modulo

v,

D = {dl, o.., ~ }

such that for an~ residue

J # 0

d i - dj ~ ~

has exactly

h

solution pairs

(di,dj)

changeable below;

(mod v)

k

residues

the congruence

(mod v)

with

difference set, cyclic difference set and

is a collection of

di

(ioi)

and

dj

in

Do

[The terms

v, k, h - difference set are inter-

the later two being used when there is some reason to stress

either the contrast with general group difference sets or the particular parameters v, k, k

involved.]

As an Lv~ediate consequence of this definition one has that the relation

(1.2)

k(k- 1): ~(v- i)

necessarily holds among the parameters Given any positive integer modulo

v. (i) (ii)

(iii) (iv)

v

v, k, ~.

there are certain obvious difference sets

These are: the null set

D =

all singletons O = [O,1 ..... v -

D = [i},

0 < i < v - 1

l}

D = [0,i ..... i - I, i + i, .... v - i]

0
These difference

sets are called trivial and are quite often either ignored or

treated only as limiting cases. are

v, k~ ~ = v, 0, 0;

If one introduces

v, l, 0;

v, v~ v

the additional parameter

that these trivial difference assumption

Note that the parameters

n ~2,

and

v, v - l, v - 2

n = k - ~,

equation

sets arise if and only if

which is often made implicitly,

trivial difference

for these difference

n = 0

sets

respectively.

(1.2) above shows or

lo

Hence the

serves to exclude all the

sets.

Some non-trivial

difference

sets are:

D I = [i, 2, 4]

rood 7,

D 2 = [0, 3, 5, 6]

mod 7,

D 3 = [0, i, 3, 9)

mod 13, mod 19 o

D 4 = ~i, 4, 5, 6, 7, 9, Ii, 16, 17]

v, k, ~

Here the parameters

are

7, 3, 13

7, 4, 2;

13, 4, i

and

19, 9, 4

respectively.

B.

Shifts, Equivalence,

Complements

Given the difference set

set

D = [dl,.O., ~ ]

[d I + s,...,d k + s} --- D + s

taken modulo

mod v, v

is also a difference

called a shift (or cyclic shift) of the original set t,

prime to

v,

the set

[tdl,.o.,td k] --- tD

set with the same parameters having the same parameters with

t

prime to

v,

For any particular be no difference equivalent

then

v, k, h.

and if D. l

and

D. j

set of parameters

and

(rood v)

v, k, h

satisfying

~

pairwise

(This result is discussed

v

~ > 0,

sets

for any integers

t, s

equation

difference

sets.

(1o2) there may or several in-

In fact, Gordon, Mills and

difference

in V. A. below)°

it is

are two difference

there exist values of

inequivalent

set~

the

is also a difference

set (up to equivalence)

sets having these parameters.

that there are at least

Dj

s

Again, given any integer

are said to be equivalent

sets, a single difference

difference

Di

D.

taken modulo

D. = tD. + s l j

Welch (1962) have shown that given any

meters.

If

then for any integer

v, k, h

such

sets with these para-

Among the examples given in I. A. above note that

D1

and

D2

are related~

that is

D I + D 2 = {O,l,o..,v - I} o

For this reason D

D1

is a difference

and

set.

v*, k*~ v

and

For m o s t purposes

m e n t a r y difference than

v/2.

are said to be complementary difference

set w i t h parameters

set w i t h parameters one sees that both

D2

h*= n

D*

is a difference

Recalling that

n = k - h

it is sufficient to consider only one of a pair of comple-

sets.

This is frequently done by insisting that

of the four - v, k, h, n. Using

k = v/2

k

be less

does not occur)°

the parameters

v, n

are the most fundamental

For this reason it is sometimes u s e f u l to express (1.2) and

n = k - h

k " k* = k(v - k) = n(v - i)

h + h* = v - 2n

its complement

If

are invariant under complementation of the difference

As w i l l become abundantly clear,

Since

h

v, v - k, v - 2k + h.

(Equation (1.2) shows that

in terms of them.

v, k,

sets.

and

k > 1

and

k, h

it follows that

h. h* = h(v - 2n - h) = n(n - i) .

for non-trivial difference

sets~

it follows

that

_>

n(n-

> v-

2n- i

and so

n

The two extreme cases

2

+n

v : 4n - i

+ i >v>4n-

and

v = n2 + n + i

sets of the Hadamard type and to the difference projective planes

io

(so-called planar difference

cussed more fully in Chapter IV - Difference

correspond to difference

sets associated with certain finite sets).

Both these types are dis-

Sets of Special Type.

C.

Block Designs, Incidence Matrices, An arrangement of (i) (ii)

(iii)

v

The Incidence Equation

objects into

b

sets (called blocks) such that:

each block contains exactly

k

different objects

each object occurs in exactly

r

different blocks

any pair of objects occurs in exactly

is called a balanced incomplete block design.

~

different blocks

Evidently the relations

b k = vr

(1.3)

r(k - i) = h(v - i)

are satisfied b y the parameters

b, v, r, k, h.

pairs involving any one fixed object.) number of objects equation

v

(1.4)

(The last follows by counting the

If the number of blocks

the design is said to be s~mmetric.

(1.3) shows that

k = r.

b

equals the

In a symmetric design

Thus for symmetric designs equation

(1o4) takes

the same form as equation (1.2). Every difference set gives rise to a symmetric block design whose objects are 0,1,...,v - 1

and whose blocks are

D,D + 1,...,D + (v - 1).

Note that any cyclic

relabeling of the objects of such a design permutes the blocks cyclically in a cycle of length

v.

Such a block design is called cyclic.

Thus every difference set

corresponds to a cyclic sy~netrie balanced incomplete block design. the difference set

D1

of

For example,

I.A. above corresponds to the block design:

BI = DI

- [i, 2, 4]

B 2 = D I + i =- [2, 3, 5] B3 = D I + 2 - [3, 4, 6]

B4 = D I + 3

- I0, 4, 5]

B 5 = D I + 4 - [i, 5, 6] B 6 = D I + 5 -= [0, 2, 6] B 7 = D I + 6 -= [0, l, 3]

(rood 7)

Associated with any balanced incomplete block design is a zeros and ones called the incidence matrix of the design. putting wise.

aij = 1

b × v

For the example above [recalling that the ith block is j - l]

A =

of

A

It is constructed by

if the jth object appears in the ith block and

that the jth object is

matrix

aij = 0

D + (i - l)

otherand

the incidence matrix is

0 0 0 i 0 i l

1 0 0 0 1 0 l

1 1 0 0 0 1 0

0 1 1 0 0 0 1

1 0 1 1 0 0 0

0 1 0 1 1 0 0

0 0 1 0 1 1 0

From the block design definition above it is clear that the associated incidence matrix satisfies the so-called incidence equation

ATA:

where

I

is the identity of order

same order.

(r-

v

and

(1.5)

:):+

J

is the matrix of all ones of that

From this it follows that the determinant of

ATA

is given by

:ATAI = [r + (V - l)~] (r - :):-i

(l.6)

If the block design is symmetric, this becomes (using equation (1.4))

IATAI = k2(k - ~)v-i

from which it follows that the incidence matrix balanced incomplete block design is non-singular.

A

(1.7)

of a non-trivial symmetric Using this fact it can be shown

(Ryser, 1950) that the incidence matrix of a syn~netric block design is normal. is, that

That

(1o8)

ATA = AA T = (k - ~,)I + ~7

for symmetric designs. Since

IATAI = IAI 2

one can conclude from equation (1.8) or (1.7) that a

symmetric block design (or a difference set) can only exist for even n = k - ~

is a square.

v

if

The study of equation (1.8) has led to other significant

existence criterions for symmetric block designs and hence for difference sets. Since Chapter II is devoted to difference set existence questions these results are deferred until then.

D.

The Characteristic Function and its Autoeorrelation Function Corresponding to the difference set

(i = 0 , . . . , v wise.

l)

given by

a. = 1 1

if

D i

is the binary sequence is a member of

D

and

{a i] a. = 0 1

other-

This is called the characteristic function or incidence vector of the

difference set

D.

[Of course it appeared in I. C. as the first row of the

incidence matrix associated with the difference set. ]

Considered as a binary

sequence, it is quite natural to inquire about its autocorrelation function v-1 Ra(J) =

~

a i ai+ j

(i + j

taken modulo

v)

i=O

and normalized autocorrelation function

Pa(j)~ !v Ra(J)' Since

{a i]

is the characteristic function of a difference set, the autocorrelation

ftunction is particularly simple, i.eo

Ra(J) = I k

if h

j -= 0

otherwise

modulo

v

Frequently the binary sequence

fb " i }v-1 0

where

b i = 2a i - 1

is considered

instead of the characteristic function (note the transformation merely replaces the zeros of

[a i}

with minus ones).

This sequence

{b i]

has autocorrelation

function

J

v

if

j ~ 0

\

V - 4(k - ~)

modulo

v

otherwise

Autocorrelation functions like these are said to be two-level and binary sequences which possess them have found extensive application in digital communications. [See Golomb et al, 1964 for some of these].

Of course, it follows immediately from

the definitions that the only binary sequences which have two-level autocorrelation functions are those associated with cyclic differences sets.

E.

Multipliers If

t

is prime to

difference set

D,

then

v

and if t

tD

is some shift

is called a multiplier of

D + s D.

of the original In terms of the

associated block design of the difference set (see I.Co above) the mapping (mod v)

is an automorphism.

That is~ if

A

is the incidence matrix of the associ-

ated block design, there exists permutation matrices order

v

x~tx

P, Q

determined by

t

of

such that

PAQ = A .

All known multipliers

(1°9)

v, k, h - difference sets have non-trivial multipliers

t ~ 1

modulo v).

(i.e.,

The question as to whether this must be so is open.

The collection of multipliers of a given difference set forms a group called the multiplier group of that difference set.

One of the most useful results in the

theory of difference sets is the so-called "multiplier" theorem.

This theorem

guarantees the existence of multipliers under certain circumstances.

Theorem i.io is prime to

v

(Hall and Ryser~ 1951 ).

and if

with these parameters

p > h,

then

p

If

p

is a prime dividing

n,

is a multiplier of all difference sets

v, k, h.

Chapter III discusses this result and its generalizations.

For the present,

merely note that for all known non-trivial difference sets the condition superfluous.

p > h

is

Again since (another open question) all known non-trivial cyclic

difference sets have

(n,v) = 1

difference sets every divisor

F.

if

one has that for all known non-trivial cyclic t

of

n

is a multiplier.

The Hall-polvvnomial, w-multipliers Instead of the difference set itself, it is often convenient to deal with the

polynomial

e(~):x

dl

+...

%

+x

.

(l.lO)

This pol~nqomial has been called the Hall-pol~nomial of the difference set, the generating polynomial of the difference set or the difference set pol~omial. In terms of this polynomial the difference set property is

e(x)e(x-1)

k ~. =

x

d. -d. I j

=

n

+

h(1 + x

xV_l ) +

o"

+

(mod x v - i) o

(l.ll)

i,j

If

~v ~ 1

is any vth root of unity this congruence yields

(1,z2)

e({v) 8({v I) = n.

This shows that a non-trivial difference set is intimately connected with the factorization of If

t

n

in the field of vth roots of unity.

is a multiplier of the difference set

D

then

e(x t) ~ xSe(x)

More generally, prime to

w,

if

w

divides

v,

(mod x v- l) .

define a w-multiplier to be any integer

for which there exists an integer

e(x t) ~ xSe(x)

Clearly

s

s

(mod

wI

w-multiplier for every divisor

w

of of

w.

t,

satisfying

xw - l ) .

may be assumed to be non-negative.

wl-multiplier for all divisors

(1.13)

(1.14)

Further, a w-multiplier is a

Thus, an ordinarymultiplier

is a

v.

On occasion it is possible to demonstrate that a hypothetical difference set must have a w-multiplier even when it is not possible to show the existence of a multiplier.

This often leads to non-existence proofs which can not be deduced

from the strict multiplier theory.

G.

Group Difference Sets A difference set in a group

distinct elements of

G

G

of order

v

is a set

{gl~..°~gk]

of

such that the equation

-1 gigj = g

has exactly

h

solutions for every

g

in

G, g ~ 1.

Group difference sets are called non-Abelian, Abelian~ or cyclic according to whether the group is non-Abelian, Abelian or cyclic.

The difference sets considered

above (I.A., etc.) correspond to group difference sets for cyclic groups (i.e., they are cyclic difference sets under this terminology). sets constitute a generalization consideration.

Thus group difference

(due to Bruck, 1955) of those previously under

Every group difference set gives rise to a symmetric block design

in much the same manner as demonstrated for cyclic difference sets in I.C. above. But not every symmetric block design corresponds to a difference set in some group G.

[For example, the

v, k, h = 31, 10, 3

design (listed in Hall, 1967, p. 293)

l0

could only correspond to a cyclic difference set since the only group of order 31 is cyclic°

But this design is obviously not cyclic. ]

occupy a t r ~ y

Thus group difference sets

middle ground between symmetric block designs and the cyclic dif-

ference sets of concern here. The main reason for introducing general group difference sets into this discussion at all is that some of the major outstanding problems are only of concern for cyclic difference sets.

Thus, in subsequent chapters a few facts about

general group difference sets are mentioned in connection with these problems.

The

purpose being to point out that the difficulties arise only because of the cyclic nature of things and thus cannot be resolved solely by techniques which apply more generally. An example of a group difference set, which is not cyclic, is the set D = {a, b, e, d, ab, ed] where

a

2

= b

2

= c

2

=

d2

in the Abelian group of order i6 generated by = i.

This set has parameters

a, b, c, d,

v, k, )~ = 16, 6, 2.

II.

EXISTENCE QUESTIONS

The main questions regarding difference sets are: do you construct them?

When do they exist?

How many (inequivalent ones) are there?

How

Even though there

is considerable overlap between the areas defined by these questions, there seems to be some value in treating them separately.

Thus, this chapter is primarily

concerned with conditions necessary for the existence of difference sets. most part only number theoretic results involving the parameters their divisors are considered.

For the

v, k, ~, n

and

Some existence tests of a more constructive nature

are presented in Chapter I I I - Multipliers and Constructive Existence Tests.

A.

The Main Existence Problems As the title of this chapter implies, the existence question for difference

sets is unsolved.

That is, given parameters

v, k, ~

it is (in general) impossible

to decide (short of an exhaustive search) whether or not a difference set with these parameters exists.

Nevertheless,

significant progress has been made.

Perhaps the most important test is the obvious relationship

k(k - i) = ~(v _ i) or

k2

= ~v + n .

(2.1)

A sub-problem of this general existence question is the curious fact that no difference sets are known which have

(v,n) > 1

though no proof of this has been given.

[or equivalently

Here one must be careful to distinguish

between cyclic difference sets and general group difference sets. exist group difference sets with parameters with

(v,n) > l;

v, k, k, n = 16, 6, 2, 4.

(v,n) > l,

(k,v) > 1],

For~ there do

the example given in I.G. above has

Thus, if there are no cyclic difference sets

the proof must be intrinsically cyclic.

12

Another outstanding existence problem arises when one notes that there exists an infinite number of difference sets with pJ + l, pJ + l, 1 struction).

for all primes p

Z = 1.

Specifically

(see Section V.A. for details of their con-

It has been conjectured that for every

finite number of difference sets.

v, k# k = p2j +

Z ~ 2,

there exists only a

This conjecture is wide open; it has not been

either proved or refuted for any single value of

Z.

Of course the same conjecture

can be made for symmetric block designs and again the problem is open.

B.

The Bruck-Ryser-Chowla Theorem As pointed out in Section I.C. above, the Incidence Equation~

i.e.,

ATA = nI + ~J

holds for all symmetric block designs. of the design, order

v).

J

is the

v × v

(Here

A

(2.2)

is the

matrix of all ones and

v x v I

incidence matrix

is the identity of

Associate a linear form with each row of the incidence matrix

A~

according to the rule

v Li(x ) =

~

aij x. J

j=m

where

x = (Xl, .... Xv)

takes the form

is a vector of indeterminates

x.. J

Then equation (2.2)

13

L~(X) + ... + L ; ( X ) =

n(~

+ -.. + 4 )

(2.3)

+ h(Xl + "'' + Xv)2

The study of these equations ((2.2) and (2.3)) has produced a number of existence criterions for symmetric block designs as well as for certain more specialized configurations. Let

B = nI + ~J

and write

ATA

as

ATIA.

Then (using the language of

quadratic forms) equation (2.2) shows that if a block design exists, then the identity matrix I represents fortunately, when

A

B

with a

0, 1

transformation matrix

the theory of such matrix representations

Un-

is not fully developed even

is allowed to have arbitrary integer coefficients.

permited rational coefficients,

A.

However, if

A

is

the Hasse-Minkowski theory of rational equivalence

of quadratic forms [see Jones (1950) for an exposition of this theory] provides necessary and sufficient conditions for the existence of such a transformation

A.

Specifically

Theorem 2.1 (Bruck-Ryser-Chowla) ATA = nI + ~J,

when

k(k - l) = h(v - 1),

(i)

for

v

even,

(ii)

for

v

odd, the equation

integers

A v x v

n

x, y, z

rational matrix

A

satisfying

exists if and only if

is a square z

2

= nx

2 +(-i )(v-l)12y2

has a solution in

not all zero.

Thus Theorem 2.1 provides necessary conditions for the existence of symmetric block designs and hence for difference sets.

In f a c %

there is no parameter set

v, k, ~

satisfying Theorem 2.1 for which it is known that no symmetric block design

exists.

That is, conceivably the conditions of Theorem 2.1 are sufficient not only

for the existence of a rational matrix

A

but also a

O, 1

matrix.

If one

restricts attention to cyclic difference sets however, this is no longer the case. For [as is shown later, section II.E] there is no cyclic difference set with parameters

v, k, k = 16, 6, 2

even though these parameters do satisfy Theorem 2.1,

14

as they clearly must since an example of such a block design can be derived from the non-cyclic difference set given in section I.G. above. It should be pointed out that Legendre

[see Nagell (1951) Theorem 113] provided

a simple effective test for the solvability of diophantine equations of the type appearing in (ii) above (see Note 2 below for a statement of this test).

Thus

criterion (ii) is an effective criterion even when a solution of the diophantine equation is not obvious. consider

As an example of a parameter set excluded by this theorem

v, k, k = 43, 7, i;

this leads to the diophantine equation

z 2 = 6x 2 _ y 2

Necessity of the conditions of Theorem 2.1 can be proved without recourse to the Hasse-Minkowski theory (see below). h = i n

Even sufficiency is available for the case

[Hall (1967), p.lll] and also whenever

n

is a square.

In particular, when

is a square the rational matrix

A= Jni+

gk n j _ _ v

satisfies equation (2.2).

Proof of Theorem 2.1

[necessity only, Chowla and Ryser (1950)].

(2.2) it follows that

(det A) 2 = k2(k - h)v-l;

is a square.

v,

For odd

thus when

v

is even,

From equation n = k - h

the number-theoretic result that every positive integer

is representable as a sum of four integral squares

[see, for example, Nagell (1951)]

and Euler's identity

(b

where

2 2 2 2 2 2 ~ 2 2 2 2 + b 2 + b 3 + b4)(x I + x 2 + x 3 + x ) = Yl + Y2 + Y3 + Y4

(2.4)

15

Yl = blXl - b2x2 - b3x3 - b4x4 Y2 = b2xi + blX2 - b$x3 + b3x4

(2.5) Y3 = b3Xl + b4x2 + blX3 - b2x4 Y4 = b4Xl - b3x2 + b2x 3 + blX4

are required.

system of equations solved for the efficients, When Xv = YV

n = b 2I + b 22 + b 32 + b 24

With

(2.5) is

x.'sl

n 2.

Thus Cramer's

as linear combinations

the denominators v ~ i (mod 4),

integers) the determinant

(b i

of which are

the relation

rule shows that the system may be

of the

Yi'S

with rational co-

2

n .

2 2 2 n = bI + b2 + b3 + b

together with

and

2 2 2 2 2 + 2 2 n(x2 + Xi+l + xi+2 + xi+3) = Yi + Yi+l Yi+2 + Yi+3

for

i = 1,5,...,v- 4

the independent

can be used to transform equation

indeterminants

yl,...,y v

Ll(Y),...,Lv(Y )

yl,...,y v. YI'

and

given by

w = x I + x 2 + .-o + x v

Since (2.7) is an identity in the

Yi'S

(2"7)

are rational linear forms in it is valid for all values of

in particular for the value

cev2

+ ..o + C J v

for i-

cI

cI ~ 1

(2.8)

Yl = e2Y 2 + ..° + C J v

for -i-

where

(2.6)

(2.3) into an identity in

2 2 2 7,~w2 L (y) + ... + L (y) = Yl + "'" + Yv-I + nYv +

where

of the

cI

cI = 1

16

n

LI(Y) =

Z

cjyj .

J=l

For this value of

Yl

however,

identity in the variables

L~(y) = y~;

Y2, o.o,yv.

thus equation (2.7) reduces to an

Proceeding in this manner with

y2,...,yv. 1

in turn, yields the identity

L2v(y) = ny2v + %~w2

where

Lv(Y )

integer

x,

and

w

are rational multiples of

Yv"

Now let

Yv

be a non-zero

which is a multiple of the denominators appearing in

then in integers

Lv

and

w,

x, y, z (x ~ 0) the equation

z2 = n x 2 + by2

(2.9)

has a solution. When Xv+ I

v ~ 3 (mod 4),

add

is a new indeterminate.

nx

2 v+l

to both sides of equation (2.3) where

Proceeding as before yields the identity

! 2 2 = 2 2 L (y) + ".. + Lv(Y ) + nXv+ I Yl + "°" + Yv+l + hw2

where

Ll(Y),o..,Lv(Y), Xv+ I

Again choosing

yl,...,y v

and

w

are rational linear forms in

yl,...,yv+ 1.

judiciously implies the identity

2 = 2 nXv+l Yv+l + hw2

where

Xv+ I

and

w

are rational multiples of

non-zero integer

z

which is a multiple of the denominators of

yields a solution in integers

Yv+l"

x, y, z (z ~ O)

2

2

Taking

of the equation

2

Yv+l Xv+ 1

to be a and

w

17

Combining this equation with that of (2.9) completes the proof of the necessity of the conditions of Theorem 2.1. Condition (i) of Theorem 2.1 was derived independently by Sch[tzenberger and by Chowla and Ryser (1950).

Condition (ii) was first established,

only, by Bruck and Ryser (1949) and then generalized to arbitrary

~

for

(1949)

h = 1

independently

by Chowla and Ryser (1950) and Shrikhande (1950). As pointed out in section I.C. the incidence matrix

A

of a symmetric block

design not only satisfies equation (2.2) above but also must be normal (Ryser, 1950)o

That is, it must satisfy

(2.io)

AA T : ATA = (k - h)l + ZJ.

Thus, when Theorem 2.1 was first established,

there was some reason to hope that

adding the normality condition would further restrict the possible parameter sets v, k, h.

This was shown not to be true by Albert (1953) for

and Ryser (1954) for general

h.

k = i

That is, the conditions of Theorem 2.1 are

sufficient to guarantee the existence of a normal rational matrix equation (2.2).

Moreover,

and by Hall

this solution

A

A

satisfying

also satisfies the condition

AJ = kJ

which is trivially necessary for block designs° Hall and Ryser also showed that any set of initial

0, i

rows, consistent

with equation (2.10) above, can be completed to a normal rational matrix satisfying that equation. more general result.]

[See Hall (1967) p. 275 for the proof of a somewhat

Clearly, specifying an initial set of

to equation (2.10) yields the same result. this to the case where

r

rows and

s

of equation (2.10) and those imposed by for symmetric block designs. rational matrix Note i.

A

A

%1

columns subject

Eo C. Johnsen (1965) has generalized

columns are given, subject to the provisions AJ = JA = kJ,

which are clearly necessary

He showed that even here there always exists a normal

satisfying equation (2.10) as well as

AJ = JA = kJ.

The definition of the incidence matrix of a block design varies with

different authors and even with different works of the same author. the matrix designated by

AT

Some say that

above is the incidence matrix instead of

A.

Since

18

symmetric block designs yield normal incidence matrices for the designs themselves. proofs.

But it is sometimes

Note 2.

a factor in the mechanics

If

has a solution

a, b, c

are squarefree

in integers,

quadratic

residues

quadratic

residue of

ax

of

integers which are relatively prime in

2

+ cz

2

m

= 0,

without

be divided out. i,

2

divide that

ez ~ g

there

but since

m a y be divided out. manner yielding

Co

Integral

above.

significant

z.

represents

x

g

is said to be a

such that

Given a ~

are

x

2_

= g

diophantine

equation whose

modulo

equation coefficients

Then,

any factor common to with greatest

a, b, c

con~non divisor

if the equation has a solution

are squarefree

and

(a,c) > i

or

(a~ b, c) = 1 ax

2

+ by

(b,c) > i

2

can

g

must

this implies

22 + cg z I = O.

can be handled

Thus

g

in the same

an equation of the desired type.

Solutions

to the Incidence

are discussed

section II.D.

-ac, -ab

(g,m) = l,

are squarefree

Thus the equation becomes Clearly

-bc,

diophantine

Similarly

a~ b, c

a, b

ax 2 + by 2 + cz 2 = 0

Clearly any square factor m a y be dropped from

change.

Since no further existence figurations

[If

other~ise. ]

is an associated

(a~b) = g > 1.

divides

if and only if

respectively.

nonresidue

Thus assume

and assume

equation

if there exists an integer

satisfy the restrictions a, b, c

not all zero,

a, b, c

and called a quadratic + by

of the

a reference.

pairs and not all of the same sign~ the diophantine

2

in no errors

Thus one should be careful to check the incidence m a t r i x definition being

used when consulting

m

this results

Equation

tests for difference

in this section,

The material

sets or more general

the reader m a y wish to skip to

cited here not only is of independent

a survey of the present

con-

status of the integral

interest but also

solution problem;

one

would hope that a complete

solution of this problem would provide new existence

criteria for block designs

and difference

As noted previously and symmetric matrices

sets.

there is a natural correspondence

which links each form

between

quadratic

forms

19

n

ci$.x.x. I ~

f(x) =

c.lj = c.. Sm

(2.11)

i,j=l

with the matrix of its coefficients

C = (cij).

are restricted to the real numbers.

Given two n-variable forms

their matrices

C, D, f(x)

transformation taking

In this discussion the coefficients

is said to represent

f(x)

into

g(y).

g(y)

f(x), g(y)

and

if there exists a linear

That is, if there exists a substitution

n xi : ~

sijY j

i = l,.o.,n

j=l

transforming

f(x)

into

g(y).

If

S = (sij),

this means that

s~cs : D

in which case

C

efficients then

is said to represent f(x)

D.

(2.12)

If

is said to represent

S

g(y)

is restricted to rational corationally

(C

represents

rationally) with analogous statements for other coefficient restrictions. f(x)

represents

integer

m~

g(y)

D

Further,

rationally without essential denominator if, for every

there is a matrix

S

of rank

rational elements with denominators prime to

n

such that m.

sTcs = D~

If two forms

where

f(x), g(y)

S

has

repre-

sent each other rationally without essential denominator they are called semiequivalent and said to belong to the same genus. If

f(x)

determinant

represents ± I,

g(y)

integrally and the transformation matrix

has

then clearly

g(y)

represents

formation matrix of determinant

± i.

Two such forms are called equivalent and

said to belong to the same class.

f(x)

S

integrally with trans-

Thus equivalent forms are clearly semi-equivalent;

in fact a genus will in general contain several classes of forms. Turning back to block designs again, consider the incidence equation

20

ATA= (k- ~)Z + ~ = B .

This says no more than that the form v x v

2 2 f(x) = x I + ... + x v

with a

0, 1

(associated with the

g(y) = (k - h) (y~ + .-- + y~) +

identity matrix I) represents the form

h(Yl + "'" + Yv )2

(2.13)

transformation matrix

A.

The Bruck-Ryser-Chowla Theorem (Theorem 2.1 above) establishes necessary and sufficient conditions for the existence of a rational matrix equation (2.13).

A

satisfying

Unfortunately the theory of integral representations of quadratic

forms is not yet complete, even though a great deal of work has been done on such problems

[see Jones (1950), Watson (1960) and O'Meara (1963) for this work.]

How-

ever the main concern here is equation (2.13), not all possible integral representation problems, thus a complete theory is not required for these purposes. Goldhaber (1960) studied the integral representation problem posed by equation (2.13) for a restricted set of parameters

Theorem 2.2. (k,n) = l,

If

v, k, h

v, k, h.

He proved

satisfy the Bruck-Ryser-Chowla theorem and

then there exists a form in the genus of

I

which represents

B

integrally. However,

for

v ~ 8,

the genus of the identity matrix contains more than one

equivalence class of forms (Magnus, 1937), thus no immediate conclusion can be drawn regarding integral representation from this result. A

0, 1

matrix

A

satisfying equation (2.13) with

k(k - l) = h(v - l)

is

the incidence matrix of a symmetric block design~ although this is not immediately apparent. (i.e.,

For, with the incidence matrix definition of section I.C. above

a.. = 1 ij

if and only if block

i

contains object

yields the fact that every object occurs in exactly objects occurs in exactly contains precisely contain

k

b0~...,bv_ 1

h

distinct blocks.

k

equation (2.13)

blocks and that any pair of

So, the fact that every block

objects is yet to be established. objects respectively.

j),

Let blocks

B0,...,Bv_ 1

Then, from the conclusions already

drawn, it follows that

b 0 + b I + ... + by_ I = kv

21

b0(b 0 - I) + bl(b I - i) + "'" + bv_l(bv. 1 - i) = hv(v - i) = vk(k - i) .

So v-1

v-i

i=O

i=O

[bi(b i - i) - k ( k -

1)]

from which it follows that

v-i

v-i [bi(b i - i) - k(k - i) + b.l + 2k2 - k - 2kb.l ] = 0 .

i=O

But

i=O

Z(b i - k) = O = Z(b i - k) 2

block contains precisely Note that if

A

one of these can be a

Theorem 2.3.

k

implies that

objects,

b.I = k

satisfies equation (2.13) then so does 0, i

matrix.

then

A

or

i.

That is, every

as was to be proved. -A.

Of course, only

Ryser (1952) has shown:

If a normal integral matrix

k(k - I) = h(v - i),

for all

-A

is a

A

0, i

satisfies equation (2.13), for matrix and hence is the incidence

matrix of a syn~netric block design. As pointed out in section I.C. above, the incidence matrix of a symmetric block design is normal~ thus it satisfies: (i) (ii)

ATA = (k - ~)I + ~ AA T = (k - h)I + ~J

(iii)

AJ

: kJ

(iv)

JA

: kJ

Ryser has shown that a nonsingu!ar also one of (iii),

matrix,

satisfying one of (i), (ii) and

(iv) satisfies all four of them and also the relation

k(k - i) = h(v - !). needed;

v x v

[In Chapter V, the fact that (ii), (iii) imply (i), (iv) is

so a proof of this much of Ryser's result is included here.

Since

22

IBI = (k - h)v-l(vh - k + k), v h - h + k ~ O.

the nonsingularity of

A

means that

k - h ~ 0,

By assumption

AA T = (k - h)I + k/,

Multiplying this last by

A -1

shows that

AJ

k ~0

j2 = vJ.

kJ

and that

transposing both sides of this same equation yields Further, note that

=

,

A-Ij = k-Ij.

JA T = kJ,

since

Whereas j = jT

So

A T = A-!(AA T) = (k - ~)A -I + )~-ij = (k - h)A -1 + Ak-ij kJ = JA T = (k - ~)JA -I + hk-Ij 2 = (k - ~)JA -I + ~k-lvj.

Thus~

=

which serves to define

k-k

m.

Thus

k -

J = mJA

J =mJ

and

vJ = j2 = (mJA)J = mJ(AJ) = mJ(kJ) = mkvJ

which provides that

mk = i

JA -I = k-ij.

established.

or

Thus

m = k -i. J = k'ijA

As shown above

From or

JA -I = mJ

JA = kJ;

and

m = k "I

it follows

that is (iv) has been

A T = (k - ~)A -I + }~-ij;

multiplying through b y

yields

ATA

by (iv). promised.]

But this is (i).

=

(k - ~)I

+ k~-ljA

=

(k - ~)I

+

So (ii), (iii) have been shown to imply (i), (iv) as

Using a different part of Ryser's result, note that an integral

solution of equation (2.13) which also satisfies one of

AJ = kJ~ JA = kJ

is

A

23

normal and by Theorem 2.3 must be the incidence matrix of a symmetric block design. Let

A

multiplied

be a matrix through by

satisfying

-1,

Thus, given such a matrix

XX T = B,

the resultant A,

then if any column of

matrix

C

A

still satisfies

there is no loss of generality

is

XX T = B.

in assuming

that

its column sums are nonnegative.

Theorem 2.4.

(Ryser,

column sums,

satisfying

free and if

k - ~

1952)

If

AA T = B

A

with

is odd, then

A

is an integral matrix, k(k-

l) = Z ( v -

1),

with nonnegative

if

(k,Z)

is square-

is the incidence matrix of a symmetric block

design. The condition n = k - h

is even and

Theorem 2.5. v x v

k - h

Let

odd is necessary,

~ = !.

v = n

2

matrix w i t h nonnegative

of two types: even,

(i)

A

consist of

have sum zero.

In particular

n + 1

A

column sums satisfying

has a single

entries

There exists

an

+l

of an Hadamard matrix and also for

n = 10.

The orders of Hadamard matrices are limited to

IV. B. for more information

on this

In addition to the solutions

n

block designs

n = 2a

v = n

2

zeros~

n = 1,2 t

and

4t

be an integral A

is of one

entries

n

columms

which is the order

of order

for positive

n

with

integers

t.

subject].

of type

(ii) solutions

+ n + l, k = n + l, h = 1

is

+l, n + 1

and the remaining

(ii) given above, for the case

E. C. Johnson

(1966A)

n -= 2 (mod 4).

there are known to be both types of solutions;

that type

n

[see the note at the end of section

always exist as well as Hadamard matrices

b e e n conjectured

A then

n2 + n

(ii) for every

has found infinitely m a n y type (ii) solutions when

2

(1, -1 square matrices

It is not known whether they exist for all

Further,

and let

AA T = B,

zero and

and

of type

+ n n/2)

[see Hall (1967), p. 286].

+ n + l, k = n + l, ~ = 1

A

determinants

exist when

is the incidence matrix of a block design or (ii)

one of the columns of

columns

for many counter examples

of order

always exist for even

satisfy the Bruck-Ryser-Chowla

for such

n = 2 a. n

It has

whenever

condition.

D.

The Theorem of Hall and Ryser d1 If a cyclic difference

set exists,

then its Hall-polynomial

e(~)

:

x

+

-.-

% + x

satisfies the congruence

e(x)@(x -I) m n + ~(i + X + .o" + X v-l)

Thus, for any divisor

w

of

v,

(2.14)

(mo~ x w - i)

(2.i5)

(mo~ x w - i)

(2.16)

one has

w-1 @(x) ---b 0 + blX + .-- + bw_IX

e(x)e(x -I) ~ n +

(mod x v - l)

(1 + X + "'" + X w-l)

~v W

where

bi

is the number of

[equate coefficients

dj

in

in congruence

D

satisfying

w-i

~

b.2

=

n

+

1

~v

~

~

w

'

i=O

j = l,o..,w - i.

This implies

2.16]

w-1

for

d. -= i mod w. J

bibi_j

-

hv w

i=O

(Here the subscript

i-j

is taken modulo

w.)

So if

S

is the cyclic matrix defined by

S

b0

b I ... bw_ I

bI

b 2 ... b 0

=

bw. I b 0 ... bw_ 2

it follows that

ssT=

D

(2.17)

25

where

D

has

n + (hv/w)

on the main diagonal and

kv/w

in all other positions.

Applying the m e t h o d of Chowla and Ryser (see the proof of Theorem 2.1 above), it follows that in order for there to exist a rational matrix

S

of odd order

w

satisfying equation (2o17) it is necessary that the equation

Z2 = ~

have a solution in integers in this solution = z + kx

and

y ~ 0.

2

+

x, y, z,

(_l)(W-l)/2(~v/w) y2

not all zero.

If

(2.1s)

n

Thus, for this case, the integers

~ = y(hv/w)

is not a square, then ~ = zk + nx,

are not all zero and they satisfy the diophantine

equation

[2 = ~ 2 + ( - n ( w - 1 ) / ~ 2 .

Of course this equation always has a non-trivial solution when

(2.19)

n

is a square.

Thus

Theorem 2.6 for odd

v,

integers

(Hall and Ryser, 1951)

then, for every divisor

x, y, z,

w

If a non-trivial difference set exists of

v, equation (2.19) has a solution in

not all zero.

From the derivation above it is obvious that equation (2.19) has a non-trivial solution for w satisfied. only when

= v

whenever condition (ii) of the Bruck-Ryser-Chowla

Thus Theorem 2°6 can be stronger than the Bruck-Ryser-Chowla Theorem v

is odd and composite.

v, k, h = 39, 19, 9and

z 2 = 10x 2 - 39Y 2

respectively. z

2

Theorem is

Here with

w = v = 39,

and they have solutions

However, for

= 10x 2 + 13y 2

w = 13,

the two equations are x, y, z = l, l, 1

these parameters does exist however. 40;

z 2 = 10x 2 - 9 y 2 and

2, l, 1

the equation of Hall and Ryser is

and it has no non-trivial integral solutions.

difference set exists with parameters

matrix of order

An example is furnished by the parameter set

39, 19, 9.

Thus no cyclic

[A symmetric block design with

It is easily derived from any Hadamard

Hall (1967, Chapter 14) discusses construction methods for

26

Hadamard matrices.] Hughes (1957) used the work of Hall and Ryser to prove a more general theorem about the structure of certain symmetric block designs.

However, for difference

sets the result of Hughes says no more than Theorem 2.6 above.

Note°

Again only the conditions relating to rational solution of equation

(2.10) were developed, whereas it was known that b. < v/w 1

for

i = 0,1, o.o~W - 1.

S

was integral, cyclic and that

In particular perhaps some improvement in the

status of the integral representation problem for quadratic forms (see section II.C) would provide stricter existence tests.

E.

Results of Mann, Rankin z Turyn and Yamamoto By far the most stringent difference set existence tests yet discovered arise

from algebraic number theoretic consideration of the equation

e(~v) e(~v l) =

which must hold for a ~

v th root of unity

(2,20)

n

~v # i

if a difference set is to exist.

Most of the results cited below are attributed to Mann (1964)~ Turyn (1965) and Yamamoto

(1963) even though the work in Mann (1952 , 1965, 1967), Rankin (1964) and

Turyn (1960, 1961) is directly related.

In fact many of the results were discovered

independently by more than one of these authors°

A few of the theorems require the

knowledge of a multiplier or w-multlplier for the hypothetical difference set under question (see sections I.Eo and I.F. for these definitions).

Chapter III discusses

conditions on the parameters under which certain multipliers are known to exist° The results of this section are all stated in terms of the parameters v, k, h, n

and their divisors;

thus no algebraic number theory is required to

either understand the nature of the results or to apply them.

However,

knowledge of algebraic number theory is required for the proofs.

some

In order to

minimize this requirement some basic facts about cyclotomic fields (i.eo~ algebraic number fields generated by the roots of unity) are presented at the end of this

27

section (Theorems 2.18, 2.19 and 2.20). used.

Unless otherwise specified,

of unity and

K(~d)

~d

They will be referred to when they are will be an arbitrary primitive d th root

will denote the cyclotomic field it generates over the rational

field. If then b

a

a, b, c C

are integers

(c ~ O)

is said to strictly divide

and

b.

ac

divides

b

while

a c+l

does n o t ,

The same terminology is used if

a

and

are ideals°

Theorem 2.7 trivial

n

for which

modulo

Proof.

Let

strictly divide

w,

P

n

be a divisor of

~(~w).

v

and assume a non-

t > i.

Let

If there exists an integer

p

f > 0

is strictly divisible by an even power of

be a prime ideal divisor of

p

in

K(~w)

and let

be a prime such that p.

pi

Then by (2.47) of Theorem 2.19 below and equations pi

(1.14),

also strictly divides

Thus, by equation (2.20), n

w > i

(p,w) = i0

then

(2.20) it follows that

that

Let

v, k, h-difference set exists with w-multiplier

divisor of tp f m -i

(Mann, 1964)

p2i

strictly divides

is strictly divisible by an even power of

n

and this implies (Theorem 2.19) p.

This theorem, which (as shown above) is an almost trivial consequence of the prime ideal structure of

@(~w)

is an extremely important result.

cyclic difference sets are concerned,

For, as far as

it contains both the Bruck-Ryser-Chowla

theorem (odd v) and the theorem of Hall and Ryser.

In order to establish the

dependence of these results on Theorem 2.7 it is convenient to know the connection between diophantine equations of the type

x

2

= oy

2

+ Bz 2

5, ~

and the so-called Hilbert norm residue 'symbol

integers

(~,G)p.

(2.21)

The basic fact is that

28

equation (2.21) has a solution in integers

x, y, z,

(G,~)p = +l

G, ~

for all finite primes

fact is written

(~,~)~ = +l].

p

and

not all zero~ if and only if,

are not both negative

See Jones (1950, p. 26 ff.) for this as well as the

other properties of the Hilbert symbol which are needed.

Corollary 2.8 exists and let

(Yamamoto,

re

has a solution in integers

e

(~,~)p : (~,~)p

(2.23)

(~,~)p(~,T)p : (~,~)p

(2°24)

1963)

x

(p = ~ allowed):

(2.22)

Assume that a

2

v, k, ~ - difference set

for an odd divisor

strictly divides

r

These are

(~,~)p : ± z

q* = (.l)(q-l)/2 q

prime such that

[this last

n,

q

of

v.

Then if

r

it follows that the equation

.(q-l)/2 2 2 + (-1)" - qy = z

x, y, z,

is a

not all zero.

(2.25)

Or, equivalently,

that the

Hilbert norm residue symbol

(2.26)

(re,q*)p = + 1

for all primes

Proof.

p.

Since

assumed to be a prime. x,y,z = l,O,r e/2.

(qlq2)* = q{ q~ If

e

(x,y,z = a,brJ,cr j)

rx

r = q ~ 3(4).

x,y,z = a,b,Co If

q

m a y be

is even, equation (2.25) has a solution

Thus attention may be restricted to odd

case (2°25) has a solution

has a solution

it follows from (2.24) that

r = q ~ 1(4)

2

+ (-1

Now

e = 2j + i.

In this

whenever the equation

~"(q-1)/2qy 2

x,y,z = i,i,0

then since such a

: z

2

(2.27)

is a solution of (2.27) when q

has a representation

(see

Nagell, 1951)

q : s2 + t 2

x,y,z = s,t,q

is a solution of (2.27).

note 2) requires

where

r, q*, -rq*

s

and

t For

are integers, it follows that r ~ q

Legendre's test (section II.B.

to be quadratic residues of

q, r, 1

respectively.

Of these conditions, the last is trivial and the second follows from the first by quadratic reciprocity (Nagell, 1951). residue of

q

(Theorem 2.7)

As for the first, if

then, by Euler's criterion~ e

would be even.

r

were not a quadratic

r (q-l)/2 e -1 (mod q)

and hence

This contradiction establishes the existence of

a non-trivial solution for equation (2.27) and hence for equation (2.25), i.eo, the corollary has been established. In terms of the Hilbert symbol, the theorem of Hall and Ryser becomes (n,W*)p = +l

for all odd divisors

of equations (2.23), (2.24), (2~26).

w

of

v

and this is an immediate consequence

Using this for

w = v, v

odd, yields

+l = (n,V*)p = (n,V)p (n,(-l)(V-l)/2)p

while

k 2 = Zv + n

provides

+l = (n, hV)p = (n,h)p(n,V)p .

Together these equations show that

(n,h)p = (n,V)p = (n,(-l)(V-1)/2)p.

Hence

(n,Z)p (n,(-l)(V-l)/2)p = (n,(-l)(V-1)/2k)p = +l.

Ioeo, the Bruck-Ryser-Chowla condition for odd

v

has been established as a direct

consequence of Theorem 2. 7, In particular, if

v,k~ Z = 241, 16, 1

the Bruck-Ryser-Chowla equation

30

z

2

= 15 x

2

Hall-Ryser However

+ y

2

has a solution

condition

360 m -1

z = y = i, x = O.

is only stronger

modulo

241;

As 241

for composite

v,

is prime and the

it must also be satisfied.

thus Theorem 2. 7 shows that no difference

set

exists. Another

result of Mann (1964)

Theorem 2. 9 .

Let

n

is a square (i)

(ii) (iii)

(iv) (~)

qv

1

be a w-multiplier

w > 1

set for some divisor Then

t

of

v.

23 n = noq

or

modulo

Let

t f E -1

t

for some prime

ql / q

Since

some integer

residue

tI

so is

divisor

and

also divides

2 n = nln 2

2

2 = n1

and

Let

q

n2 = q

k 2 = hv

+ n,

n 2 > 1. n

tl I / -1

modulo

ql

fl"

thus

8(~w l) = ~

O(~w)

for

=

q

So k - ~

n

d

for

w

it follows

2 n = nln 2

w

th

roots of

where every prime

so from now on assume

that

w

v of

were

and

from

and hence n2

w

and

2 n = nln 2 n2

are

and apply the same process

w; dq s

would show

w = qS (s >_ l)

in the field of

or

playing the role of If

(2.28)

m a y be assumed to be square free.

be a prime divisor

necessarily.

then the same process with assumption

n2

the theorem is proved,

above with the odd prime m

w

of

2 n = nI

By Theorem 2.1 it follows

odd.

tf;

then

= n ~wr

ideal factorization that

n2

v

Then

unity (Theorem 2.19) , it follows

necessarily

f.

In this later case

and all integers

is w-multiplier,

From this and the prime

n 2 > i.

for some integer

q

is another prime divisor of

t

r.

of

[8((w)]2

with

q.

w

s > 1

is a quadratic

if

Unless

modulo

v, k, ~ - difference

4

for all multipliers

of

of a non-trivial

is odd

w = qS

Proof.

is:

this yields

n = m q.

with

and

q = i

n = 4q n -= v = 0

2

d > 1

modulo

q q

Thus

(d,q s) = l,

which contradicts with

as

prime.

As

that

q2

the

31

divides

n;

thus with

is not assumed that As before, in

nO = nl/q

the desired form

2 3 n = noq

emerges, where it

(no, q ) = 1. K(~q)

there is an equation analogous to (2°28), namely

[e(~q)] 2 = n ~q

for some integer

~ > 0o

By definition

(2.29)

q* = (-l)(q-l)/2q

Gauss (see Nagell, 1951, section 53) that with

and it was shown by

~q = e 2~i/q

= z+ ~q+ ~q4 + ~q9 + . .. + ~( q-l)2

Hence

~

is an algebraic integer of

automorphism defined by Further, if into

t

whenever

Since

~

q,

t

lit is invariant under the field is a non-zero square modulo

then the mapping

is an algebraic integer of

Y = e(~q)/nl~q~q*

is an element of this field.

=

=

n--~

i.e., since

~,

is not a square modulo

- ~q*. ]

the fraction

~q~

K(~q).

Y

n(-l) (q!l)/2

satisfies the equation

K(~q)

modulo 4o ~

satisfy As

Y

x 2q = i; is a

t ~q

~q~ K(~q)

q.

takes

it follows that

Indeed, since

(2o31)

[ (-l)(q-l)/2

x 4q = i,

an algebraic integer but also a root of unity in in

(2.30)

it follows that it is not only K(~q).

But all roots of unity

thus it follows from equation (2.31) that

2q th root of unity, it is

± ~

for some

q

th

q m i

root of unity

Thus

@(~q) = ~ q nl"~'~q*

and as

is a q-multiplier it follows from congruence 1.14 that

Now if

is not a square modulo

q,

(2.32) s 8(~ ) = ~q

e(~q).

it follows from this equation together with

32

(2.32) that

t

t ~q -~ ~q

since, as noted above, contradiction, so

t

nl~q. = ~(±

maps

(2.33)

~ nl~q~. )

W~q* into

- ~q*.

But (2.33) is a

necessarily is a non-zero square modulo

q.

Of the conclusions of Theorem 2.9 only the last one remains to be verified. If the prime

ql ~ q

were another (necessarily odd) prime divisor of

which there existed a multiplier modulo

ql'

tI

and integer

fl

such that

then, by the process used above, it would follow that

23 n0q

(since it cannot be both

and

23 m0ql).

nor

q

fl tI n

v~

for

~ -i was a square

This contradiction completes the proof

of Theorem 2. 9 . Note that neither

tI = t

Consider the parameters 2

dividing

v,k,h = 813, 29, 1.

nO

is excluded by this theorem.

Theorem 1.1 (section I.E) shows that

is a multiplier, hence also a q-multiplier for

q = 3.

Thus Theorem 2. 9 shows

that no such difference set exists. Before establishing any further existence tests it is convenient to have some results concerning congruence relations in cyclotomic fields°

Let

C

be a number

theoretic function (i.e., a function which is zero except on the integers) and define the difference operator

A(O)

by

a(~) c(i) : c(i + 0) - c(i)

where

p

is a rational number not necessarily an integer.

exists such that

Theorem 2.10 decomposition of

•(n) C(i) = 0

for all

(Yamamoto, 1963) N.

Let

m

Let

integers of the cyclotomic field d

be a divisor of

N

then call

N

K(q)

and let

~

C

~i """ Ps~s

N = Pl

be relatively prime to

number theoretic function with period

let

i,

N,

whose values and let

If an integer

periodic of period

be the prime power let

C(i)

C

be a periodic

are algebraic

f ( x ) = ~i_-1 C(i)x i.

be an integer of

n

K(~m).

Further,

n.

33

Then, in order that

f(~)

m 0 (mod G)

for all divisors

r

of

d,

it is

necessary and sufficient that

tI ts -tl-1 -t -I P! "'" Ps A(NPl ) ... D(Np s s ) C(i) " 0

for all

let

i

and for all

Proof.

(i)

u = 0.

Now

Let

tl,...,t s

such that

s = i, N = p~, d = pU

~-i i P ~-

~i

i=O

j=0

tI ts Pl ''" Ps

(mod J)

d.

divides

and, proceeding by induction on

U,

~i+jp ~-I C(i + jp~-l) ~N

f(~N) =

p~-l_l i

I 1

i=o

j=l

[C(i + JP~-l) - C(i)] ~N+JP~-I

~-l since

~P

~i+jp ~'l N

= ~p for

K(~m) ; thus

is a primitive

pth root of unity.

0 _< i < p~-l, i _< j < p f(~N) - 0 (mod ~)

for all these values of

i

and

~(pZ)

integers

form an integral basis for

if and only if j.

The

C(i + jp~-l) _ C(i)

K(~mN)

over

0 (mod (~)

This condition is equivalent to the desired

one

for all (2)

i. Let

t, 0 < t < u.

Thus the theorem is proved for s = l, u > 0

s = I

and

u = 0.

and assume that the theorem is true for all integers

Let

p

g(x)=p

~-l

I i=O

-1

c(i) x~p

3~ and note that

p

~-i p -i

1

f(x p) i=O

Since

i=O

f({N) m O (mod G)

modulo

G

~l c(i + jp~-l)

C(i)xiP __-

implies [see part (i) above] that

mod(~,x N - I) .

t f({~ ) m 0 (mod ~)

for all

t equivalent to 0 < t < u - i. equivalent to

f(~

) m 0 (mod ~)

and

0 < t < u - i,

for all

t

such that

0 < t < u.

Now assume s.

s > i

where

k

0 < t < u

is

for all

t

such that

and

pt+l h(p~-t-2) C(i) m 0 (mod 5)

that is, to

pt A(p~-t-l) C(i) m 0 (mod ~)

Thus, the theorem has been proved for

s = i.

and assume the validity of the theorem for smaller

~i ~2 ~s N = NIN', N I : Pl ' N' = P2 "'" Ps ' d = dld',

Put

d I =(Nl, d), d' = (N',d).

exist integers

g({~ ) ~ O (mod 5)

pt A(p~-t-l) C(i) ~ 0 (mod G)

such that

values of

such that

By the induction hypothesis these last two congruences are

t

(3)

t

t

for all

and

C(i) ~ C(i + jp~-l)

it follows that

Thus the condition

N1

(mod x N - i) o

j=O

f(x p) ~ g(x)

r = rlr'

x ip

rI

Any divisor

divides

j, k

dI

such that

determined modulo

NI-I f(x) -

N' -i Z

and

r r'

of

d

can be uniquely written as

divides

d'.

i ~ N'j + Nlk (mod N) N'.

For any integer with

j

i

there

determined modulo

Hence

C(N'j + Nlk ) x

N'j+NIk

(mod x N - i)

j:o k=o NI-I r f(~N ) =

Z j=O

N'-I ~ k=O

NI-I Nlr

~N'jr < i k r = C(N'J + Nlk) ~N

c*(~ N j=O

~N'rj

,j)

~N

35

NW_l

where

C*(y,j) =

Z

C(N'j + Nlk)Y k.

k=0

N'

Now

IN

= ~

is a primitive

NI th root of unity and

NI IN = ~

root of unity. Further, the condition

f(~)

the same congruence for all primitive

N th roots of unity.

for some

IN

~ 0 (mod ~)

is a primitive

for some Thus

IN f(~)

N 'th

implies ~ 0 (mod ~)

implies that

NI-I Z

C*(~r' ,j)

~rlJ

(mod ~)

o

j=O th

for all primitive

NI

qo

C*(q

Note that the

(mN',N1) = lo

roots of unity r'

,j)

~

and for all primitive

are algebraic integers of

Thus applying the theorem for

s = 1

N'

K(~N,)

roots of unity

and that

to this case, i.e., to the

polynomial

NI-I

~,

c*(n

rW

,j) x j

j=0

yields the result that

f ( ~ ) ~ 0 (mod ~)

for all divisors

r

of

d

if and only

if

tl -tl-i r' Pl aj(NlPl ) C*(~ ,j) ~ 0

for all ~j(p)

tI

such that

tI Pl

divides

dI

(mod Q)

and for all divisors

(2.34)

r'

of

indicates that the difference operator applies to the argument Congruence 2.34 may be rewritten as

d'o j.

Here

36

N t- I

tI Pl

-tl-1 •j(NlP 1 ) C(N'j + Nlk)n r'k -=0

~

(2.35)

(rood G)

k=0

for all

tI

and

r'

such that

t1 Pl

divides

dI

and

r'

divides

d'.

Apply the

induction hypothesis to the polynomial

N'-I

tl

~

Pl

-tl-1 aj(NlP 1

) C(N'j + Nlk ) x k

k=0

which can be done since

N'

has

s - 1

distinct prime divisors, since the co-

efficients of this polynomial are algebraic integers of (m,N') = 1.

K(q)

and since

Thus congruence 2.35 holds if and only if

t I t2 ts -tl-i -t2-1 -t -i Pl P2 "'" Ps ~j(NlPl ) ~(N'P2 ) ... ~ ( N , P s s ) C(N'~ + ~ k )

ti

= Pl

for all

i

-tl-!

ts

"°" Ps ~(NPl

and for all

-t s -i

) ... &(NPs

ti,...,t s

such that

) C(i) - 0

tI t Pl ... ps s

(mod ~)

divides

d.

That is,

the theorem has been established. Folliwng Mann (1967) note that:

Corollar~ 2.11.

Let

~ > 0

be an integer and let

periodic, number theoretic function of period 0 < C(i) < M

N,

C

be an integer valued,

whose values

C(i)

satisfy

and

N-1

o /

c(i){~ ~ o

(~od ~).

i=0

Then, if

N

is the product of exactly

s

distinct prime powers, it follows that

37

< 2 s'l M .

Proof.

Apply Theorem 2.10 with

m = d = l;

this yields

A(Np[1) .,. a(~p~l) C(i) ~ 0

which must hold for all integers (2.36) must be non-zero. congruence Z C(i)~ C(j)

For some

[For otherwise

for any arbitrary value of

~ 0.]

with

magnitude

i.

Consider

0 ~ C(j) ~ M;

(2.36) at

~

i 0.

(mod ~)

i,

say

i0,

(2.36)

the left side of

Theorem 2.10 could be applied to this and this would contradict the condition The left side contains

exactly half of which have negative

of the left side of (2°36) is bounded by

2s'~,

2s

terms

signs.

ioe.,

Thus the

~ ~ 2s-~

as

was to be shown. Corollary 2.11 will be put to good use in the proof of the next existence test.

But first a lemma and a definition are needed.

Len~na 2.12o K({s) ,

and let

w-i Z a.x i, where the a. are algebraic i=0 1 l (s~w) = (m~w) = 1 for some integer m. If Let

A(x) =

A(~)

for

0 < j < w - i,

then

a. -= 0 l

integers and (2.37) holds only for

Proof.

--- 0

modulo

modulo

m

for all

1 < j _< w - l,

m

i.

(2.37)

If the

a.

l

then

a 0 -= a I ~ -.. =- aw_ I

modulo

Assume that (2.37) holds for

0 < j < w - I,

integers of

m.

i.e., that

are rational

38

1

1

1

ao l

. . . 1

#,

1

al

<

-bO l t i I

a2

=

I

bl

1

b2

J

I •

•

°

o

o

•

•

•

•

•

•

,

•

o

(2.38)

I

1 <_l

where

b. m 0 1

modulo

m

for all

io

The determinant of the system of equations

(2.38) is van der Monde and not zero, for it is know.to be

i>j

.

Since

F(x)

(for

x = i)

x w-I . .+

i>j

.

+.x +. 1

.

(x

~)(x

~)

o. . (x - <w-1 )

it follows

that

w . (i .

.~)

q,)(1 .

This shows that only primes dividing by Cramer's rule, it follows that

w

. . o

(1

~w - 1 ).

may occur in the determinant above.

a. m 0

modulo

m

for all

So,

i.

I

Suppose the modulo that

m.

+

I _

m

for 0

are rational integers and suppose that

Consider the polynomial

~ + ~w ~ 0

modulo a.

ai

modulo m

[certainly possible since

0 < j < w - 1

modulo

m

B(x) : A(x) + ~F(x),

for all

where

~

(m,w) = 1].

is chosen so Then

and hence (by the first part of this lemma) i,

that is

!

a 0 - a I -= ~o. = aw. 1

as was to be proved°

A(1) = ~ ~ 0

modulo

m

B(4 ) ~0

39

Let

p

w = p wI p

f

be a prime and let

with

~ -i

(P,Wl) = i.

modulo

Wl,

then p

w

2

are self-conjugate modulo

w.

divides

follow from

n~

divides

self-conjugate modulo

exists.

Let

v

and

O(~w l) = n

Theorem 2.13

w

that

w,

divide

for some divisor

w.

then

If all the m

m

m

is said m

in the

is self-conjugate

in this field is fixed under

w >l

m

is self-conjugate modulo

e(~)

~ 0

modulo

m.]

w~

it would

Note that if

it is also self-conjugate modulo any divisor of

(Turyn, 1965)

m2

such that

Thus, if a difference set is under consideration for which

w

e(~)

i.e., let

[From the prime ideal factorization of

then every prime ideal divisor of

complex conjugation. m

f > 0

roots of unity (see Theorem 2.19) it follows that, if

modulo

w,

is said to be self-conjugate modulo m

to be self-conjugate modulo th

strictly divide the integer

If there exists an integer

prime divisors of an integer

w

p

n

Assume a non-trivial

and suppose that

of

v.

If

m > 1

(m~w) = 1

then

m

is

w.

v,k, h-difference set is self-conjugate modulo m < (v/w)o

If

(m,w) > i

then

m _< 2r - 1

where

r

e(4) Let

divides

(2.38)

is the number of distinct prime factors of

Proof. that

(V/W)

Since =- 0

is self-conjugate modulo

modulo

w = VlV 2 m.

m

m

for

where

Further,

1 < j <w

(v2,m) = 1

w

(m,w)o

it follows

(as noted above)

- 1.

and where every prime divisor of

vI

let

w-1

e(x) ~

~

ai x i

mod ( ~

- 1)

i=O

where w = v 2.

0 _< a i _< v/w Here

e(~)

First consider the case

necessarily. ~ 0

modulo

m

for

i < j < w - i

(m,w) = i,

i.e.,

and by Lenzna 2.12 it

also

4O

a.

follows that all the coefficients

are congruent modulo

m.

Let

~

be the

1

smallest

such coefficient°

Then

w-1

2

w-1 a.x~

= ~

i=O

where

0 ~ a i - a ~ v/w

zero, then divides

e(~)

ai - ~

Z

n

i

(a i - ~) x i

i=O

and not all the

for all

2

i=O

and hence

Assume now that

w-1 +

ai - ~

are zero.

[For, if they were all

would also be zero~ a contradiction].

it follows that

(m,w) > i

m ~ a i - ~ ~ v/w

has exactly

r

Since

m

as was to be proved.

distinct prime divisors.

Consider

the polynomial

w-i

~(x) =

where,

as usual,

~v I

v2"l i

Xi =

~ a i~vl i:o

is a primitive

v~±

st

~

~ j--o

b xj

j

root of unity and

J+v2 + o bj = aj ~ i

Since

m

+ aj+v2 ~Vl

is self-conjugate

modulo

~J+(Vl-l)v2 "" + aj+(Vl-l)v2

w

and

vI > 1

j; a

0 < ~ < v 2 - 1. further b j,

b. ~ 0

(mod m)

By Le~na 2o12 this implies that for some

j

since

n ~ 0o

yields

m < 2 r-I (v/w)

"

it follows that

v2 ) ~ 0

for

Vl

bj = 0

modulo

m

for all

Applying Corollary 2.11 to such

41

as was to be proved. This theorem shows, for example, that if (n,v) = lo and

[For if

n = k - k.

p

divides

(n,v)

then

v p

is a prime power then

2

divides

Thus the theorem may be applied with

v,k,Z = 56, ll, 2.

since

k2

m = p, w = v.]

particular no cyclic difference set may have parameters Consider the parameter set

n,

Since

=

Av + n

In

v,k,h = 16, 6, 2. n = 9

is a square, all

previous theorems allow the possibility that a difference set might exist with these parameters. If

But it is ruled out by Theorem 2.13, since

(a,c) = l, a b ~ 1

x (1 ~ x < b)

then

b

Theorem 2.1~. let

ql

modulo

c

and

ax ~ 1

modulo

c

is said to be the order of a modulo

(Yamamoto,

strictly divide

v.

1963)

Let

q = 4t + 3

33 m -1

modulo 28.

for any c,

written

ord c a = b .

be a prime divisor of

Assume that any prime divisor

p

of

n

v

and

satisfies

one of the conditions: (1)

order of

p

modulo

q

(ii)

order of

p

modulo

q~

(iii)

is even is

q~-l(q . 1)/2

p : q.

Then, if there exists a non-trivial

v,k,k - difference set, the diophantine

equation

2 + qy ,

0 -< x ,

has a solution in integers

x, y.

Sn = x

Proof. group of

Let

K(~w)

2

w = q~, e ( ~ )

= ~

0 _< y

< v q -~ ,

_

and let

over the rational field;

is a primitive root modulo

w~

If a prime

~

+

~

takes

satisfies

from Theorem 2.19 that all its prime ideal factors in automorphism even;

s2.

If

p

y

f

(Theorem 2.19) that the p-component of

such that V

in

~

into

where

s

p

are fixed by the modulo

pf = -1 (mod w).

K(~)

s ~w

(ii) or (iii), it follows

K(~w)

satisfies (i) then the order of

thus there exists an integer

_< 2vq -~

be a generator of the Galois

then p

x

w

is also

This implies

is fixed under complex

42

conjugation and thus (from integer

2 G

a.

Thus in

~

K(~)

= n)

all the prime ideal factors of

Hence the principal ideals

where

~

is a unit of

it must be the principal ideal

(~)

K(~w).

(To"2 )

and

T

(pa)

for some

are fixed under

are the same, i.e.,

~ = ~T

0`2

It follows from (2.20) and Theorem 2.20 that

thus must be a root of unity in

K(~),

i.e.,

q = £ 4

for some integer

j

where

E=+I. Let

N : ~(w) = q~-l(q _ 1),

EN/2 = g

then

N/2

is odd since

q ~ 3 (4).

So

and

1 = ~l+~+'"+J-2 = N/2 -w <j(l+~+'''+J-2)

shows that

g

is a w

th

root

is replaced by its shift

of unity;

D + u

then

thus h = T

g = + i.

i-~

(2.39)

If the difference set

is replaced by

Thus there will be no loss of generality in assuming that

~ = + i

provided that

it can be shown that

(i - s2)u e - j

has a solution i - s

2

u.

If

is prime to

q.

q ~ 3

modulo q~

then (2.40) clearly does have a solution since then

When

q = 3,

3~

the 3-component of

3-component of

I - sN

is

equation (2.39)

[i.e.,

j(l - sN)/(l - s 2) ~ 0 modulo 3 ~]

j,

since

s

G 2. n = ~

So

~

~ = + I

i - s

is a primitive root of

which of course guarantees a solution

without loss of generality,

u

is the norm of an integer of w = (-I + ~ - ~ )

the complementary difference set

D*

K(~-q).

has

is 3 ~.

3

and the

Thus 3

for (2.40) in this case alsoo

and thus

/ 2 ;

2

provides that

~

thus

divides So,

is fixed by the automorphism

is an algebraic integer of the quadratic subfield

rational integers and

(2.40)

So

T = a + b~

K(-~q) where

4n = (2a - b) 2 + b2q.

T* = - a - b~,

that

and a, b

are

[Note that

~ = a - b - b~

43

and that

~* = b - a + b~.]

Thus by using

~

or by replacing the difference

by its complement or by doing both it can be assumed that

a > 0

and

set

b > 0.

From (2.30) it follows that

l W

i ~(i) ~q = ±

=

i=l

Here

~(i) = i

or

q, {q = e 27~i/q

if

- Zi= ,~-i 0 B(i)

j xi

according ±

~

i

is a quadratic residue or nonresidue

depends upon the value of residue of

x w - i,

w-i f(x) =

as

is a quadratic modulo

yields

x = ~;

~(q~-l)

q

j

and

~

in the relation

otherwise)°

Let

. Z-I

B(i)x i - \ a + b

~(j)x 3q

:

j=l

for all

io

C(i)x i i:0

so apply Theorem 2.10 with

C(i) = 0

of

then the polynomial

i:O

is zero for

iq~_ I ,(i) ~w

i=l

and the sign

~-i ~qw = ~qJ (+ e(~)

0

~i

s = d = i

In particular,

let

and

~ = 0.

This

i = 0, q ~ - l , . . . , q ~ _ q ~ - i

then

B(O) - a = B ( q ~ - l r ) T b ~ ( r )

and since

4(1) = i, ~(q - i) = 0

B(-q~-l).

As

a ~ 0, b ~ 0

Applying this same process to

and

it follows that 0 ~ B(i) ~ vq -~

~

Combining these results establishes Yamamoto

= b - a + be

B(0) - a = B(q ~-I) 7 b = this implies that

yields

the theorem with

shows further that when

v =w = q

r = l,...,q - i

O ~ a , b ~ v q -I.

la - b I ~ vq "~.

x = 12a - b I

and

the only non-trivial

y = b. difference

sets which satisfy the conditions of Theorem 2.14 are equivalent to the set consisting of the quadratic residues of in partieular~

q

or to its complementing

Theorem 2.14 shows that no difference

set

D*.

D Thus~

set exists with parameters

44

v,k,k = 239, 35, 5

(a parameter set not ruled out by the previous results of this

section).

Theorem 2.15. prime divisors of where

~

(Yamamoto, 1963) v

and let

q~

Let

rm

denotes Euler's function.

q = 4t + 3

and

r

v

with

strictly divide

be distinct odd ($(q~)~ ~(rm)) = 2,

Assume that any prime divisor

p

of

n

modulo

r

satisfies one of the conditions:

(i)

Ordqp ~ 0 (mod 2)

(ii)

order of

p

and

modulo

q~

OrdrP ~ 0 (mod 2) ~ 0(mod 4), is

½ ~(q~)

and order of

p

m

is

~(rm), (iii)

p = q

and order of

p

Then, if there exists a non-trivial

4n = x 2 + qy 2 ,

0 ~ x,

has a solution in integers

modulo

rm

is

$(rm).

v,k,h - difference set, the equation

0 ~ y ~ 2vq-~r-m,

x + y

4vq-~r-m

x, y.

Note that, by Theorem 2.7, these hypothesis are only satisfied if square.

is a

Thus as a non-existence test, applied after Theorem 2.7, it can be

restricted to those cases. for parameters

21, 5, 1.

The integers With

v,k,h = 306, 61, 12

P = 7, q~ = 9, rm = 17

3, 6, 7, 12, 14

q = 7, r = 3

(ii) and the equation has solution meters

n

the prime

x = 4, y = 0.

form a difference set 2

satisfies condition

On the other hand, the para-

are not associated with any difference set since

satisfy (ii) of Theorem 2.15 and the diophantine equation

has no solution.

Proof.

Let

group of

K(~)

p

~

where

[respectively K(~R)]

Q = q~, R = rm, w = q~rTM

and

G ( ~ ) = Vo

Note that the Galois

over the rational field is generated by two automorphisms

Krespectively K(~Q)]

p]

fixes every element of the field

and generates the Galois group of

over the rationals.

then there exists an integer

If f

p

K(~Q)

is a prime divisor of

such that

pf ~ -i

modulo

n

~

and

K(~R) [respectively

which satisfies (i), w;

thus (as in the

45

previous proof) the p-component of the ideal for some integer (~)

is

(pb)

Finally, if of

p

a°

Similarly, if

for some integer p

b

is the principal ideal

(pa)

satisfies (iii) then the p-component of

(a consequence of Theorems 2.7 and 2.19).

satisfies (ii) it follows from ((P(q~), $(rm)) = 2

modulo

divisors of

p

(~)

that the order

w

is

½ q0(w); thus there are two (Theorem 2.19) prime ideal

(p)

in

K(~w)

quadratic subfield

and since

OrdQ p = ½ (P(q~) they originate in the

K( ~-q ).

Hence the ideal

(~)

is fixed by

p,

fixed by

o2

and originates in

i_~ 2

K(~-q)

.

Thus V

= ~, 1 - p

implies (by 2.20) that unity in

K(~).

of unity.

Now

o2,

if

= 55 = 1

l-p

is an

,i

and

q = 3

5 = e ~R

then

and

5

K(~-q)

are units in

1-p

K(~);

9, 5

of unity and

hence

is a

q~th root

= 81-o~ = + 1.

Thus

~

of degree

2~(r m)

i, j

with

g,g' = + i a.

q = 3(h) has no units except

D + ~r TM

but allows

5

~ = + 1.

+i

unless

[Shifting

5. ] Shifting again by a multiple of

to take the shape

g'

g i ~3a

or

for

unless

Thus

q = 3;

[This last follows

D

q~

q = 3. ]

E = +l

for a judicious choice of

assume, without loss of generality, that has no effect on

is

over the rationals.

By the same proof as above, see equation (2.39), it follows that

m r

are roots of

51"~2

may occur for some integer

that by considering the shift

this

is a root of unity in the subfield fixed by

for some integers

8 = g §3~R

from the fact that

rmth root

K(~R, ~-q )

t ~a~i

~, B

and that (Theorem 2.20)

T(1-c2)(l'P);

K(~Q)

i.e., in the field

= g~

~

But both are

a root of unity of

= 5 where

~

and

one may

by a multiple of does not disturb

9

q = 3.

l.C 2

First consider the case an integer of fixed by belongs to

p,

K(~R, ~-q ) i.e.,

T2

K( ~r*, ~Q)

q ~ 3.

Since

and since

1-p

is an integer of where

K(J~-q) S,

T

= g, = + l K( ~

).

over

a

T in

K(~)

it follows that

it follows that Also

T l-p2 = l,

as before.

~ = K(~-q, ~r*).

then

In this case, for some ifiteger polynomial satisfied by

= ~ = 1

r* = (-l)(r-1)/2r

integer of the biquadratic field does not belong to

T

If

g' = -1

necessarily generates K( ~

), x

2

That is,

- a = 0

[i.e. ~

over

T

2 so ~

is

is T is an

if K( ~

).

is the irreducible

and thus [Mann (1955) Chapter 12] the

46

relative different of (r,n) = 1

T

~

is prime to

algebraic integer of

2 T '

K(~-q)

is

27.

q = 3o

r

is odd,

@

: n

belongs to

T

K({R; ~

K(~)

r;

a contradiction.

satisfies ).

Since

= K(~3)

[here

T

= I

extension

~q of

a subfield of

K( ~-3 ).

K(~R, ~ - ~ )

Now

a ---0 ( 3 ) ]

r

~' = 1

and

7

thus its relative different is

T~

x3 - b = 0

is cubic over

Thus

T2

determines a cubic

K( ~

).

~

a - 0 (3)

and

r

since

On the other hand

72

b

of

K( ~

);

(r,n) = I

and

belongs to

q ~ 3

Thus in every case

T = e(~w)

theorem~ this implies that [with according as modulo

e

K( ~

and, since

3

plays ~' = i

).

is an integer of T = a + b~

).

K( ~'q ).

w = Zq-ll ~(e)~q,

is a quadratic residue or nonresidue of

As in the previous ~(e) = i

or

0

q~ 8(x) - Z iw-i = 0 B(i) x i

xw - 1 ]

f(x)

=

w-l~ , ~ /, Bki)x i -

{' \ a + b

i=O

has a zero at that

K( ~

[see

is

since

72

K({3)

no essential role in that argument, it may be invoked here to show that is an algebraic integer of

2

9

for some integer

which is prime to

Thus

and thus

72

So this case has been reduced to the conditions for

7

is an

it follows that either

divides the diseriminant of

374,

Contradiction.

it follows that

or

properly containing

satisfies an irreducible equation

r ~ 3 = q.

Thus

T 2(l-P) = {3 2a

van der Waerden (1949) p. 171 for this fact].

and that

and

K( ~q-q).

Since

belongs to

since

Since

is implied by conditions (i)~ (ii)~ (iii)~ it follows that the

discriminant of

Let

over

x = ~.

~±

~r(e)

m ~-i x er q

e=:l

for all

io

=

w-1

Ci[ i ")"x

i=l

This implies (Theorem 2.10 with

A(wq -I) ~(wr -I) C(i) -- 0

)

s = 2, d = i

and

~ = 0)

That is,

c(i) - c(i + q~-lm) ~ c(i + q~r m-l) - C(i + q~m-Z

+ q~-lT)

(2.41)

47

for all

i.

Combining equations

B(0) - a - B(q~-!rmj)

for

Now

j = l,...,q - io

i = 0, q~'irm'...,(q- 2)q~-irm

yields

% b~(j) = B(q~r m-l) - B (q~r m-I + jq~-irm)

When

j = I

and

j = q - 1

equation

(2.42)

(2.42) becomes

B(0) - a - B(q~-ir m) W b = B(qlr m-l) - B(q~r m-I + q~-irm)

(2.43)

B(0) - a - B(-q~-l~ TM)

(2.44)

0 < B(i) _< v/w

Ib 1 _< 2v/w

(2.41) for

for all

i.

= B(q~ m-l) - B(q~ m-I - q~-irm).

Thus equation

follows from the same reasoning

from subtracting

(2.44) from (2.43).

(2.4~;) shows that

Consideration

the equation

and

applied to the equation which results

resulting equation analogous to (2.44) produces x = 12a - bl, y = Ibl

lal _< 2v/w

of

~* = b - a + b~

Ib - a I < 2v/w.

4n = x 2 + qy2

and the

Thus with

has a solution of the proper

type and the theorem has been proved. Yamamoto notes that when

Im v = w = q r

the only difference

all the conditions

of this theorem have parameters

are all equivalent

to the set

Theorem 2.16. divisors of

v,

(Yamamoto,

let

q

1963)

Let

q = 4t + 3

be a quadratic non-residue

where

divisor

satisfies one of the conditions:

(i) (ii)

of

n

v,k, h = 21, 5, i;

thus they

[3,6,7,12,14}.

(q0(q2),~0(rm)) = 2 p

sets satisfying

q~ , rm

strictly divide

and

of v.

r

r = 4s + I

be prime

and let

Assume that any prime

ordqp ---0 (mod 2) and ordr(P) ~ O(mod 2) ~ O(mod 4) order of

p

modulo

q~

is

@(q~)

and order of

p

modulo

rTM

~(rm). Then, if there exists a non-trivial

4n = x

2

+ qry 2,

0 < x,

v,k,~ - difference

0 < y < 2vq-~r -m,

se%

the equation

x + y < 4vq-~r -m

is

48

has a solution in integers

x, y.

Again, as a non-existence test, applied afte9 Theorem 2.7, this can be restricted to those cases where

n

is a square.

The integers

0,1,2,4,5,8,10

form a difference set with parameters

v,k,k = 15, 7, 3.

satisfies (ii) for

and the equation is satisfied by

q = 3

and

r = 5

v,k,h = 286, 96 , 32

On the other hand, the parameter set tes%

since

q = ll, r = 13, p = 2

Here the prime

2 x = y = io

is ruled out by this

satisfies (ii) and no solution to the

diophantine equation exists.

Proof. e(~)

= 7

field,

and let

~

%

p

generate the Galois group of

generating the Galois group of

generates the Galois group of of

n

Q = q~, R = rm, w = q~rm,

As in the proof of Theorem 2.15 above, let

K(~R)

K(~Q)

and fixes

K(~)

over the rational

and fixing K(~Q).

K(~R)

If

p

while

p

is a prime divisor

which satisfies (i) then it follows, exactly as in Theorem 2.15, that the p-

component of the ideal While if

p

(7)

is the principle ideal

satisfies (ii) it follows from

(pa)

for some integer

(~(q~),C0(rm)) = 2

that

ord

a. p = ½q0(w);

w

thus there are two (Theorem 2.19) prime ideal divisors of since

OrdQ p = ~(q~), o r ~

TM)

p = £0(r

(p)

they do not arise in

in

K(~w)

and

K(~Q)

or in

K(~R).

Since [see Mann (1955), Chapter 13] they must arise in a quadratic subfield of K(~w)

it can only be Hence the ideal

1-~p

= N

K( ~ (T)

is a unit of

= + 1.

D

Since

the field fixed by

is fixed by K(~w)

unity in this field, i.e., difference set

),

~ = + ~J

for some integer

by a suitable shift (e(q~),~(rm)) = 2,

2

and

O2

K( ~

).

Thus

D + ~q~ + ~'rm

both ~0

j.

~2

and so

and

p

By replacing the

it may be assumed that

2

are powers of

~ 1-°2 = ~yl-p2 = + 1.

~p. Thus

In ~

is

and as such lies in the biquadratic field

= K( ~r, ~-q). Furthermore, ~p Suppose T = -T, then since

2 ~/ ~

is fixed by

~p

is an integer of

1,(1 + $r)/2, (1 + ~q-q)/2, (1 + ~r + ~ it follows that

and originates in

and by (2.20) and Theorem 2.20 it must be a root of

fact they are both even powers of fixed by both

~p

~p.

~ = (c ~-q + d ~fr )/2

+ -~qr)/4

and thus belongs to ~

K(-~qr).

and since

form an integral basis for

for some rational integers

c, d

0,

such that

49

c =- d

modulo

2.

Thus

4n = 4 ~

=

c2q + d2r,

and since

equation

solution in integers

not all zero.

x,y,z = 2 ~ n ,

c, d

II.B note 2) shows that this only happens when which contradicts

the hypothesis.

Thus

x

2

square, this implies that the diophantine

q

n

- qy

2

is necessarily - rz

2

= 0

But Legendre's

a

has a test (see

is a quadratic residue of

7 ~p = 7

i.e.,

7

is an algebraic

r, integer

K( ~ ) .

of

So

7 = a + b~

is the Gaussian

+ 1

: 0

or if

i

for some rational integers s qr-1 . i -i=l ~(1)~qr"

sum

/2

where

if

~

Here

if

i

is a quadratic non-residue of

m

w-i ~

f(x) =

f(x)

= 0;

(2.30)].

Let

~ 2B(i)x i - \ 2 a

2B(O) - 2a - 2 B ( j q # - i r

for

j = l,j'

q

TM) +

j : i,

(j'

for

i

yields

~*

+ qry 2

la-bl

i.

modulo

~-i m-i r

and

r

[This of

xw - 1

=

~r, and let

C(i)x i

s = 2, d = i,

- 2)q~-lr TM

~ b - 2 B ( q # r m-1 + j q # - i r m )

]a I _< 2v/w. of

q)

Further,

_( 2v/w.

q

otherwise.

That is, equation

i =0, q~-lrm,...,(q

alsoo

that is

i=0

for all

a quadratic non-residue

associated with 2

1951);

representations

Apply Theorem 2.10 with

it follows that

Ibl _< 2v/w

4n = x

r, = -i

~(j)xjq

b = 2B(q#r m - l )

Theorem 2.15, that

equation

and

8(x) ~ Zi= lw-1 B(i) x i

&(wq -1) •(wr -1) C(i) = 0

for

q

+ 2b

x = ~.

and combining these equations

From which,

~(i) --

j=l

has a zero at

thus

and

of the analoguous

i=0

then

2wi/qr

~ = (-1 + -~qr)/2

is a quadratic residue of both

of both

is a consequence

given by equation

~qr = e

where

is the Jacobi symbol (see Nagell,

(i,qr) > 0, = I

representation

a, b

yields

_+ 2b @(q + j r )

.

Combining these equations

provides,

as in the proof of

the analogous equation

Thus with

(2.41) holds

x = 12a-bl,

(for

y = Ibl

j = i) the

has a solution of the proper type and the theorem has been

5O

proved. Yamamoto notes that in this theorem when q, r

are twin primes (i.e., either

v = w

q = r + 2

or

then in fact r = q + 2)

v = qr

where

and the only non-

trivial cyclic difference sets which occur are the so-called twin prime difference sets;

these have parameters

v,k,h

q r , ( v - 1)/2,

=

(v-3)/4.

These difference sets

are discussed further in section V.D. below. Let

p

n = k - h divides so

p

be a prime divisor of both it follows that

k

r

the relation

divides

n

p

v

divides

and

k

and

k ( k - i) = h ( v - i )

also.

Suppose

p

r+s

n, h

then since also.

pr

implies

pS

strictly divides

v

(ii)

r < s

implies

pr

strictly divides

v.

the p-component of

v,

pr

strict ~

is the p-component of

r > s

p<~

In fact if

shows that

(i)

Thus in these cases, with

k 2 = hv + n, strictly

divides

n~ s > O.

it follows that

h;

Then


Finally (iii)

r = s

implies

Now, in this case, let where that

kI

and

2 r+3

nI

pr

p = 2.

are odd.

divides

v

divides

v.

Then

is a square;

Further

since

n k2

-

2 2 k I - n I -= 0

~2r n 2 k = 2rkl ' n = ~ I

thus

2 = by, n = 2 2r-(k2I - nl) modulo 8.

which shows

The next result shows that,

in fact, no cyclic difference set has such parameters.

This quite specialized

result has application to some of the difference sets of section IV.C. below~ as well as elsewhere.

Theorem 2.17. v, k, h

such that

Proof. divides v, From

let

(Turyn, 1965)

k w =

k -= 0

22t (t _> I)

No cyclic difference set may have parameters strictly divides

n

if

2 t+l

By (i), (ii) only case (iii) need be considered. and

h

while

2t+s-j modulo

for all integers

r.

2 t+~

divides

(0 < j < t + s) 2t

v.

and let

Let

Here

2 t+s (s > 2)

A(w/2)CJ(i) =- 0

e(~)

2t

v.

strictly

strictly divide

e(x) =- Zi=oW-i cJ(i)x~

and Theorem 2.19 it follows that

So, by Theorem 2.10,

divides

modulo =- 0

modulo

x

w

modulo 2t;

i.e.

- i. 2t

51

cJ(i + w/2) ~ cJ(i)

and thus

cJ+l(i) = cJ(i + w/2) + cJ(i) m 2cJ(i)

modulo

2t

modulo

2t

modulo

2t

In particular

ct(i) m 2ct-l(i) ~ ..o ~ 2tcO(i) m 0

for all

i. Z(i) = ct(i)/2 t,

Let

then for

w-i

w = 2s

w-i

(2.4~) i=O

for all

m ~ 0

modulo

of this equation becomes Surmuing (2.45) for

i=O

w,

~2t 2 n : m n I.

where kI 2

When

where the odd integer

m = O,l,...,w- i

w kI

divides

the right side

m

satisfies

k : 2tk I.

yields

w-i

w-i

2 (2 s - 1)n~ = 2 s kI +

z(i)z(j) i~o

i# j

Since this last sum is zero for all

i { j

m= 0

it follows that

w-i Z2(i) = n 2 +

2 2 kl - n I 2s

i=O

Since

2s

strictly divides

is thus even. contradiction.

2 - n 2I kI

On the other hand

[see case (iii) above] the sum

Z~i=O -I Z(i) = k I

9-10 Zi=

z2(i)

is an odd sum of integers,

Thus no such difference sets exist, as was to be proved.

52

All parameter sets

v,k,h = 4N 2, 2N 2 - N, ~

- N

to process cyclic difference sets by this theorem.

for

N = 2m

are shown not

Among others, the case

N = 28

is not excluded by previous results. The following facts of algebraic number theory are appended for easy reference. Proofs may be found, for example, in Mann's book (1955).

Theorem 2.18. degree

~(d).

The numbers

~(r)

K(~d)

The conjugates of i~ ~d'

rationals. any

The field

If

~(d)-! °'''~d

(r,s) = 1

is a normal extension of the rationals of

~d

are the

~(d)

form an integral basis for

the field

consecutive powers of

K(~rs )

~r

i ~d

numbers

where

K(~d)

is of degree

~(r)

(i,d) = 1.

over the over

form an integral basis for

K(~s)

K(~rs )

and

over

K(~s)O Theorem 2.19 .

is of the principal ideal

(p)]

f

mapping

K(~d)

d = pi

(2.46)

(P) = P! "'" Pg

when

(d,p) = 1

(2.47)

p

Pi

are conjugates.

modulo

g

is

d.

q0(w)/f and

f

when

is the order of

irreducible modulo

~d

g

is

~(d)/f

Pi"

d = paw, (p,w) = i

p

of (2.47), (2.48) can be determined explicitly. irreducible equation satisfied by

Further,

The field automorphism determined by the

fixes each of these prime ideals

(P) = (P1 "'" pg)~(pa)

where

is given by

when

is the order of ~d ~ ~

in

[that

(p) = (i - ~d )~(d)

where the distinct prime ideals where

p

The prime ideal decomposition of the rational prime

modulo For, if

w.

The ideals

f(x)

p,

~a modulo

PI' "°"Pg

denotes the

over the rationals then, with the

f(x) ~ (fl(X) ... fg(x)) (p)

(2.48)

p

f. i

53

and

Pi = (P'fi(~d))"

Theorem 2.20.

Any algebraic integer which lies, together with all its

conjugates, on the unit circle is a root of unity.

III.

MULTIPLIERS AND CONSTRUCTIVE EXISTENCE TESTS

The mere fact that parameters

v, k, h

have passed the existence tests of

Chapter II does not insure the existence of an associated difference set.

Of

course the parameters may well belong to a known family of difference sets (see Chapter V for a discussion of these) in which case there is no problem.

Often

enough~ however, the parameters neither belong to ~ known family nor are they excluded by the results of Chapter II.

When this happens an attempt is usually

made at constructing a difference set with the given parameters.

The straight-

forward process of examining all sets of

0, l~...~v - 1

soon becomes prohibitively large.

k

elements taken from

Thus~ more efficient methods have been devised;

for the most part these methods depend upon the existence of a multiplier of the hypothetical difference set in question.

This chapter is concerned first with

general theorems about multipliers (multiplier results specific to certain types of difference sets are found in Chapters IV and V) and second with constructive existence tests~ whether based on multipliers or not.

A.

Multiplier Theorems The question of whether every cyclic difference set necessarily possesses a

non-trivial multiplier meters

v, k~ h,

multiplier.

t

(i.e.,

t ~ i

modulo v)

is open.

Thus, given para-

one cannot in general assume the existence of a non-trivial

However, under certain circumstances, multipliers can be shown to

exist - the main result of this type is due to Hall (1956):

Theorem 3.1. of

n~

where

Let

D

(nO,v) = i

there is an integer

jp

be a and

v~ k, h - difference set and let n O > h.

If, for every prime

such that

Op

~ t

(modulo v)

p

nO

be a divisor

dividing

nO ,

55

then

t

is a multiplier of

Proof.

D.

From

e(x)9(x -1)

m n + ~(1 + x + ...

+ x v-l)

(mod x v - 1 )

comes the factorization

9(x)8('x-I) e n : non I

when + x

fi(x)

V-I

which

is least positive;

@(~)e(~ -I) = n0n 1. ~,...,~u-1

(3.l)

fi(x)

is any one of the distinct irreducible factors of

over the rational field j

modulo

K.

Let

{ =

form an integral basis [here K({)

be that root of

fi(x)

for

then congruence 3.1 yields the factorization

In the cyclotomic field

algebraic integers of

e2Wlj/v

T(x) = i + x + ...

u

K(~)~

the algebraic integers

is the degree of

fi(x)]

l,

and the

can be associated with the residue classes of poly-

nomials with rational integral coefficients modulo

fi(x)

[see~ for example, Mann

(1955) Theorem 8.6 for a proof of these facts]° Now if (p,v) = l, leaves

P

P

is any prime ideal divisor of the rational prime

then the automorphism fixed

(i.e.,

P~ = P).

determines an automorphism of n0o

Hence

nO

divides

K(~)

e(~)e(~ -t)

~

of the field

Si(x )

desired.

fi(x)

of

Thus

~

~P, ~ ~ St

which fixes all the prime ideal divisors of as well as

9(~)8(~-1),

modulo

has rational integral coefficients.

each irreducible factor

and if

determined by

[See Mann (1955) Theorem 8.1.]

e(x)9(x -t) ~ n0Si(x )

where

K(~)

p

T(x).

so

(3.2)

fi(x)

There is such a congruence for

A similar congruence modulo

T(x)

is

Suppose

@(x)9(x -t) = n0Rj(x ) + A(x)Fj(x)

(3.3)

56

where

Fj(x) = fl(x)f2(x) ... fj(x)

and

with rational integral coefficients. 3.2°

have no common factors.

as well as

Rj(x)

This is immediate for

Assume equation (3°3) holds for

Fj(x)

A(x)

j

and consider

Thus their resultant

j = 1

j + i. z

are polynomials from congruence

Now

fj+l(X)

and

[van der Waerden (1949)

p. 83-87 establishes all the resultant theory needed here] is a non-zero rational integer, and there exist integral polynomials

C(x)

and

D(x)

such that

(3.~)

c(x)Fj(x) + D(x)fj+l(X) : z.

But Fj(x)

z

can be expressed as a product of factors and

~

is a root of

unity, and if

~

fj+l(X).

is a primitive

appropriate exponents

y, Z.

Thus

~

~ - ~ and

8

[(x+l)

~Y

v-

is unit of

1]

=

is a root of

~ - ~ = ~Y(~

K(~)

xV-1 + ' ' "

~

are different

v th roots of unity,

Now

where

and

~

v th roots of - l)

- 1

for

is a root of

+V

X

thus since with

- 1

divides

(n0,v) = 1 i = j + 1

v.

Hence

z

will divide an appropriate power of

has been assumed, then also

(nO, z ) = 1.

v,

and,

From congruence 3.2

follows

C(x)Fj(x)8(x)e(x -t) = n0C(x)Fj(x)Sj+l(X) + C(x)B(x)Fj+l(X).

Multiplying congruence 3.3 by

D(x)fj+l(X )

(3.5)

and adding the result to equation (3.5)

yields [by (3.4)1 ze(x)@(X-t) : n0S(x ) + G(x)Fj+l(X).

This can be combined with the trivial relation

n00(x)e(x -t) = n0H(x )

57

to yield (since

nO

and

z

are relatively prime)

6(x)e(x -t) = n0Rj+l(X) + A(x)Fj+l(X) •

Continuing in this manner provides the desired equation

@(X)e(X - t ) = n0R(x ) + A(x)T(x).

Since

A(x)T(x) ~ A(1)T(x) modulo x v - I,

equation (3.6) yields

e ( x ) e ( x " t ) m n0R(x ) + A(1)T(x)

Let

x = i

in this congruence, then

any difference set and altering

R(x)

(nO,v) = 1

(3.6)

(mod x v - l ) o

k 2 = n0R(1 ) + vA(1). this implies

A(1) m ~

Since

k 2 = n + Zv

modulo

nO .

in

Thus, by

if necessary 3

e ( x ) e ( x - t ) ~ noR(X ) + hT(x)

(mod x v - 1) o

(3.7)

Now every coefficient on the left side of this congruence is non-negative and since no > h

all the coefficients of

this congruence provides Since

R(x)

are non-negative alsoo

k 2 = noR(1 ) + hv;

thus

x = 1

R(1) = n 1.

(t,v) = l,

e(x)6(x -1) m e(xt)e(x -t) ~ n0n I + ZT(x)

thus

Further with

e(xt)e(x-t)e(x)e(x "l) e [n0n I + ZT(x)] 2,

As

(mod x v -

R(x)T(x) m R(1)T(x) m niT(x )

comparison of these two results yields

x v - l)

while (3.7) gives

[n0R(x ) + %T(x)] [n0R(x-1 ) + %T(x)]

for this same product.

(mod

modulo

l)

x

v

- ±,

a

58

R(x)R(x -I) ~ n~

This implies

[since

R(x)

(mod

x v - 1).

has non-negative coefficients] that

single non-zero term, i.e.,

R(x) m nlx's (mod

x v - i)

R(x)

has only a

for some integer

s.

Thus

congruence 3.7 implies

e(x-l)e(x t) ~ n x s + ~T(x)

Multiplying this last congruence by

e(x)

(mod

and simplifying yields

o(x t) ~ xSe<x)

i.e.,

t

x v - 1).

(mod x v - l)

is a multiplier of this difference set as was to be proved.

This theorem was first established for cyclic projective planes

(i.e.,

h = 1

only) by Hall (1947) and later extended (Hall and Ryser, 1951) to the case of general

h

in the form stated in I.E. above.

generalization. known case.

The conditions

nO > h

and

Theorem 3.1 represents a further (n0~v) = 1

are superfluous in every

For~ no cyclic difference sets are known with

prime divisor

p

of

n

(n,v) > 1

and every

is a multiplier of every known cyclic difference set.

Morris Newman (1963) extended this result slightly by showing that the odd prime

p

is always a multiplier whenever

n = 2p

and

(7p,v) = 1.

Turyn (1964)

generalized Newman's result.

Theorem 3.1A

(Turyu)

that for every prime

p

Let

dividing

n = 2n 0 nO

t

nO

odd and prime to

there is an integer

pJP -= t

Then

with

modulo

jp

v.

Suppose

such that

v .

is a multiplier of every difference set with these parameters, provided

merely that

t

is a quadratic residue of

7

whenever

7

is a divisor of

v.

59

(7,v) = i

Note that in particular this eliminates the condition result.

For if

residue of

7

n = 2p a (a

odd), the assumption that

implies that

contradiction that

a

t3 ~ p3jp ~ -i

(mod 7).

t ~ pop

from Newman's

is not a quadratic

So Theorem 2.7 provides the

is necessarily even.

Mann and Zaremba (1969) investigate the situation when is not a quadratic residue of

7.

7

divides

v

and

However they do not resolve it completely.

particular, they find no case where

t

t In

is not a multiplier.

On occasion it is possible to establish the existence of a w-multiplier for some divisor multiplier.

w

of

v

Frequently this is of importance in constructive existence tests.

[Section III.C.

contains an example of such a case.]

under the conditions that

t

even when it is not possible to show the existence of a

(n0,w) = l, n O > hv/w, pJP ~ t

is a w-multiplier.

However the condition

thus this is not of much use.

Theorem 3.2° n

Let

w

there is an integer

jp

By retracing the proof above (mod w)

n O > hv/w

it can be established is rarely satisfied;

The following related result is more useful:

divide

v

and suppose that for every prime

p

dividing

such that

Jp p

with

(t,w) = i.

parameters

Then

t

~ t

(modulo

w)

is a w-multiplier of every difference set with these

n, v.

Of course the proof of this theorem is almost identical to that of Theorem 3.1 above.

The congruence analogous to (3.7) above, being

9(x)~(x -t) =- nR(x) +

hv

(i + x + ..- + x w-l)

(rood

x w - i)

(3.8)

W

with above.

R(1) = 1.

R(x)R(x -I) ~ 1 (rood xw - i)

But this implies that

must be that 3.1.

From which

R(x) = x -s

R(x) = + x "s

for some

s

is deduced in the manner and since

R(1) = l,

it

From which point on the proof concludes as for Theorem

6O

A surprisingly important fact about cyclic difference sets is that:

The?rem 3.3.

Minus one is never a multiplier of a non-trivial cyclic

difference set. This fact was known for several years prior to any publication of its proof. This accounts for the anomaly that it is often referred to in publications which predate the papers

[Johnsen (1964), Brualdi (1965) and Yates (1967)] containing

proofs.

Proof.

The result (proved in section III.B. below) that any multiplier o f a

difference set Assume that

D

-1

belongs to

D

x - y ~ d

is a multiplier and that only if

and

tions of

d

fixes at least one shift

-x

(= v - x)

(-y) - (-x) ~ d

unless

x = -y.

for

Hence,

D + s D

is fixed by

does also. x, y if

in

D;

d ~ 2x

0 ~ x ~ v/2. of Thus

D

for some

So

d

x

in

for every

x

in

D

y ~ x (y

(0 ~ x ~ v/2)

h

in

D)

v/2

v.

is represented at least

in

D

(such

d

is necessarily even. occurs provided

Whence

y ~ x - v/2 k - i

y - (-y) ~ d

also.

x ~ y + v/2.

Thus,

also belongs to

times.

Hence

D.

h ~ k - i

So, only trivial cyclic difference sets have

as was to be shown.

has been shown (Mann~ 1952) that

B.

x

occurs an even n~nber

such that

It is, of course, not necessary that a multiplier

set only when

d

x - (-x) ~ d

the element

and the difference set is trivial. -1

for some

then

Thus

which can only happen for even

Thus the difference

multiplier

then

x

can appear an even number of times as a difference of elements

only if there exists a 2x m 2y,

D,

D.

i.e., that

they are distinct representa-

0 # d ~2x

of times as the difference of elements of

-1,

Consider the differences

must exist if the difference set is non-trivial),

If

of the difference set is used.

n

2

t

divide

n.

However,

is a multiplier of a non-trivial difference

is even.

Difference Sets Fixed by a Multiplier The use of a multiplier

t

for constructing a difference set is greatly

it

61

facilitated by the assumption that the set is fixed by the multiplier (i.eo, that tD ~ D modulo v).

As mentioned in section I.E. every multiplier

set determines an automorphism of the associated block design. the incidence matrix of the block design, then the multiplier tation matrices

P

and

Q

(Q

takes coltunn

x

t

That is, if t

into column

of a difference A

is

determines permu-

tx

modulo

v)

such

that

PAQ = A

since (see section I.C.)

A

and thus

(3.9)

A-1pA = Q-1

Hence (by

is non-singular for non-trivial designs.

well known facts of linear algebra)

Tr(P) = T r ( A - % A )

where

Tr(X)

elements).

denotes the trace of the matrix But

Tr(P)

Theorem 3.4.

fixed by

then

t

[Similarly, if

is the number of objects fixed by the

tx ~ x

modulo

also fixes the t

D

v,

which is

d

shifts

E + j(v/d),

shifts of

is one such shift, then all others are of the form

D

t,

w

modulo

E

for of w.

E + J(w/8)

Hence

there exists

In fact if

is a w-multiplier for some divisor (t - 1,w) = 8

(t - 1,v) = d.

with multiplier

shifts fixed by the multiplier.

necessarily fixes exactly w

Tr(Q)

Given a difference set

(t - 1,v) = d t,

(i.e., the sum of its diagonal

So, by equation (3.10), the number of shifts fixed by the multiplier

is the number of solutions of

exactly

X

is the number of blocks (or shifts of the difference set)

fixed by the multiplier, whereas multiplier.

(3.1o)

= Tr(Q -i) = Tr(Q)

is a shift

j = 0,1,...,d- 1. v

then If

t

E

modulo

modulo

w,

for

j = 0,1,...,8 - i.] Suppose suppose that

tl, t 2 D

is also fixed by

Theorem 3.5.

are both multipliers of the same difference set

is fixed by t 1.

If

t !.

Then

tl(t2D ) = t2(tlD ) = t2D;

D

and

that is,

t2D

So

t l, t 2

are multipliers of the same difference set then

t2

62

permutes the shifts fixed by

t I.

Thus if any mu!tiplier

t

fixes only one shift of the difference set then

that shift is fixed by all multipliers.

Theorem 3.6.

(i)

Hence

If there exists a multiplier

t

such that

(t - l,v) = i,

then exactly one shift of the difference set is fixed by al__!multipiiers. (k,v) = i

(2)

If

then there exists at least one shift fixed by all multipliers.

Part (2) follows because difference set has

(k,v) = i

insures that exactly one shift

e I + e 2 + --- + e k m 0 (mod v).

E

of the

Such a shift (being unique) is

certainly fixed by any multiplier. Some additional fixed shift results which apply only to the case contained in section IV.A.

h = 1

are

The fact that every multiplier fixes at least one shift

of the difference set was shown by MeFarland and Mann (1965).

Part (2) of Theorem

5.6 is due to J. Jans and the remainder of this section restates for arbitrary

h

results of Hall (1947).

C.

Multipliers and Diophantine Equations Suppose it is known (by Theorem 3.1 or otherwise) that a hypothetical

(Theorem

difference set has a multiplier

t.

Then

set is fixed by the multiplier.

Thus, there is no loss of generality in assuming

that the difference set is a union of sets modulo v.

3.4) some shift of this difference

{a,ta, ...,tra-la],

where

tma -= a

So

Lemma 3.7. union of sets

If a

v, k, ~

[a, ta, .... tm'la),

difference set with multiplier where

tma ~ a (modulo v),

distinct elements and forms a difference set with parameters The sets

[a,ta,...,tm-la)

t

exists, some

has exactly

k

v, k, ~,.

are often called blocks fixed by the multiplier

t.

The number of distinct elements in each of these fixed blocks is always a divisor of the order of

t

modulo

v (this follows, for example, from Theorem 60 of Nage!l,

1951) and is in fact always equal to that order unless prime all the blocks (save

[0])

(a,v) > i.

Thus when

v

fixed by a multiplier contain the same number of

is

63 distinct elements

m

( = the order of

set can exist only if

k = jm

or

t

modulo

jm + i,

v).

In this case a difference

for some integer

j.

Lemma 3.7 is not always easy to apply (for sometimes there are many unions of these blocks having cant information

k

almost trivially.

By Theorem 3.1 above, 151 is prime and

k = 51

with

6

14]

and

{0},

modulo 151)

or

{3, 6, 12],

so a difference

modulo

[9, 15, 18],

set candidates

These are equivalent

15,

Now set

contradic-

that

2

{7, 14] are

is a

and two

{3, 6, 7, 12,

(see section I.B. for this

sets, as is easily verified.

set exists,

is a multiplier.

Theorem 3.1 establishes

only non-trivial one known in which every residue If a difference

15; i

it provides signifiv, k, ~ = 151, 51, 17.

151]

is

0

Thus the only difference

(7, 9, 14, 15, 18]. difference

modulo

21, 5, i,

the fixed blocks are

elements each.

definition)

76

were congruent to

For the parameter values

multiplier~

Consider the parameters

t = 76 [ ~ 214 ~ (17) 35

m ( = order of

could only exist if tion.

distinct elements) however frequently

[This difference d. l

has a c o ~ o n

then its Hall-polynomial

set is the

factor with

9(x) = x

dl

+ -'. + x

v.]

¢~

satisfies the congruence

e(x)e(x -1) ~ n

Thus, for any divisor

w

+ ~(1 + x

of

v,

+ ... + xV-1)

(mod

xv-

1).

(3.11)

one has

e(x) ~b o +bzX+... +bw_ZZ - I

(mod Z - Z )

(3.12)

e(x)e(x -I) ~ n +

(mod

(3.13)

where

bi( _< v/w)

yields

(comparing

Lemma 3.8.

~v

(i + x + .-. + x w-l)

is the number of coefficients

in

in congruence

If a difference

there exists integers

dj

set exists,

b i (i = 0, ...,w - l)

D

satisfying

x w - i)

dj m i nod w.

This

3.13).

then, for every divisor

w

satisfying the diophantine

of

v,

equations

64

w-i

w-i

Z

b. -- k~ l

b.2= n + i

hv -~'

(3o14)

0 < b, < v/w

i=0

i=0

and

w-i

~, b.b..= i l-j

(3.15)

;~v/~

i=O

for

j = l,...,w-

i.

(Here the subscript

i - j

is taken modulo

w.)

An example of the application of Lemma 3.8 is provided by the parameters v, k, h = 70, 24, 8. thus also for

Here

w = 5, 7.

2

is a w-multiplier

Consider a shift of

with

O(x)

fixed under the multiplier

2.

The residues modulo

impart certain restrictions

on

O(x)

modulo

Mod 35

0

w = 35 modulo 35

x5 - i

(Theorem 3.2) and x 35 - i

which is

break into fixed sets which

and modulo

x 7 - 1.

Mod 7

Mod 5

0

0

1,2,4,8,16,32,29,23,11,22,9,18

4(1,2,4)

3(1,2,3,4)

3,6,12,24,13,26,17,34,33,31,27,19

4(3,6,5)

3(1,2,3,4)

5,10,20

3, 6, 5

3(0)

15,3o,25

1,2,4

3(o)

7, 14, 28, 21

4(0)

1,2,3,4

Thus

0(x)

~ o o + Cl(X + x 2 + ... + x 18) + c3(x3 + ... + X 19) + C5(X5 + x I0 + x 20)

+ c15(x15 + x 30 + x 25) + c7(x7 + xl 4 + x 28 + x 21)

(mod

x 3 5 - i)

65

where

c. (0 < c, < 2)

where

is the coefficient of

(mod

x 7 - i)

e(x) ~ b 0 + bl(X + x 2 + ~

(mod

x 5 - l)

and

+ x 4)

0 ~ b. ~ 14.

b 0 + 4b I = 24, b 02 + 4b~ = ]28.

Then

For

equation (3.14) becomes

has only two solutions with a0 = 6

c I = c 5 = O. x~x

3

and hence If

w = 5

equation (3.14) becomes

b 0 = 8, b I = 4

implies (from the residue table above) that

Thus

Further

e(x) ~ a 0 + al(x + x 2 + x 4) + a3(x3 + x 6 + x 5)

0 ~ a. ~ i0

w = 7

x i.

c7 = 1

is the unique solution. and hence that

a 0 ~ 4;

these are

a0 ~ 4 .

a0, al, a 3 = 6, 2, 4 then

For

2 = 96 + a 3)

a 0 + 3(a I + a3) = 24, a~ + 3(a

This

c O = 2.

When

a I = 2, a 3 = 4

a I = 4, a 3 = 2

then

c 5 = 2, c I = I, c 3 = c15 = 0.

or

which 6, 4, 2.

c15 = 2, c 3 = i, Since

transforms this second solution into the first (and takes the difference

set into an equivalent one) it is only necessary to consider one solution. without loss of generality, modulo

Thus,

x 35 - i

e(x) ~ 2(1 + x 5 + x I0 + x 20) + x + x 2 + x 4 + x 7 + x 8 + x 9 + x II + x 14

(3.i6) + x 16 + x 18 + x 21 + x 22 + x 23 + x28 + x 29 + x 32 •

It can be seen (after some searching) that no difference set polynomial satisfy congruence 3.16. 24, 8

D.

Thus no difference set with parameters

@(x)

can

v, k, h = 70,

exists.

Polynomial Congruences If a difference set exists then, for every divisor

exist polynomials

ew(X), G[w](X )

o(x) ~ e[wl(X)

w

of

v~

there must

with rational integral coefficients such that

modulo

xw - l

(3.17)

66

O(x) ~ 8w(X )

where

fw(X)

fw(x)

(3.18)

is the irreducible polynomial satisfied by the primitive

of unity over the rational field. non-negative. [@w(X),

modulo

Furthermore,

the coefficients of

w

th

roots

O[w](X )

are

Conversely (by the Chinese Remainder Theorem), the set of polynomials

all w's dividing

8(X) -

v]

uniquely determines

V

e(x)

8w(X ) BV, w(X )

(mod

modulo

x v - i.

x v - i)

In fact,

(3.19)

wlv

where

V r

rlw

and

b

is the MSbius function.

S[w](X) = ~

r

X

- i

Similarly

Sd(X) Bw, d(X)

(rood ~ - l)

(3.21)

alw

Proof.

By the Chinese Remainder Theorem, congruence 3.19 can be established

by merely verifying the conditions imposed by congruence 3.18.

That is, by

verifying that

v

@w(X) Bv, w(X ) - Or(X )

wlv

for an arbitrary divisor

r

of

v.

modulo

fr(X)

(3.22)

67

[To facilitate this as well as for future reference, recall that if g(x), h(x)

are integral polynomials and

field, then

g(x) m h(x)

any root of

f(x).

~

f(x)

is irreducible over the rational

if and only if

g(r) = h(r)

where

r

is

Further recall that

xm- 1 =

where

modulo

f(x)

f(x),

I~ fd(X) dlm

a~d

fd(~) =

[~ ~jd

(xh- l) ~(d/h)

(3.23)

is the Mobius function.]

Now if

r

and

s

m x x

I - i

s

are divisors of

m/s

modulo

0

modulo

then

fr(X)

when

r

fr(X)

otherwise

divides

s

=

- 1

This follows from (3.23) for the left side at

e2wi/r

when

s Bm'd(X)-

m,

r]s,

r's not dividing r

s sm

]d

divides

_ =

so

m

s

and can be seen by evaluating

Thus, when

d

divides

~ r]s,

modulo

m

fr ( x )

Id

fd(x) Bm'd(X) ~ I

m

modulo

0

modulo

fr(x)

for

r /d

by the standard property of the ~6bius function.

From this congruence 3.22 is

immediate and so congruence 3.19 is established.

For future use~ note that when

d

divides

m

68 m

x -i

where

(')

[x f~(x)]

denotes derivation with respect to

verified by substituting the various gruence 3.24. unity;

(mod

m

th

x.

xm - l )

This congruence may be

roots of unity for

The only difficulty arises when

(3.25)

x

x

and using con-

is a primitive

d th root of

but l'Hospital's rule then shows that the right side is indeed

m

as

desired. A constructive existence test procedure can be based on congruence 3.21. this congruence is used to construct integral polynomials modulo

fd(x).

~d(X)

it requires the knowledge of

for all divisors

d

of

w

such that

O(x) m ed(X)

Neglecting for the moment the problem of finding an exhaustive

list of candidates for [G d}

~[w](X),

If

(i.e., one

ed

Od(X),

note that given a set of integral polynomials

for each divisor

d

of

w)

there is no guarantee that

the polynomial computed from them by congruence 3.21 will have integral coefficients, much less that these coefficients will be non-negative.

Of course, if these co-

efficients are not non-negative integers, then there is no difference set corresponding to this set of used to construct

Od(X)'S.

e[w](X)

Furthermore, if congruence 3.21 is being

for some divisor

expect that integral polynomials

e[d](X)

have been constructed previously.

w

of

v,

it is quite reasonable to

(for all divisors

d

of

w)

will

Thus, the assumption that they are known is

not restrictive.

This assumption allows the use of the following lemma, which

helps screen the

ed(X)

candidates by imposing the condition that

e[w](X )

have

integral coefficients.

Lemma 3.9. d

of

w,

Let

w

be a positive integer and suppose that, for each divisor

an integral polynomial

integral polynomial

O[w/p](X )

6d(X )

is given.

Further, assume that an

is known, for each prime

that these are consistent with the given

O[w/p](X) ~ ed(X)

ed(X)'S,

i.e.,

p

dividing assume that

modulo fd(~)

w,

and

69

for all divisors

d

of

w/p.

Then, necessary and sufficient conditions for the

existence of an integral polynomial

e[w](X),

e[w](X) ~ ed(X)

for all divisors

d

of

w,

modulo fd(x)

Proof.

p

Let

dividing e[w](X)

(3.26)

are that a-i

%(x) ~- e[w/p](:~) for all primes

such that

w.

Here

mod(P,fwPz (x)) w = paw I

with

p

prime to

(3.~) w I.

be an integral polynomial satisfying congruence 3.26.

By the Chinese Remainder Theorem

e[w](x) ~ e~[w,/P ](x)

mod(x w/p - i)

e[~/p](X) ~ ew(x)

mod[x ~/p - l, r(x)].

e[w/p](X) -= % ( x )

mod[p, fP 1

SO

Thus a-i

(x)]

i.e., congruence 3.27 has been established. Conversely, assume

e[w](X)

Remainder Theorem from the given

is the polynomial provided by the Chinese ed(X)'S

(it may not have integral coefficients)~

then by congruence 3.24

[9[w](X ) - ew(X)] BW,w(X ) ~ 0

(mod x w - 1)

7O

(w) If

s

divides

modulo

s[e[w](X

w,

x s - i,

) .

9w(X)]

Xs - 1 x - 1

-0

(mod

xW-1)

the Chinese Remainder Theorem guarantees that

.

(3.28)

8[s](X) ~ e[w](X)

and so

w

8[w](X

)

x

x

w

s

- 1

- e r

](x)

x

Ls~

-i

x

s

-

1

(mod

xw -

1)

- i

thus (3.28) becomes

(;)s

We[w](X) ~ Wew(X) -

Let

w = P2 "'" Pj~

sEe[s

w

by

w -

~ = w/pw.

a P

Then, since

w

w

(3.29)

is glven by

a2 a. P2 "'" PjJ

~(q2r) = 0

for any prime

only non-zero terms in the sum (3.29) are those for which Hence dividing through by

(mod x w - i).

ewC~) xS-i

where the prime power decomposition of

w

and define

3(x)

s

q,

the

is divisible by

yields

"-n

T)

p~S[w](X) ~ p~Sw(X) r

r[e[r~](x)

- ew(x)]

x x

7T

w

r£o

- 1

(3.30)

-1

r~pTr

modulo

x

- i.

Thus the theorem is proved provided it can be shown that the sum

on the right side of congruence 3.30 has all its coefficients divisible by By the second part of equation (3.23), with

w I = w/pa~

pF.

71

a-I f

a-i

(x~) =~l(X p )---fPwl (x) P2"'" Pj

modulo

p.

Thus congruence 3.27 implies

%(~) - e[w/p](~) ~ o

mod

[p,f (xW)]

which in turn implies~ by congruence 3.24, that

mod(p,Z - l)

[e[w/p](X) - ew(x)] Bpv, v(~ ~) ~ 0

or

[e[w/p](X ) -@w(X)] ~

~(~)r

r'~l

As

e[r~](x) ~ e[w/p](X)

x

modulo

mod(p,x w - 1).

xW-re0 1 - 0

(3.31)

-i

x rw - i

for all divisors

r

of

~

this be-

comes

Z rlTT

Z

l

x

-1

o

mod(p,Z

l)

where it should be noted that the polynomial on the left side of this congruence has integral coefficients by assumption. Y(x),

of degree less than

~~. ~ (~) r rpp~

w,

So, the unique integral polynomial

defined by

r[e[r~] (x) - ew(x)]

xwrw- l x -1

(moaZ-

l)

72

has every coefficient divisible by

pv.

Thus

e[w](X ) = ew(~ ) _ (pv)-i

~(x)

has integral coefficients, as was to be shown. Thus, provided a complete list of candidates for each divisor

d

of

w)

ed(X)'S

is available (for

congruence 3.21 (or its computationallymore convenient

formulation, congruence 3.29) can be used together with Lemma 3.9 to construct integral polynomials

@[w](X)

and ultimately, if these have non-negative coef-

ficients, the difference set polynomial Of course, for any divisor d ~ l

e(x)

itself.

of v, the major source of polynomials

ed(X )

is the equation

ed(~)ed(~ -1) = n

which must hold for any primitive

d th root of unity

spondence between the polynomials

ed(x)

factorizations of

n

over the field of

an algebraic integer which satisfies

(3.32)

~.

Thus there is a corre-

and a restricted set of principal ideal dth roots of unity.

c~ = n

If

ed(x) ~ ~ x j ~, ai xi

(c)(~) = (n)

n

will be constructed.

ideal in the field of

(3.33)

in the field of d th roots However, for small

it can be done.

As an example of this method, the 7, 12, 14}

8d(X)'S associated

fd(x).

modulo

of unity is, in general, an extremely difficult problem. v, k, ~

is

are given by

Determining all such principal ideal factorizations of

parmmeters

i

(the bar denotes complex conjugation)

then, by a theorem of Kronecker [Theorem 2.20 above], the only with the principal ideal factorization

c = Z ai~

Here

v, k, h = 21, 5, 1 e[l](X) = k = 5

difference set

{3, 6,

and since (2) is a prime

3rd roots of unity [Theorem 2.19 above],

e3(x) = ~ 2x a

73

for some

a = 0,i, 2.

By congruence 3.27

e[l](x ) = ~ : a ~ ± 2

thus

@3(x) = + 2x a

necessarily.

mod(3, x - i)

By congruence 3.29

eE3](x) = 2x a + i~ [5- 2xa] xx 3--i1 _ ~ a + 1 + x + x 2

So, by shifting the difference set if necessary,

_ l) .

(mod x 3

813](x) = 3 + x + x 2

with the

shift fixed modulo 3. In the field of

7 th roots of unity the ideal (2) splits into a product of

two prime ideals [Theorem 2.19 above] and since, as Reuschle (1875, p. 7) lists, (i + ~7 + ~ ) (i + ~7 + ~ ) = 2, (3.32), the ideal ~73)2.

(87(~7))

these ideals are principal.

can only be (2)

Thus, by equation

62

(1 + ~7 + ~7 )

or

(1 + ~7 +

Since these last two could only correspond to equivalent difference sets,

only one of them need be considered. (b = 0,1,...,6) using

or

4

If

(@7(~7)) = (2)

then

e7(x) = + 2x b

and congruence 3.27 shows that the sign is negative.

@7(x) = -2x b

a contradiction.

However

eEv](X),

in congruence 3.29 yields negative coefficients for

Thus, without loss of generality, one may assume that

e7(x) = ± (1 + x 4 + x6) 2 x c

for some

c = O,1 ..... 6.

5 - + (i + x 4 + x6) 2 x c e + 9

By congruence 3.27

mod(7, x - i)

thus the negative sign prevails and from congruence 3.29

e[7](x) -z _xC(l + x + 2x 3 + 2x 4 + x 5 + 2x 6) + 2(i + x + ''' + x 6)

The different values of (3,7) = l, c

c

(mod x 7 - i).

correspond to different shifts of the set.

can be specified arbitrarily without affecting

uniquely specifying the shift one may assume that

(c = 5)

e[3](x);

Since thus by

74

e[7](x ) = 2 + x 3 + x 5 + x 6

In the field of two prime ideals ~i)

= 2,

be (2) or

these ideals are principal.

1>2

(I + ~ i + ~

d = 0,1,...,20.

3.27;

When

d = 3e.

~21(x)

21 st roots of unity,

~[3](x)

For

thus

P = 7

P = 3,

the ideal

.18,2

(i + ~21 + ~21 ) " @21(x)

e[21](x)

for some

e = 5.

Thus

[3, 6, 7, 12, 14} Of course,

is the desired difference

x 21 - i.

(mod

x 21 - i)

set.

there are much simpler ways to construct this particular difference

set (Lemma 3.7 for exsmple), all the principal

but the method is perfectly general and works whenever

ideal factorizations of

unity can be determined.

@d(X) = ~

(n)

in the fields of

d th roots of

Thus there are parameter sets where this method is

easier to apply than any of the others. is a square and

modulo

3.29 yields

e(x) : Q[21](x) e x 3 + x 6 + x 7 + x 12 + x 14

i.e.,

can only

3.27 rules out the minus sign and shows

this same congruence yields

by congruence

+

The first two of these do not

= (I + x 12 + x18) 2 x i5 ~ x 9 + x 15 + x 18 + 2(x 3 + x 6 + x 12)

Computing

•

(@21(~21))

= ~ (i + x 12 + x18) 2 x d

congruence

2

(i + ~1221 + ~21 )18 (i + ~ i

As before,

12

or

= 3 + x + x

the ideal (2) splits into a product of

[Theorem 2.19 above] and since

satisfy congruences

that

and

~n x s

As Turyn (1960) noted,

for all divisors

d

of

v

the cases where

n

are particularly

nice. Note i.

In the course of the exsmple above the fact that a trial

had a negative, eT(x ) = -2x b.

though integral,

@[7](x)

coefficient was used to exclude the possibility

This is a consideration

outside the range of L e n a

might be led to suspect that the complete collection of conditions

3.9.

Thus one

(3.27) imposed

by Lemma 3.9 was not sufficient to guarantee the existence of a difference This is false. the problem,

That is, given that the polynomials

i.e., that

@d(X)

set.

used are meaningful

for

75

Sd(X)Sd(X'l)

=

In k2

then any integral polynomial polynomial of a

e[v](X)

when

d % 1

(3.34) when

x = 1

computed b y the above process is the Hall

v, k~ k - difference set.

integral polynomial of degree less than

e[v](X)O[v](x-l)

x = ~d

For, whatever else it is, it is an

v

~ n + ~(1 + x + " "

which satisfies

+ x v-l)

(mod

x v - I)

SEv](1) ~ k as follows from equations (3.26) and (3.34).

Thus with

9[v](X) = Z a'xil

(a i

i n t e g e r s ) i t follows t h a t (compare constant c o e f f i c i e n t s )

2 2 2 a_o + al + ''" + av--I = k

a 0 + a I + "-- + av_ I = k

and the only solutions to these diophantine equations have conditions

a i = 0,i.

So the

(3.27) of Lemma 3.9 together with the trivially necessary conditions

(3.34) are not only necessary but also sufficient for the existence of a v~ k, h - difference set. Note 2.

Since this method and that based on Lemma 3.8 are both aimed either

at establishing the non-existence or at the construction of successive polynomials 9[w](X),

they are often combined (i.e., for a particular value of

method is easier takes precedence).

w

whichever

Generally speaking however (as seen in the

example given in section III.C.) the successful application of Lemma 3.8 requires the knowledge of a multiplier or at least a w-multiplier.

[Thus, in view of the

multiplier theorems of section Iii.A~ Lemma 3.8 is more likely to be useful when

76

v

has a relatively large divisor

w

prime to

n.]

On the other hand, multipliers

play no obvious role in the method of this section [where known they can be used to restrict the possibilities for

9d(X)] ;

thus the two approaches tend to comple-

ment each other. Note 3.

The successive construction of the

~[w](X)'S

as a means of deter-

mining whether or not a particular difference set might exist has been used almost from the beginning of the study of difference sets.

Indeed, the use of the

algebraic number theoretic implications of congruence 3.32 in this, is also quite standard. 1961). ]

[Perhaps the best documented examples are in the works of Turyn (1960, Nevertheless the explicit determination of the relations (3.19)~ (3.20),

(3.21) upon which the method is based as well as Lemma 3.9 is quite recent and is due to H. C. Rumsey, Jr.

IV.

DIFFERENCE SETS OF SPECIAL TYPE

Various groupings of difference sets have been studied more extensively than others.

These groupings usually consist of all (or all cyclic) difference sets

with a certain fixed property.

For example, those having

may be constructed by some special process.

k = 1

or those which

If the common property is of a

constructive nature, the grouping is usually called a family of difference sets. (These difference set families and their special constructions are discussed in Chapter V.)

The present chapter concerns itself with some groupings of difference

sets which have received special attention but are not of the familial type.

A.

Planar Difference Sets The incidence matrix of a non-trivial symmetric block design with

also the incidence matrix of a finite projective plane.

h = 1

is

That is, if the blocks

are called lines and the objects are called points, the incidence matrix details the structure of a system of

(i)

2

+ n + 1

each line contains exactly n + 1

(ii)

v = n

n + 1

points and

v

lines such that

points and each point is on exactly

lines

any two distinct points are contained in one and only one line;

any

two distinct lines contain one and only one point in common (iii)

there exist four points no three of which are on the same line.

This last condition serves to exclude certain trivial configurations.

[An

introduction to the study of finite projective planes is provided by Albert and Sandier (1968).

See Dembowski (1968) for a comprehensive survey. ]

An open question, which has received a great deal of attention is that of determining the values points, exist.

n

for which finite projective planes~ with

They are known to exist whenever

n

2 v = n +n+l

is a prime power and known

not to exist whenever the associated Bruck-Ryser condition (see section II.B. fo~

78

this) is not satisfied.

For all other values of

projective plane is undecided.

[In particular

n,

the existence of a finite

n = lO

is undecided;

i.e.,

it is

not known whether or not a symmetric block design exists with parameters v, k, h =lll,

ll, 1.]

Primarilybecause of the interest in this problem~ the existence question for / cyclic symmetric block designs with k = 1 (i.e., difference sets with k = l) has been pursued extensively.

These difference sets are called plana r or simple.

Planar difference sets do exist with parameters h = 1

for all prime powers

tion details].

[Singer (1938),

see section V.A. for construc-

On the other hand, not all finite projective planes correspond to

cyclic difference sets; ~ny

pJ = n

v = p2j + pj + l~ k = pJ + l~

those that do are called finite cyclic projective planes.

of the results originally developed for planar difference sets were

subsequently generalized to the case of arbitrary sections of this survey.

h

and as such appear in earlier

In order to increase the readability of this section some

of these results are repeated here;

others are merely referred to when needed.

Three areas of interest regarding these planar difference sets are discussed in this report. n = p3

p

Singer's construction process for planar difference sets with

prime, is presented in section V.A.

The question (still open) of

whether there can exist multiple inequivalent planar difference sets for prime power

n

is mentioned in section VI.A and elsewhere.

Finally,

results concerning planar difference sets for general values of restricted to prime powers) are given.

in this section, n

(i.e., not

Of course, since all known planar difference

sets are of the Singer type, these results are mainly rules which establish the non-existence of planar difference sets with certain parameter values. If

t

is a multiplier of a planar difference set, then

t

determines an

automorphism of the associated symmetric block design (as noted in section I.E. above) and hence an automorphism of the associated finite cyclic projective plane v;

thus

t

the points of

is also said to be a multiplier of the plane ~

under its cyclic automorphism is

for an arbitrary automorphism

~

v.

[If the ordering of

Po, PI~...,Pv.1,

to be a multiplier of

7,

then, in order

it is necessary that

79

there exists an integer

t

according to the rule

such that the points of the plane are permuted under Pi ~ Pti "]

All non-trivial planar difference sets have non-trivial multipliers as is easily seen from Theorem 3.1 and the parameters In particular,

all divisors

(Hall, 19h7) and

3

t

of

n

v~ k, h = n

are multipliers.

2

+ n + l, n + l, 1.

[In fact, the primes

2

(Mann, 1952) are multipliers of a planar difference set if and

only if they divide

n.

This is not true in general;

multiplier of the Singer set

v, k, k = 21, 5, 1.]

for

ll ( ~ 25 )

Since

(k,v) = 1

is a for all

planar difference sets, Theorem 3.6(2) shows that there exists at least one shift E

of the difference set (i.e., line of the plane) which is fixed by all the

multipliers.

Actually,

combining a result of Evans and Mann (1951) with one of

Mann (see Hall, 1947):

Theorem 4.1.

At least one and at most three shifts of any planar difference

set are fixed by all the multipliers. fixed shift

E;

When

n ~ 0,2

this shift contains the object

does not contain

0

when

n ~ 2

modulo

3.

0

If

modulo for

3

n ~ 0

n ~ i

modulo

there is a unique modulo 3

and there may be one or three shifts fixed by al__~lthe multipliers. determined by the pair of objects determined by the object pairs multiplier satisfies

Proof.

Since

t ~ i

Vl, 2v I

O, v I

modulo

and

n ~ O, 2

modulo

shift is unique and no divisor of the blocks e.g.,

[0])

[a, ha, n2a].

fixed shift n,

(save

E.

When

O, 2v I

v

O, 2Vl;

3

then

n

modulo

also divides

3

then

t ~ i

(n - i, v) = i

n

n - i. contain

3,

0

thus the fixed

3

elements each,

belongs to the unique so

v = 3v I.

Since

(t,v) = i

necessarily,

Here

fixes the three objects

and hence also the three shifts containing the object pairs

Vl, 2v I.

3.

This implies that all

(n - i, v) = 3;

modulo

or

yields the first part of

(n - i, v) = i;

Thus determining whether or not

as well as any other multiplier

O, v I, 2v I

The shift

are fixed if and only if every

it follows that

fixed by the multiplier

n ~ i

v = 3v I

3.

v = (n - l)(n + 2) + 3

When

and

is always fixed, whereas the shifts

Thus applying Theorems 3.4 and 3.5 to the multiplier the theorem.

then

3

O, Vl;

the only other multipliers

8O

possible have

t e 2

modulo

thus the only shift fixed by containing

Vl, 2v 1.

With

3. n

These fix

0

and interchange

vI

with

2Vl;

and also fixed b y such a multiplier is the one

Thus the theorem is proved.

n = 2,3,4,7

the planar difference sets of Singer (see section V.A.)

provide an example for each of the possibilities listed in Theorem 4.1.

Corollar~f 4.2

(Evans and Mann, 1951).

trivial planar difference set with t

modulo

while if

v

n ~ 2

Proof. case 3 E

divides one of modulo

Since

divides

v

3

n, n + i. then

So

n ~ O,

fixed by all the multipliers. v

uniform size

As

~.

~

If

If

divides

2

t

is a multiplier of a non-

+ n + i

modulo

Furthermore

prime, then the order

n ~ 0

modulo

3

n ~ i

then

modulo

~

of

divides

3

(for in this

and hence there is a unique shift (since

v

is prime), all the non-

are distributed into disjoint blocks E

3

~

n + I.

is prime it follows that

v).

zero residues of

v = n

2

[a, ta,...,t~-la}

of

is necessarily a union of these blocks with the possible

addition of the object

O

the only question is whether or not

O

belongs to

E.

So the corollary follows from Theorem 4.1. Another important result, used but not explicitly stated by Hall (1947), was stated b y Mann (1952) as

Theorem 4.3. and if

If

ti~ t2, ts, t 4

tI - t2 ~ t 3 - t4

modulo

v,

are multipliers of a planar difference set then

(t I - t2)(t I - t3) ~ O

Proof.

modulo

v.

Theorem 4.1 guarantees the existence of a shift

E

fixed b y all

multipliers.

Such a shift must contain, together with each of its elements

the elements

tle, t2e , t3e, t4e;

thus

tle - t2e ~ t3e - t4e

modulo

v.

e,

81

Since

h = I

element

e

this is only possible if of

E

tie = t2e

or

i ~ e i - ej

ated equations

modulo

(4.1) for

v

for some

e. i

and

Applying Theorem (4.3) with

modulo

el, ej

e. j

of

2

and

2j - 2 ~ 0

modulo

divisor

of

t

v.

n

thus subtracting the associ-

2j + 1

i.e.,

t I = l~ t 2 = 2, t 3 = 2 j - l, t 4 = 2 j 2

(4.1)

t I = l, t 2 = 2, t 3 = 2 j, t 4 = 2 j + 1

for which they are not distinct multipliers,

planar difference sets having

E;

v.

establishes the theorem.

the only planar difference sets having

Similarly

So for every

it follows that

(t I - t2)(t I - t3)e ~ 0

Now

tle = t3e.

and

2j - 1

shows that

as multipliers are those

2j - 1 ~ 0

modulo

v.

yields the result that the only as multipliers are those for which

When Theorem 4.3 is combined with the fact that every

is a multiplier for any planar difference set (Theorem 3.1),

it provides a quite effective non-existence test for these difference sets in the case where

n

is composite.

Many of these cases fall within the scope of the

following corollary:

Corollary 4.4.

(Evans and Mann, 1951)

Let

al, a 2, a 3

be non-negative

integers for which

aI a2 a3 q - Pl = P2 - P3 '

where

q' PI' P2' P3

are prime divisors of

no planar difference sets with

Proof.

Since

aI Pl < 3q,

n~

plq~

v = n

2q

aI

2

n

and

a2 P2 < 3q

Pl % q % P2"

Then there are

+ n + i.

it follows that the hypotheses imply

a2

n2

which contradicts Theorem 4.3 and establishes the corollary.

82

This corollary modulo

6

[let

Another

shows,

for example,

that no planar difference

set has

n ~ 0

q = 3, Pl = P2 = P3 = 2, a I = a 2 = I, a 3 = 0].

efficient

non-existence

test for planar difference

composite

n

is provided by the next theorem.

collection

of some of the more easily recognizable

Of course,

these special

full generality.

[Actually

cases were established

[See Hall (1947)~

Mann

(1952),

it is no more than a

special

before

sets having

cases of Theorem 2.9.

Theorem 2.9 was known in

Hall and Ryser

(1951) and Evans

and Mann (195m)]. Theorem 4.5. and

q

t

be a multiplier

be prime divisors

ing conditions

are met

of

n

n

and

t

has even order modulo

(ii)

p

is a quadratic

(iii)

n ~ 4

or

6

(iv)

n m i

or

2

8

modulo

4

2 nI

or

modulo

2 n I + n I + i.

The elementary

[see, for example, (i)

modulo

If the order of

(iii)

~ery If

n m 4,6

modulo

8.

is

(iv)

~

and

of

q

P m 3

(n,v) of

modulo

then

2f = i

q 8

modulo

4

and

is of even order

p

residues

t f =- -i

modulo

is used in this

q,

then

n

is even

symbol q

so Theorem 2.9

suffice.

is of even order modulo

So the Jacobi

least one prime divisor residue,

q

(1951)].

t

non-residue

p

Then if any of the follow-

number theory of quadratic

Nageli

set and let

a square:

2 nI + nI + i

modulo

and the fact that (ii)

of a planar difference

respectively.

non-residue

n ~ nI

Proof.

v

is necessarily

(i)

(v)

proof

Let

of

~ v

(let

p = 2)

= -i.

for which

q.

~us 2

and

v ~ 5,3

there exists at

is a quadratic

non-

i.e., this reduces to (ii).

the reciprocity

law for the Jacobi

symbol and the ~ p o t h e s e s

2

)=-l

dove

83

since

p

divides

divisor (v)

q

Since

n

2

nI

2 n~ + n~ + l; As an example

n

13

excludes

and

2

nI

it

+ nI + i

shifts.

case

D

if a planar difference

(Hall, v

2 vI = nI + nI + 1

1957).

designate

v.

nI

Since

+ nI

p

of even order modulo

+ 1.

is of even

some prime divisor

n I = 3.

set is to exist with

or

12

modulo

13.

This shows that n ~ 3

or

9

Among others this

an individual multiplier

Let

t

the finite

(t - 1,v) = v I t.

subplane

Vl

Of course

having

of

is a multiplier

nI + 1

be a multiplier

t

set

m a y fix more than

vI

may be

vI > 1

0,v'~...,(v I - 1)v'.

let Let

Then

E

representation

~v' = e i - ej

is unique

(i ~ ~ < Vl).

it follows

that

representations

of

~v'

unless

also fixes

these elements

of

is uniquely

v'

from

E

D

vI > 3

then

determine

a

every multiplier

h

D

in section III.B. are which is fixed by

is of course also a planar difference

v',

non-zero multiple

Further,

be a shift of

of

E

but if

for general

as well as all multiples from

3

then the fixed elements

E = [el,...,en+ I}

t.

or

shifts of

Do

Vl"

v = VlV' ,

the multiplier

1

vI

together with the fixed shifts

The first assertion was established

Assume

plane generated by

and exactly

points on a line.

of the subplane

of a planar difference

cyclic projective objects

and the fixed objects

cyclic

E;

2

modulo

n = 35.

fixed by the multiplier

above.

~ 0

In fact

and let

Proof.

of a proper prime

Theorem 4.1 shows that at most 3 shifts of any planar difference

Then there are exactly

D

v

(v) of Theorem 4.5 when

p ~ 2,4,5,6,7,8,10,11

Theorem 4.6. set

that

is also a divisor of

are fixed by all the multipliers, 3

follows

it is necessarily

consider

the possibility

Whereas

the existence

thus this case reduces to (i) and the theorem is proved.

must be a square

modulo

This insures

as in (iii).

or

2 nI + nI + 1

order modulo of

v

~ nI

So any divisor of

q

of

n.

t

represented

being multiples

of

v'

But since

set. t

e. - e. = te. - te. e.

and

as a difference also.

e..

So the fixes

E,

are distinct Hence every

of elements

So the elements

of

of E

84

divisible

by

v'

determine

parameters the form

E + iv'

fixed by

t

for some

constitute

as was to be proved° shifts fixed by the subplane Bruck v = n

2

t

Vl

j,

If

s

Since

t

with

nI + 1 of

Vl"

Vl' ~l

t

with

is of

together w i t h the shifts points on each line

D then

(Theorem 3.5) hence

Thus all assertions

[For, each line of

v I > 3)

s

s

permutes

the

is a multiplier

of

of the theorem have been established° plane

v,

with

2 v I = n I + n I + l,

with

contains

n - nI

Thus the number of points of

v

2 n = n1

then

points of

v

which

which lie on no line of

2 2 2 + n + i - (n I + n I + i) - (n - nl)(n I + n ! + i) = (n - nl)(n - nl)

n > nI 2 nI

than

on exactly

necessarily,

it follows that

there exists a point

2 nI + nI + I

lines joining n + i

be improved to

P

P

lines of

of

~

n ~n~o

which

to the points of ~

that in certain special cases

it must be that (including

n _> n 2 + n I + 2o

If

n

~i

are distinct

n ~n~

+ nlo]

divide

n,

see H. Neumann,

For any integer

fixes every object and shift that

are thus

nO + 1

Corollary

4.7

(Hall~

is a line fixed by

t

and as

Roth

P

lies

(1964) shows

that of Theorem 4.6) this inequality m a y

plane examples

nI

divides

(For an arbitrary

are known where

is a multiplier t

fixes.

whenever

If

E

then Theorem 4.6 shows that

objects

So the

nI

subplane

does not

1955).

j > 0, t j

(t j - 1, v) = v 0

~i"

In all known cases of Theorem 4.6

finite projective

.

is actually greater

lies on no line of

but the question of whether this must be so is open.

of an arbitrary

E

Vl

is another multiplier

has a proper subplane

for

is

2

if

fixed by

(1955) has shown that if a finite projective

+ n + l,

n

n,

set (non-trivial

Since each shift fixed by

the objects

among themselves

are not points of

Vl

difference

a cyclic subplane

also.

2 n ~ n I + n 1.

or

a planar

2 v I = n I + n I + l, k I = n I + l.

of

E

1947)o and if

fixed by

If v0

t

t j.

t

is, and

tj

is any shift fixed by

v 0 = n02 + n O + 1

certainly t

and

and that there

So

is a multiplier

is a divisor of

v

of a cyclic plane such that

7,

(t x - 1 , v ) =

if v0

85

for

x = j

nO + 1

but not for

points on

permuted by

t

E

v = m set

+

D1

of

D

mr

fixed b y

(Ostrom,

+ l, n = m r ,

Since

m3

These

lengths

1953).

of

2

nO + 1

divide

and there are exactly

points of

E

are thus

jo

If a planar difference

(r,3) = 1

vI = m

is also a multiplier

Proof.

t j.

with

with parameters

2 v0 = nO + nO + 1

then

in cycles whose

Coroilar~ 4.8 2r

x < j,

set

D

has parameters

then there exists a planar difference

+ m + l, n I = m.

Furthermore,

every multiplier

D 1.

is obviously

a multiplier

Theorem 4.6 once it can be established

of

D,

the result

follows from

that

(4.2)

(m 3 - i, m 2r + m r + i) = m 2 + m + 1 = v I.

From

(r,3) = I

it follows that

thus the only question (n + 2) + 3 3v I

shows that

is whether

value of

n

divides the left side of equation

some multiple

(n - i,v) = i

is the only candidate.

and this implies that

vI

or

If it were

vI ~ 0

modulo

3v I 3.

for which this happens.

3;

of

vI

thus then

Thus

Now

(m - l,v) = 1 m ~ 1

v ~ 0

So equation

does alsoo

modulo

modulo

or 3

9

(4.2), v = (n- i). 3.

So

necessarily

and there is no

(4.2) is valid and the corollary

has been established. Assume v. t

Then of

q

D

modulo

a planar difference

D

exists and let

is said to be a type I divisor of

having q

set

be

lower order modulo

~.

q

w

or

is the modulus of a proper subplane

Mann,

1951):

if there exists v.

some multiplier

Let this order

Then

where w

be a prime divisor of

than it does modulo

(t ~ - 1,v) = w ~ 0

properly

v

q

divides

v.

modulo

By Theorem 4.6 ~l"

q

2 w = nI + nI + i Thus,

and either

in particular

w = 3

(Evans and

86

Corollary modulus q

4. 9 .

If no proper factor of

of a planar difference

is also its order modulo Prime divisors

as it does for

v

q

then

n

3,

q = 3

q ~ i

set, then the order of every multiplier

t

modulo

for which every multiplier

q ~ 1

q

~ = 1

prime divisor

3.

so

q

(Ostrom, of

v,

If

~ = I

q = 3.

If

of

D

(i.e.,

1953).

n ~ i

~ = 3

then

~

of

modulo 3

If a planar difference group of

q,

type Ii~ so

t 3 - i =- 0

modulo

Corollary

of

q

modulo

v.

i.e.,

v

n q

q

of

modulo

q

modulo

q

and

divides

D

set

~(q)

i.e.,

D

has a type ii

is cyclic and its order

of the multiplier of

Thus

that

v).

D

Theorem 4.11

modulo

t I (t 3 - l) ~ 0 t3 - 1 ~ 0

1953).

q

be distinct

and if then

modulo

modulo

is isomorphic

q.

qo

Since

But

q

is of

the distinctness

set with

q - i. n ~ pJ

is sought

the search to the cases where

q

of

of

with a subgroup of the non-

Theorem 4.5 insures that no multiplier

any of the prime divisors

(Ostrom,

t!, t 2

it is cyclic and has order dividing

When this is done~

D

Then there exists a multiplier

t 2 ~ tlt 3

in restricting

group of

Hence this image group is

For let

4.8 shows that if a planar difference

even order modulo

q.

also and this contradicts

group of

is no loss of generality

not a square.

modulo

this implies

So the multiplier

zero residues

residues

it would follow that

necessarily

q

have the same image. tI ~ t2

t 2 ~ tlt 3

modulo

(tl, q) = 1

there

then

The image under reduction modulo

such that

tl, t 2.

n2 + n + i ~ 0

For if

then the multiplier

No two multipliers

multipliers

tI ~ t2

3.

and this implies that the order or

q;

modulo

is of course a subgroup of the non-zero

t3

has the same order

q - i.

Proof.

cyclic.

t

3o

Theorem 4.10

divides

is the

v

and

modulo

modulo

q

it should be noted that the only prime divisors

modulo

i.e.,

+ n + I ~ 3

by the prime

are called type II divisors.

n3 - i ~ 0

divides 2

are

divisible

v.

and

Before going further n2 + n + i

v

v.

n

is

is of

Thus

Suppose that there exists a planar difference

87 2

set modulo

v = n

necessarily

distinct primes.

for some

i, qi

multipliers

+ n + 1

where

3h

where

Theorem 4.12

h

and the

Then the order

is odd and

3h

n

S

and all the prime divisors

qi

v

cyclic group of multipliers

divides

n

(When

n ~ 0

divisor of

modulo

3.

pliers

n + 1 n ~ 1

when

n ~ 2

modulo

3,

residues size

of

v

S.

are distributed

Thus the unique

is necessarily

a union of these

Thus the theorem is proved.

3.

By Theorem 4.1 the object

Combining

a result of 0strom

Theorem 4.13. v = VlV2, v I > 1 w i t h respect to

by

t

and

S

divides

v.

v 2 > l, Suppose

then

S

Proof.

divides

Let

are multiples

E

of

n - nI

0

further that

type II divisor

and

[a, ta,...,tS-la]

is added if and only if

(t j - 1,v) = Vl, Then

addition of n ~ 0

(1951) yields

set with

v2

is of type II

where

j < S

and

2 vI = nI + nI + 1

t

and

v2

are primes,

nI ~ 1

and

tJ;

it contains

nI + 1

objects which

(see the proof of Theorem 4.6 above). E~

let

a(t ~ - i) ~ 0 of

t

vI

S

divides

(n - nl)/V 1.

objects of

q

vI

let

4.1) shift fixed by all the multi-

where every prime divisor of

be a shift fixed by

v2

Then

sets

(1953) with one of Evans and Mann

If in addition

S;

is of type II the

Suppose that there exists a planar difference and

n - nlo

v.

of the

is always a type I

v

into disjoint

is a generator of the cyclic group of multipliers.

modulo

3

S

sets together with the possible

modulo

other

then

3

group is cyclic of order

(Theorem

0.

n - nI

modulo

Since every prime divisor of

the object

divides

of Theorem 4.11 are satisfied

are of type II, then the order

By Theorem 4.10 the multiplier

be a generator of this group.

of uniform

~(qi ) = qi - 1.

v)°

Proof.

non-zero

are not

is not a square and that

divides

If the hypotheses

of

qi

of the cyclic group of

1953)o

when

(Ostrom,

... qj

Suppose further that

is of type II.

is

v = qlq2

v 2.

Thus

~

If

be the least power of

modulo ~ = S

v I v2 and

hence S

t

a

such that

t~ - 1 z 0

divides

is any of the

n - n I.

modulo If

v2

at ~ a some is

88

prime then

(Theorem 4.10)

S

divides

v~~ - 1.

2 n 2 + n + 1 = (n I + n~~ + l)v 2

Since

(n - nl)(n + n I + i) V~

-

i

=

vI

Now

vI

(since

divides

n - nI

n I ~ 1).

divides

by assumption,

Since

(n - nl)/V 1

S

divides

thus

n - nI

n + n I + 1 =- 2n I + 1 ~ 0 and

vI

is prime it follows that

and M a n n to

sets with

(1951) to

n < 100 have

n

parameters

n, v

Let

respectively.

Corollary

(b)

If

n = 4,6

modulo

8

then

(c)

If

n -= 1,2

modulo

4

and if

(d)

If

n - nI

n2

modulo

tests for

P = 3

then

(e)

If

p

is a quadratic non-residue

of

(f)

If

t

is of order

and

if

n - 0

If

t

(g)

n + 1 (h)

Let

is of order if

n -= 2

v = vlq

3. ~

v

n

v

3o

If

n

of any multiplier

If

is not a square

~

t

be prime

set with conditions:

p

n

is a square.

is of even order (In particular

the

were used.) q

then v

n

is a square.

is prime,

v

~

then

then

is not a square

modulo

must be odd.

~

~

~

n

must divide

divides

must be odd.

set exists modulo q

divides

must be odd.

is prime,

where no planar difference

~

then

is a square.

and n

4

and if

is not a square

modulo

modulo

order n

If

of a difference

modulo

n2 + n I + 1

modulo

p, q

must be a square.

n I = 1,2~3~5,6,7

~

and let

Theorem 4.3 must hold.

n

n 2 + n I + l,

modulo

p. 209) was pushed

is subject to the following

4.4 and more generally

with respect to

(1968,

be a multiplier

Then the existence

(a)

or

that the only planar

This was extended by Evans

to Dembowski

t

v , k , h = n 2 + n + l, n + l, 1

associated

S

(unpublished).

The tests of Evans and Mann. of

a prime power.

n < 1600 and according

n _< 3600 by V. H. Keiser

divisors

v1

as was to be shown.

Utilizing mainly Theorem 4.3 Hall (1947) established difference

modulo

v 1.

Then the

G = (q°(Vl), q0(q)).

89

(i)

Let

v = vlq ,

Vl, q

exists modulo

vI

multiplier

t

of the

modulo

but

of

t

vI

modulo

divides

Proof.

are both primes and a planar difference set

but not modulo

t ~ 1 q

q.

Let there exists a non-trivial

v,n + 1,1 - difference set such that modulo

divides

n - n I (n I ~ l)

(n - nl)/V I

above.

where

q.

n - nI

If

n

is not a square then the order

and is odd.

then the order of

t

If, in addition, modulo

(f)

Consider test

and

(h).

(g)

v

thus

~

divides

So Theorem 4.5 finishes off.

t

~(Vl)

and

Consider test (i).

are prime it follows from Corollary 4. 9 that multiplier

divides

(a),o..,(e)

are a combination of Corollary 4.2 and Theorem 4.5.

Here Theorem 4.6 shows that the order of

the same as its order modulo definition.

q

v1

and is odd.

Theorem 4.3, Corollary 4.4 and Theorem 4.5 establish

Tests

t = 1

q

t

modulo

~

divides

Since

q

is a type II divisor.

of the test corresponds to the multiplier

tj

vI

is

~q) and

by

v1

The

of Theorem 4.13.

Thus Theorems 4.5 and 4.13 establish the validity of test (i). Since the only known planar difference sets have prime power

n

and in fact

may all be constructed by the process of Singer (see Chapter V), it is easy to make up perfectly reasonable conjectures about planar difference sets.

Just conjecture

that any property possessed by the Singer sets holds for all planar difference sets.

One such property of the Singer set

v,k,h = p2j + pj + l, pJ + l, 1

that its multiplier group consists of all the powers of powers of

p

p

modulo

are necessarily multipliers has been observed earlier.

only the powers of

p

are multipliers

v.

is

[That all The fact that

is non-trivial aud is due to Gordon~ Mills

and Welch (1962), see section V.A. for a proof of their more general result. ] a general planar difference set (i.e.,

n

For

not necessarily a prime power) the

analogous conjecture would be that its multiplier group was generated by the prime divisors of

n.

A reduction lemma established by Halberstam and Laxton (1964), in

the course of providing an alternate proof

(h = 1

only)

of the Gordon, Mills~

Welch result on Singer sets, may be of some use in attacking this problem. that

t, nt, n2t

Note

are necessarily either all multipliers or all non-multipliers of

9O

any planar difference set with be of reduced type modulo

n

2

+ n + 1

0 < r, s < n

Lemma 4.14. with

n > i,

and

modulo

0 < r + s < n.

j

to

n2 + n + i

Then

(Halberstam and Laxton)

then at least one of

define an integer

if

j-=r+sn

with

Furthermore,

k = n + i.

If

t > i

t, nt, n2t

and if

(t,n 2 + n + i) = i -

is of reduced type modulo

2 n

B.

+ n

+ i.

Hadamard Difference Sets Difference sets whose parameters

Hadamard difference sets.

(ii)

There are several reasons for this; among them are:

the relative abundance of such difference sets with

k < v/2

as usual,

section I.B.

h

varies between

i

and

(v - 3)/4~

see

Thus planar difference sets and Hadamard difference sets

present the extreme values of (iii)

are called

Like the planar difference sets of section IV.A., these

have been extensively studied. (i)

v, k, h = 4t - l, 2t - l, t - 1

h.

the autocorrelation function of the

l, -1

characteristic function of

these difference sets (see section I.D. for this) is minimal [~(j)

= -1

for

j ~ 0

modulo

communications applications

v].

This has led to several digital

Ksee, for example, Golomb e t a l

(1964)

or Goldstein (1964)]. (iv)

the relation between these difference sets and the~ as yet unsolved, Hadamard matrix problem (see the note below).

The known Hadamard difference sets can be classified according to the value of

v. (a)

The groupings are: v = 2 j - i, j ~ 2;

section V.A. discusses a large family of difference

sets whose parameters include these (construction details are given there).

91

(b)

v = 4t - i

is prime;

here there always exists a Hadamard difference

set composed of the quadratic residues modulo prime

v = 4t - 1

is expressible

as

v

4x 2 + 27,

set (due to Hall, 1956) whose construction

(see section V.B.) and when the there is an additional

is discussed

difference

in section V.C.

Some

others exist also. (c)

v = p(p + 2)

where

p

and

p + 2

are both prime numbers

(see section

V.D. for the details). Occasionally

v

satisfies more than one of these conditions

most part, this leads to multiple (a) Mersenne

and

(b)

(a)

overlap if and only if

primes of the form

Chowla and Lewis and

4x 2 + 27

difference

v

are

sets.

Specifically: the only

is a Mersenne prime;

v = 31, 127

and

[see Skolem,

131071

(1959) for this].

(c)

overlap if and only if

The known Mersenne ~rimes with

inequivalent

and, for the

v = 15.

(i.e., primes of the form

2 j - i)

are

v = 2j - 1

j = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281,

3217, 4253, 4423, 9689, 9941, 11213 at present,

[see Gillies,

1964;

in fact

2 I1213 - i

is,

the largest number known to be prime].

The difference v = 3, 7, 15

and

sets corresponding 31.

Now

(a) provides a difference set corresponding

to

v's

set and (b) provides is equivalent

sets arise from

While all known Hadamard difference

two more.

(c).

In particular,

sets with parameters

except for

is prime, thus

However,

the difference

to the one from (a);

thus only two

v = 31. sets have

not all of them can be constructed by the methods in (a), (b),

are inequivalent

v = 31 = 25 - 1 = 4.8 - i = 4.12 + 27

4x 2 + 27

truly distinct difference

to these

v's of types

(a),

(c),

(section V.A. - V.D.) indicated

there are exactly six inequivalent

v, k, h = 127, 63, 31

(b),

difference

(see Baumert and Fredricksen,

1967),

three of which do not arise from these constructions. It is known (Golomb, v < i000,

then

v

Thoene, Baumert)

is one of the forms

that if a Hadamard difference

set has

(a), (b), (c) with six possible exceptions.

92

These exceptions are decide

this

v = 399, 495, 627, 651, 783

and

975.

The methods used to

were~ of course, those of Chapter II and III.

Some problems of electrical network theory [Belevitch (1968), see also Goethals and Seidel (1967)] have led to the consideration, skew Hadamard difference sets.

That is, Hadamard difference sets which contain

precisely one of the residues

d, v - d

for

1 < d < v - 1.

obvious examples of such sets are the quadratic residues primes

q ~ 3 (mod 4).

Theorem 4.15.

e(x)

x

v

Let

Johnsen (1966B) has shown that there are no others.

(Johnsen)

The only cyclic difference sets which are skew

T(x) = 1 + x + ''' + x v'l

for such a difference set.

- i

and so [using

Of course, the

(see section V.B.) of

Hadamard are given by the quadratic residues of a prime

Proof.

among other things, of

q,

where

q ~ 3 (mod 4).

and consider the Hall polynomial

It satisfies

e(x)e(x "l) ~ n + h T(x)

1 + e(x) + e(x -1) ~ T(x) and

@(x)T(x) ~ k T(x)]

modulo it

follows that

e2(x) + e ( x ) + n ~ n T(x)

So, for all

v th roots of unity

mod(x v - 1).

~ % i

e2(~)

+ e(~)

e(~) = ( - i ±

since

v = 4n - 1 Since

for which Let

p % q

q

divides

v

p

roots of unity.

,.,~)/2

(4.4)

it has an odd prime divisor but

q

s+l

does not divide

be another odd prime divisor of

equation (4.4) shows that th

(4.3)

in any Hadamard difference set.

v ~ 3 (mod 4), s

+ n = o

(-1 ~ ' ~ ) / 2

Contradiction.

v

and let

q v

with and

s

q m 3 (mod 4) is an odd integer.

~ = ~p = e 2wi/p,

then

is an algebraic integer of the field of So

v

is an odd power of the prime

q,

where

93

q ~ 3 (mod 4). Let

~q = e 2vi/q

and let

z(x) =

so that

Z(~q) =

1 ~

~q

in section III.D.

divisors

d

v (=qS).

(4.5) it follows that

G i(x) = q

Here

gi = ~ 1

From (3.19),

+x

+x

~

To do this we need the complete set of Equation

(4.4) provides us with these,

el(X ) = k = (qS . 1)/2

and that

1 ~

)

(3.20) it follows that modulo

v

ew(x) Bv,w(X)

~(i) = l, ~(q) = -i and

for from (4.4),

for

-v

and

-

r = w/q.

in assuming that

2v

ew(X)

s1 = + I.

x---f:-i--!

~

rl w

~(qi) = 0

i = 1 ..... s.

for

r

r

r

x

i = 2,...,s,

So interchanging

- 1

this last summation

the order of summation and

matching terms yields

+

for all

xv - 1

wl v wll

r = w

ed'S

process

i-1 + el q(S-l)/2 z(xq

+

only involves

(4.5)

~ge shall use the constructive

and there is no loss of generality

=

Since

x[(q-l)/2] 2

+ "'" +

(Gaussian sum).

discussed

of

l

~

1

v

- - -

1 v

Oq(X)

qi i=l

1 --i xq - 1

x---?'i

{eqi(X) - eqi+l(X)} ]

.

94

Now, for

i = 0,1,...,s,

x q i ( x-V- i ' "i )

xV l -i

xq - 1

m°d(xV - i)

xq - 1

Using this fact

e(x) -= ev(X) + ~

v

-d

x v - i~ (4.6)

s-1 +

-

v

) -

aj+l(~/2)}

x q3 - 1

so the only question is whether this has integer coefficients. if

v = q,

then (4.6) becomes

e(x) ~ - 1

~

j =!

(gl = i,

+z(x) = - ~ 1

+ ~z

If

s = l~

i.e.,

as noted earlier)

+x

+x 4

+ ...

+ ~[(q-Z)/2]

2

which does indeed have integer coefficients and in fact is the quadratic residue difference set for Let

s

efficients,

q ~ 3 (mod 4).

be odd and

s > 3

then in order for (4.6) to have integral co-

it is necessary that the terms

_q(S+l)/2 2~

( ~ Z -- i I ) and

-i

q(s+l)/2 ( 6 1 z ( x ) } ( x v_- _1 )

v

(4.7)

x q- i

compensate for each other, since they are the on]~v terms of (4.6) which have the coefficient

~(s+l)/2 v

i q(s-1)/2 "

95

A l l other terms of (4.6) contain higher powers side of (4.7) contributes contributes

to

to all

(qS + qS-1)/2

other and therefore

q

S

of

q.

coefficients

coefficients.

no such difference

But the term on the left

whereas

the other term of (4.7)

So they cannot compensate

set exists.

for each

Thus Theorem 4.15 has been

established.

Note:

A n y symmetric block design

parameters course,

v,k,h = 4t - i, 2t - i, t - i

gives rise to the name Hadamard

set of a cyclic design. a Hadamard distinct matrix,

design,

-i.

a matrix of O.

orthogonal. achieves

If

-i

not necessarily

difference

That is, a

21

Such a matrix

abs. val.

det.

= (4t)2t].

the matrices, For example, modified

v,k,h = 7, 3, i

Hadamard

difference

of any two

+l's

to this

row inner products 4t

are

whose rows are mutually

a Hadamard matrix because determinantal

Thus the name Hadamard

of

in the incidence matrix of

whose distinct

to the associated block designs the

0

row and column of

the upper bound specified by Hadamard's

which,

set for the associated

square matrix of order

is called

design,

is such that the inner product

Thus, by adding a constant +l's is constructed

cyclic) with

is called a Hadamard

is used instead of

the resultant matrix

rows is

uniformly

its determinant inequality

[i.e.,

passed from the inequality

and finally to the difference

difference

set

(1,2,4]

to sets.

has associated

incidence matrix and Hadamard matrix:

-i -i -i 1 -i 1 1

i -i -1 -1 1 -1 1

i i -i -1 -1 1 -i

Chapter 14 of Hall existence Spence

(i.e.,

-i i 1 -1 -i -1 1

i -i 1 1 -1 -i -i

(1967)

-i i -i 1 1 -1 -i

surveys

of Hadamard matrices;

(1967),

and Whiteman

Goethals

(1970).

i i i 1 1 1 1 1

-i -i 1 -1 1 1 -1

i -i -i -1 1 -i 1 1

(with proofs) for results

and Seidel

i i -i -i -i 1 -i 1

i i i -i -i -i 1 -i

i -i 1 1 -1 -i -i 1

i i -i 1 1 -i -1 -i

subsequent

(1967), Wallis

cases are

n = 188,

i -i -i 1 -i 1 1 -i

most of the known results

on the

to Hall's book see

(1969 AB,

1970),

It is known that the order of such a matrix

and the first few undecided

i -i i -i 1 1 -i -i

Turyn is

236 , 260, 268, 292.

(1970)

i, 2

or

4t

96

C.

Barker Sequences ~ Circulant In 1953 R. H. Barker

munications,

considered

and minus ones coefficients

Hadamard Matrices

(1953),

in connection with a problem in digital

the question

[bi] [ ~

of the existence

of finite

with the property that their aperiodic

should be as small as possible,

sequences

comof ones

auto-correlation

That is, he asked that

v-j c.J =

~

b i bi+j

= 0

or

-i

i=l

for all become

J, i ~ j ~ v - i. customary

sequences~

and

to relax Barker's

whose aperiodic

Barker sequences. denotes

He found such sequences condition

v = 3, 7, ii.

It has

slightly and call all finite

autocorrelations

Only the following

for

e. J

are restricted

Barker sequences

to

are known:

I, -i

-i~ 0r ix (+

denotes

+i

-i)

v =2

++

v=4

+++-;

V = 5

+ + +-

+

V =7

+++-

- +-

v =ii

+++-

° - +-

v =13

+++++-

++-

+

- +-

- ++-

+-

+

together w i t h the sequences which may be derived from them by the following

trans-

formations:

b: = ~"C-l) i b 1 1 b: = (-i) i+1 b. l 1 b~ = -b.. l I

In fact,

Storer and Turyn

(1961) have shown that any further Barker sequences which

may exist must be of even length,

indeed they show that

v ~ 0 (mod 4)

is necessary.

97 Note that in terms of the

i, -i

representation of the characteristic function

of a difference set (see section I.D. for this) all these sequences correspond to difference

sets.

v = 7,11,13

For

v = 2,3,4,5

the difference sets are trivial;

the sets have parameters

v,k,h = 7, 4, 2;

ii, 5, 2

and

for 13, 9, 6

respectively. It can be shown (Storer and Turyn, 1961) that Barker sequence.

Further,

cj + cv-j = ~ ( j )

if

v - O (mod 4)

then

e. + c

,]

v-j

~ v (mod 4)

cj + Cv_ j = O.

in any

Thus, as

[the correlation coefficient defined in section I.D], any

further Barker sequence

{bi}

has autocorrelation function

Rb(J)= Ii if j ~0

modulo

v

otherwise

i.e., a two-level autocorrelation function. difference set. 2.1 and since v = 4N 2. values of

Since n, v

Since ~(j)

v

is even,

n = k - k

= v - 4(k - h) = 0

k(k - l) = h(v - l) show that

Thus such a sequence corresponds to a

for

is a square (say j ~ O

modulo

v

N 2)

it follows that

for any difference set and

k = 2N 2 - N

or

2N 2 + N.

b y Theorem

n = k - ~

Since these

these

k's correspond

to complementary difference sets, there is no loss of generality in assuming that v,k,h = 4N 2, 2N 2 - N, N 2 - N. Thus, further Barker sequences exist if and only if there exist difference sets with parameters (v,n) = ( 4 ~ , ~ )

v,k,h = 4N 2, 2 ~

= N2 ~ i

- N, ~

- N

for

and 2.17 above rule out many of these cases; with the single exception

by a constructive method;

Since

here, this is a subcase of the unsolved problem con-

cerning the existence of cyclic difference sets with

i ~ N ~ 55

N ~ i.

N = 39.

(v,n) > i.

in particular,

N ~ 55,

all the cases

Turyn (1968) excludes this case

essentially that of section III.D.

Barker sequences exist they must have

Theorems 2.13

i.e.,

Thus, if any further

v ~ 12,100.

A matrix is called circulatory or said to be a circulant if each successive row is derived from the previous row by shifting it cyclically one position to the

98

right.

For example, the matrix

is a circulant.

(+

for

.

÷

+

+

+i, -

÷

for

-i)

÷

+

This particular circulant has every entry

orthogonal to each other.

!1

and its rows are

Matrices with these properties are called Hadamard

matrices and they have been extensively studied [see the note at the end of section IV.B].

The example, thus, is a circulant Hadamard matrix.

rearrangement and scalar multiplication by only known circulant Hadamard matrix.

-i

In fact, up to

of its rows and columns~ it is the

It follows immediately from the autocor-

relation function of a Barker sequence of even length

v ~ 4,

that there is a

one-to-one correspondence between such sequences and circulant Hadamard matrices. Thus (from the Barker sequence results above), if there exists any further circulant Hadamard matrices they have orders

v ~ 12,100.

It sho~Id come as no surprise

then, that the absence of any further Barker sequences/circu!ant Hadamard matrices has been conjectured.

[Using the fact that

-i

is never a multiplier of a non-

trivial cyclic difference set (Theorem 3.3 above), Brualdi (1965) has shown that there does not exist an Hadamard matrix of order

v ~ 4

which is a symmetric

circulant.] The related problem of finding

i, -i

sequences of length

v

for which the

maxi~2~m aperiodic correlation coefficient is of least magnitude (i.e., for which

m~ lojl J is minimized) and indeed the problem of determining this minimum, at least asymptotically as a function of

v,

is unsolved.

of the known results on this subject.

Turyn (1968) provides a survey

V.

FAMILIES OF DIFFERENCE SETS

The known difference sets (with a few exceptions) can be divided into families.

This chapter deals with these families of difference sets, construction

methods specific to them, their multiplier groups and the status of some open questions related to them.

A.

Sin~er Sets and Their Generalizations.

The Results ' of Gordon~ Mills and Welch

Singer (1938) discovered a large class of difference sets related to finite projective geometries.

These

parameters:

have

N+I v

=

~

N -i

q - i

for

N > i

k =

q

'

N-I ~ ~

~ =

q

q - i'

and they exist whenever

q

-1

(5.1)

q - i

is a prime power.

In order to discuss Singer's result properly it is necessary to know some of the theory of finite fields [see, for example, van der Waerden (1949) for proofs]. For any prime power

q

there exists a finite field with exactly

q

elements.

This field is unique up to isomorphism and is called the Galois field of [written

GF(q)].

The multiplicative group of

generated by any of its

~(q- i)

GF(q)

elements of order

is cyclic; q - i.

elements are called primitive roots and if

~

ever

the residues

u

is prime to

q - i.

For prime

p,

structed from

GF(q)

irreducible over all divisors

j

by adjoining any root

GF(q). of

m.

The subfields of

GF(p). ~

These generating

GF(pm),

O,i,...~p - I p;

p

~u

when-

form a

this field is often

m GF(r)~ r = q ~

of any

elements

thus it is

is a primitive root so is

field with respect to addition and multiplication modulo taken to be the generic representation of

q

can be con-

m th degree polynomial prime, are

GF(p j)

for

f(x)

lO0

GF(qm) GF(q).

is often represented by the set of all m-tuples with entries from

In this representation addition is performed componentwise but multiplica-

tion is more complicated. polynomial

am_ 1 x

m-i

Associate with the

+ ... + alx + a 0.

m-tuple

am_l, am_2,...,al, a 0

the

Then, in order to multiply two m-tuples,

multiply instead their associated polynomials and reduce the result modulo any fixed

m th degree polynomial

f(x)

irreducible over

GF(q).

The coefficients of

the resulting polynomial constitute the m-tuple which is the product of the original two.

For multiplicative purposes it is more convenient to represent

terms of a primitive root

~;

in which case,

GF(qm)

GF(qTM) in

consists of

m

O,l,C~, 2 ~ o . o ,(~q -2

m

Multiplication then becomes a simple matter of reducing exponents modulo

q

but addition is more complicated.

Both these representations of

are used

in the proof of Singer's theorem.

Table 5.1 shows both kinds of representation for

TABLE 5.1.

0

0000

O0

1

0000

Ol

o( 2 o~

J c~

6

J 8

2 lO

c~

~5

GF(26)

with

GF(qm)

f(x) = x 6 + x + i.

i01000

~i

i001

Ol

010011

~2

0010

0 i

48

001101

~49

011010

i00111

0000

i0

i00110

o~ 3

0100

io

0001

O0

001111

~4

i001

00

~0

ii0100

ooio

i i

~51

i01011

OlOi

i 0

~2

010101

~53

i01010 010111

0010

O0

011110

a35

0100

O0

iiii00

~36

i000

O0

0000

ii

22

iii011

~Y

lOll

0 0

ii0101

~38

OliO

1 l

J4

iioi

i o

~5

i01110

0001

i0

i01001

~39

0011

O0

010001

0

iOli

i I

J6

011111

0110

O0

i00010

~41

Olii

o i

~57

iiiii0

000111

J2

iii0

i o

~8

iiiiii

001110

~43 44

iioi

i i

iiii01

i011 OllO

o i 0 1

~59 6o (~ 61

ii00

i 0

~62

i00001

ii00

00

i000

ii

0001

Ol

25 26

28 c~

011100

0010

i0

iii000

J5

0101

00

ii0011

(~46

iii001 ii0001

- i

i01

GF(26).

[In this example the primitive root

no means necessary;

~

satisfies

That is, a primitive root of

will always satisfy an irreducible polynomial of degree polynomial is said to be primitive of degree find such polynomials.

polynomial of degree

m

m

such that

over

GF(p).]

The finite projective geometry

that in

dimension

and

Thus any set of

a subspace of 0

PG(N,q)

PG(N,q),

PG(N,q)

GF(q).

of dimension GF(q)

(baN, baN.l,..o,ba0) j + 1

GF(q);

such a

The trick is to

of dimension

J.

The

N - 1

Thus there are and

over

b ~ 0, b

(N + 1)-tuples determines

(qN+l _ 1)/(q - i)

subspaces of

( J + l _ 1)/(q - l)

sub-

Any two distinct hyperplanes

so they have

v = (qN+l _ 1)/(q - l)

k = (qN _ 1)/(q - i)

GF(q), con-

are identified for all

are called hyperplanes. N - 2;

N

subject to the restriction

linearly independent

intersect in a subspace of dimension points in common.

over

a primitive irreducible

are the point ~ of this geometry and the

spaces of dimension

planes in

over

pm < 109,

(N + 1)-tuples of elements of

(aN,aN_l,...,a0) GF(q).

m

m

GF(qm)

Alanen and Knuth (1964, Table 7) list, for each prime

and all exponents

sists of all

This is by

however it is often quite convenient and it is theoretically

always possible to arrange things this way.

p < 50

f(x) = 0.

(qN-1 _ 1)/(q - l) points and

v

hyper-

points in any hyperplane.

Singer

(1928) has shown:

Theorem 5.1. PG(N,q)

Considering the points as objects and the hyperplanes as blocks,

forms a symmetric block design with parameters given by (5.1) above.

block design is cyclic;

thus the points of any hyperplane determine a

This

v, k, h

difference set.

Proof.

The discussion above shows that

A,

the incidence matrix of this

configuration, satisfies

AAT=

(k - ~)I + kT,

~

= kJ.

Thus conditions (ii), (iii) of the block design definition follow from the result of Ryser (1950) discussed in section II.C. just below Theorem 2.3.

So

PG(N,q)

102

forms a symmetric block design as indicated. It remains to be shown that there exists a numbering of the points and hyperplanes of

PG(N~q)

which demonstrates the cyclic nature of the design.

be a primitive root of

G F ( J +l)

and let

~

satisfy the irreducible polynomial

f(x) = x N + I + c ~ x N + ... + C l X + c o

over

GF(q).

GF(q).

Then each power of

Since

vi

belongs to

~

(5.2)

corresponds to a unique

GF(q)

for all

i,

the elements of

are

point of

Thus there is a one-to-one correspondence between the elements

= 0,1,...,v - l)

PG(N,q)

an exponent

and the points of

~j+vi

GF(q)

and it follows that

~(j

and

(N + l)-tuple over

0,1,~v~...,~ (q-2)v PG(N,q).

~

Let

PG(N~q)

correspond to the same

which assigns to every point of

j, 0 < j < v - 1.

Consider the mapping

~:~

i

~

i+l

~:0~0

(5.3)

or in additive notation, using equation (5.2),

: (aN,...,al, a 0) ~ (aN~ I - aNCN,.-o,a 0 - aNc !, - aNCo) •

(5.~)

This mapping obviously maps points onto points and [as is clear from (5.4)] maps subspaces onto subspaces without any loss of dimension. planes into hyperplaneso that corresponding to If H

~

H,

~J~

the mapping

is not cyclic of order

and an integer

of

Since the point corresponding to

then

ts.

chosen;

so

s, 1 ~ s ~ v - l,

So

t

~j+iv

is cyclic of order

such that

~s

fixes

form an orbit in

H

H.

divides

v

on the points.

Let

under

i

~s

be a point with

is necessarily the least positive integer such that

This is, of course, independent of which element t

is the same as

on the hyperplanes then there exists a hyperplane

i, i+s ,o..,~ i+ts = ~ i

i( ts _ !) = 0. divides

v

~

Thus~ it maps hyper-

k.

Since

v

does not divide

s

i

of

H

v

was

(by assumption) the fact

lO3

that

v

divides

that

(v,k) > l;

necessarily and

ts

implies that

and as

this contradicts the fact that ~

is cyclic of order

Singer has been established. (say

(v,t) > i

~i,~,...,Gm),

v

t

divides

v - qk = 1.

k

Thus

on the hyperplanes.

it follows v

divides

s

So the theorem of

Thus, if one lists the elements of any hyperplane

their exponents form a difference set with parameters given

by (5.1) above. Any mapping

L

from

G F ( J +I)

onto

GF(q)

which satisfies

L(b~ + c%) = bL(~) + cL(v)

for all 6, ~ from

in

G F ( J +l)

GF(q N+I) to

GF(q).

such a linear functional in

PG(N,q).

and all

b, c

in

The set of elements L

[i.e., such that

Further, every hyperplane of

functional from

GF(q)

G F ( J +l)

to

GF(q).

~

is called a linear functional of

GF(q N+l)

L($) = 0]

PG(N,q)

N = 5

then

v, k, ~

6th degree polynomial (say

are

constitutes a hyperplane

is annihilated by some linear

Thus to apply Singer's Theorem one merely

computes the null space of a single linear functional. and

annihilated by

63, 31, 15

x 6 + x + 1 = 0)

and

~

over

For example, when

is to be a root of a primitive GF(2).

Consider the linear

functional

L

which maps each element into its right-most component.

shows that

L

annihilates

i

when

i

q = 2

is one of

Table 5.1

l, 2, 3, 4, 5, 7, 8, 9, 10, 13,

14, 15, 17, 19, 20, 25, 27, 28, 29, 33, 34, 36, 37, 39, 42, 46, 49, 50, 53, 55, 57. Thus these numbers constitute a difference set with parameters

Note:

v,k, 7~ = 63,31,15.

Singer's construction can be varied in several inessential ways, the

only effect of this is to generate a difference set equivalent to the original one. For example, if a different hyperplane (or linear functional) is used this merely shifts the set. G,

If the primitive root a t, tr - 1 mod(q N+l- 1),

the equivalent set

root is needed. is not in

GF(q)

rD = {rdl,.. o,rdk)

results.

In fact not even a primitive

It is only necessary to have an element for

i = 1,...,v - 1.

[The use of

difference set as that generated by the primitive root

is used instead of

~

6( = ~u)

for which

Gi

results in the same Gu+jv,

the existence of

104

which (for some value of

j)

is guaranteed by the condition on

G. ]

Extensive use will be made of the following well-known fact:

Lemma 5.2. GF(q m)

Let

L

be any linear functional, not identically zero, from

to its subfield

linear functional from defined by

GF(q j) GF(q m)

L~(~) = L ( ~ )

and let to

for all

~

be any element of

GF(q j) ~

is of the form

in

GF(qm).

GF(qm).

Then every

L~,

where

L~

is

Moreover if

~ ~ v

then

L~ ~ L w. [Linear algebra provides the fact that there are precisely functionals from

GF(q m)

to

GF(q j)

qm

linear

and the above process constructs

qm

distinct

ones. ] Complementary to any Singer set

D

is a difference set

D*

with parameters

N+l v =

for

N > i

and

q

q

- i q-l'

k = qN

h '

a prime power.

Here

qN-l(q_l)

(5.5)

=

j = O,l,..o,v- 1

belongs to

D*

if and

m

only if

L(~)

~ 0

root of

GF(qN+I)o

where

L

is a fixed linear functional and

Call this difference set

of generality in assuming that

L(1) = l,

D(L,~).

~

is a primitive

Note that there is no loss

for this may be arranged by taking a

different linear functional° The difference set of

D(L,~)

corresponding to the example given above consists

O, 6, ii, 12, 16, 18, 21, 22, 23, 24, 26, 30, 31, 32, 35, 38, 40, 41, 43, 44,

45, 47, 48, 51, 52, 54, 56, 58, 59, 60, 61, 62. that modulo 3,4,5~6,7,8)o

9

they constitute In fact

D(L,~)

4

Examination of these residues shows

copies of the trivial difference set

gives rise to the array of Table 5.2

E = [0,2,

i05

TABLE 5.2.

where the

(i,j)

that, for each

entry is i

J

0 1 2 3 4 5 6

i~= 0

i 0 1 0 0 1 1

1

D(L,G)

0 0 0 0 0 0 0 0 1 0 0 ! I i

3 4

0 1 1 1 0 1 0

5 6

0 0 1 1 1 0 1

7 8

0 i 0 0 1 1 !

0 0 1 1 1 0 1

i 0 1 0 0 1 1

0 0 1 1 1 0 1

in the difference set

one of its shifts. of

1 2

if and only if

characteristic function of the

D(L,~).

A Representation of

E,

i + 9J

belongs to

D(L,G).

Note

the rows of this array are the

w,~,~ = 7,4,2

difference set

F = [0,2, 5,6]

or

Gordon, Mills and Welch (1962) have shown that the structure

always depends upon difference sets

E

and

F

in this manner.

Specifically, they prove:

Theorem 5.3. N > 1.

Let

L

Let

q

be a power of the prime

p

and let

N + !

be a linear functional from the finite field

be an integer,

G F ( J +l)

to the sub-

w

field

GF(q),

such that

intermediate field

L(1) = lo

GF(qm),

where

functional which assigns to each GF(qm), Set

m ~

which satisfies the relation

polynomials of y = x ~.

GF(q N+l)

be the restriction of

divides in

D(L,~)

and

N + 1.

GF(q N+l)

Let

~

L

to an

be the linear

the unique element

L0(~(~)5 ) = L(~5)

~nd

and let

~ = ~.

D(L0,B)

respectively.

Let

for all

~ = vl~. @(x)

and

Let ~(y)

[For m = i,

~(~)

5

in ~

in

GF(qm). be a

be the Hall take

~(y) = i.]

Then

G(x) ~ ~(x) ~(y)

where

L0

v = (qN+l - l)l(q- i), w = (qm _ 1)l(q- l)

primitive root of

Let

Let

(mod

x v - l)

(5.6)

lO6

~(x)

: Z

xiy ri

(5.7)

and this summation is taken over those values of

~(i)

~ 0 ,

In the example above E

0 _~ i < { ,

r. 1

and

~

for which

~(czi) : B

N + 1,m,q,p = 6,3,2,2,

is the difference set determined by

The

i

-r. 1

(5,8)

F = D(L0,B) = {0,2,5,6]

and the extension

GF(qN+I)

determine the shifts associated with the various copies of

Proof. a~L(~8),

Consider 8

in

~.

Let

GF(qm),

~

be an element of

GF(qN+I).

is a linear functional from

Lemma 5.2, there is a unique element, call it

~(~),

of

GF(qm).

in the array.

Then the mapping

GF(qm) of

F

while

to

GF(qm),

GF(q).

So, by

such that

=o(Z(O~) = ~ ( ~ )

for all

5

in

GF(qm).

[L0

is not identically zero since

is a properly defined mapping from remains is whether or not of

GF(qm)~

a

and

b

~

fixed~ and let

= L((aq

:

~

to

~, ~

belongs to

L0(1 ) = 1 GF(q)

when

GF(qm).

that m : l,

L0

L(1) = I°]

+ b~)8)

Let

= L(aqS)

Thus

The only question which a, b~ 5 be elements

be fixed elements of

GF(q N+l)

then

+ L(b~8)

Lo((a~(q ) + bZ(~))8) .

is indeed a linear functional and in fact

follows from

Let

GF(q N+l)

is a linear functional.

Lo(~(a q + b~)8)

So

(5.9)

~(i) = i.

is the identity mapping on it follows from (5.9) that

When

m = i,

GF(q); ~ = L

it

since

in this case.

lO7

w-~

5jyj

~(y) =

{ 0 if L0(~J) = 0 with

j=O

5. = J

I if Lo(~J) ~ 0

and let

v-Iz 8(x) = i=0

" a.x ~ l

{ 0 if L(Gi) = 0 with

e.1 =

~-i = ~ xi~i(y)when

i if L(Gi) { 0

w-i o]i(y)= ~, si+~jy3o

i:o

j:o

Since ~ is a primitive root of GF(qTM)

every value of ~_LL(i) is either 0 or

-r.

a power of ~, say ~ I. Now si+__~j= 0 if and only if L(G i+~j) = 0 and

I

0

L(~i+~j) = L(~i~j) = L0(~((~i)~j) =

if

~(~) : o

if

Z ( i ) ~ o.

J_ri ) L0(~

So

o

if

Z(~ i) : o

3-r.1 Thus ~i(y) = 0 if ~(i) = O; while if ~(Gi) # 0

w-i ~i(Y) =

w-i

w-i

Z £i+~J yj = ~ 5J-r. yj = ~ 5J yj+ri = y 1 j=O j=O j=O •

~(y)

(mod

Xv

- l).

r.

Hence @(x) -~(y) 7. xly m the theorem is proved.

r. 1

where the sum is taken as prescribed by (5.8), and

108

Now

~(y)

is the Hall polynomial of the

has parameter values (for

w :

@0(y )

set

D(L0,~ ) which

m > !)

m q - 1 q-1 '

m-i ~ = q ,

Gordon, Mills and Welch show that, if polynomial

w~,~-difference

$(y)

~ : qm

-2

(q-l)

(5.10)

is replaced in (5.6) by the Hall

of an arbitrary difference set having parameter values (5.10),

i.eo, if

eo(X ) ~ ~(x) ~o(y)

then (5.5).

e0(x )

(mod

is again the Hall polynomial of a

x v - i)

(5.11)

v, k, h-difference set with parameters

For let

w-1 ~o(Y) =

be a Hall polynomial and let

y = x ~.

Z j:O

Let

~JYJ

G(x)

be given by (5.7).

k

e.

x

eo(X) = n(x) ,o(y) :

Then

i

i=l

where the modulo

x

e. I v

are distinct modulo

v

by construction.

Further~ by definition,

l,

Co(y) ~o(y -I) =- (~ - ~) + ~(i + y + ... + yW-l) _ ~(y) @(y-l) °

Hence, modulo

x v - i,

Io9

eo(X) eo(x'l) = n(x) ~(x-1) ~o(y) ~o(y-1) ~ ~(x) o(x"l) ~(y) ~(y-1) = o(x) e(~-1) .

Thus

el, ...,e k

example, let

[For

form a difference set with the parameter values of (5.5).

@0(y ) = 1 + y + y2 + y5

then using

G(x) = i + x2y 6 + x3y 3 + x4y 4 + x5y 4 + x 6 + x7y 6 + x 8 4

from set

D(L,~)

above, in congruence 5.11 yields the

D(L,~)

This difference set is not equivalent

as is explained below.]

Gordon, Mills and Welch show that two (5.11), are equivalent if and only if the

difference

0, 2, 6, 7, 9, ii, 12, 15, 16, 18, 22, 23, 24, 26, 30, 38, 39, 40, 41, 43, 44,

45, 48, 49, 50, 51, 53, 56, 58, 59, 61, 62. to

v,k, k = 63,32,16

w,~,~-difference

v,k,h-difference

@0(y ) = yS @(y).

sets are shifts of each other.

sets derived from

That is, if and only if

[Since

-1

is never a

multiplier of a cyclic difference set (section 3.1) the difference sets are never shifts of each other. generate inequivalent

Thus the polynomials

v,k,h-difference

the inequivalence of the two

@(y)

and

sets by these means.

63, 32, 16

@(y-l)

D

and always

This accounts for

difference sets mentioned above.]

Several lemmas are required for the proof of this result (which is Theorem 5.12 below). Following Gordon, Mills and Welch, let C = {Cl, C2,...,c ~}

be two

w,~,u-difference

B = {bl, b2,...,b ~} sets, and let

ci

,

i

be their Hall polynomials.

[If

m = i

and

i

let

~b(y ) = ~c(y ) = i.]

Put

-D

ii0

eb(~) = ~(x) %(y),

Then

eb(X )

say

B

and

such that

and

8c(X )

C.

If

are the Hall polynomials of two

B

(t~v) = 1

ec(X) = a(x) ~c(y).

and

~

are equivalent then there exist integers

Proof.

(mod

B

(mod x v - l )

~b(y ) ~ yS 9 c ( t )

(mod

and

C

such that

(5.13)

B

r.

and

x a 8c(X ) = Z xa+tiy

Let

j

be such an

i;

and

~.

a + th m j

modulo

~.

i

tr. i ~c(yt)

for which

~(i) ~ o,

then, comparing terms in (5.12)

i a+th t rh Wc(yt) x iy r. ~b(y ) ~ x y

Since

@b(y ) ~ yS @c(yt)

Now

s

yW _ l) °

are inequivalent then so are

where these summations are taken over those values of

where

and

By construction

•

where

and

(5.12)

r

O(x) ~ x r O(xt )

eb(X ) = Z xly I ~b(y )

0 < i < ~.

x v - i).

If (5.12) holds, then there exists integers

In particular if

a

and

eb(X ) ~ xaec(Xt )

Lemma 5.4.

v,k,h-difference sets,

(mod

x v - 1 = yW _ 1

(mod

x v - l)

this yields

(5.14)

yw _ i)

s = tr h - r. + ~-l(a + th - j). 3 @b(y) @b(y -I) ~ ~ - ~ + ~(i + y + ... + y w - l )

modulo

yW . i

and this

iii

implies that

~b(y )

is relatively prime to

yW _ i.

So from (5.12), i.e., from

~(x) ~b(y ) ~ xa~(x t) ~e(y t)

(mod

xv -

e(~)

(mod

xv

i)

and (5.14)

~

x a -ys

9(x t)

-

1)

and the lemma is proved. Let

Q = GF(q)

Lema

5.5.

and let

Q*

be the set of all non-zero elements of

Suppose (5.13) holds and

be an element of

GF(qN+I).

Then

~(w)

(t,v) = i.

Let

belongs to

Q*

~ = ~r,

Q.

and let

if and only if

w

~(h t)

does also.

Proof.

Since

~

is a linear functional from

G F ( J +l)

to

GF(q

TM)

it follows

that

~(x) = ~. xiyri = ~

x i+~ri =

~

xj

(5.15)

j in S

where

S

j

is the set of all

such that

~(aJ) = i,

O ~ j < qN _ 1.

is a primitive root of

Q*,

the effect of adding

by a primitive root of

Q*.

Thus (5.15) can be written as

V~

a(x) ~

~,

v

to

(mod

xj

j

Since

is to multiply

c~v ~(~)

x v - i)

j in S'

where (5.13)

S'

is the set of all

Z(~)

belongs to

len~na is trivial.

If

Q*

~ ~ 0

j

such that

~(eJ)

if and only if

belongs to

~(r+jt)

Q*,

does also.

the result follows by putting

~ = ~.

0 < j < v. If

~ = 0

the

112

Lemma 5.6. element of

Suppose (5.13) holds and

GF(qm)

if and only if

and let

~(q t)

~

belong to

belongs to

GF(qN+l).

Let

~ ~ 0.

belongs to

Q*.

By Lemma 5.5 this is true if and only if

Here

~(~w t)

~(~)

let

~ be an

belongs to

~Q*

~tQ..

Let

which is equivalent to

~ = r,

Then

Proof.

Q*~

Then

(t,r) = i.

~(~)

belongs to

L(~w t)

~Q*

if and only if

being a member of

belongs to

vtQ *

for some

v

in

first part of this proof

~(~)

is an element of

~(~-I)

~(~ t~-t)

belongs to

~tQ.. Next suppose

GF(qm).

If

v ~ 0~

~(~) = 0.

then by the

vQ*~ contradiction.

So

v = 0

and the proof is complete.

Lemma 5.7. Suppose that (5.13) holds and elements of

GF(q N+I)

ei, ai( 1 < i < s)

(t,v) = i.

Let

which are linearly independent over

be elements of

GF(qm)

s

~1,~2, .... ~s

GF(qm)°

be

Let

such that

t

s

(}.16) i=l

Then

a.

1

Proof.

belongs to

c~ Q*,

l

Since the

linear functionals

~i

K. J

i=l

i < i < s°

are linearly independent over

over

GF(qm)

such that

i

By Le~ma 5.2 these exists elements

u. J

of

K.

if

GF(qm)

( l ~ i~ j ~ s)

i = j.

can all be expressed in terms of

GF(q N+I)

there exist

such that

Z(uj q)= ~ij"

~,

that is there

i13

Then~ by Lemma 5.6j Now

~L(~ujt ~ ) = 0

~(uj Z e i ~i ) = cj,

if

i ~ j,

and

~(~u t ~ )

belongs to

Q*.

so that

~(~u~( ZC i ~i )t)

belongs to

c~ Q*.

On the other hand, using 5.16,

i

and thus belongs to

Len~na 5°8. for

c I

=

c 2

. . . . . t

al~ a 2

t % ai ~i = 0

with

ai

shows

that

=

0~

this

in

are linearly independent over

N + 1 >mo

Then

=

a 2

Let

as was to be proved.

~i' ~2' "'" Is

be a basis

are also such a basis.

GF(qm)o a I

Suppose (5.13) holds with

c~ Q*

(t,v) = 1.

then

and suppose of

is an element of

GF(qm);

c s

Lermma 5. 9 . G F ( J +l)

aj

~t,~t,...,~ ts

over

Let

~1~2 =~v,f:,...,~ s

So

Suppose (5.13) holds with

G F ( J +I)

Proof.

ajQ*.

Apply Lemma 5°7 with

. . . . .

GF(q m)

a s

=

0o

Hence

and form a basis for

(t,v) = i.

Let

w

OF( +I)

be an element of

(1 + ~)t = al + a2 t

for some elements

Q*.

Proof.

First assume that

a basis for

G F ( J +l)

over

~

is not an element of

GF(q m)

C 1 = C 2 = l, e 3 = o.o = c s = 0;

so

which contains i + ~ = Z c i ~i"

GF(qm).

Then there exists

~l = l, ~2 = ~° Moreover

Let

~'~''°"~s t

is

also a basis (Lemma 5.8) so

(I + ~)t = Z ai~ ~

with to

ai

in

GF(qm).

By Lemma 5°7,

a3 = a4 . . . . .

a s = 0,

and

al, a 2

belong

Q~.

NOW suppose

~

is in

GF(qm).

Let

~ be an element of

GF(q N+l)

such that

114

~({) = ~ and so

and

does not belong to

<(w) = w;

~(M) = O. element Q*

{

Now M.]

thus

N + 1 >m Now

for some elements

Z(~)=

~(~) = ~. Q*o

~(~ it ) Thus

M

~(1) = 1

[Such a

{

exists since

does not belong to

b, e

+ {)t)

(1) ~(~(1 + {)t)

(2)

Q*.

belongs to (1 + w) t

Q*

~(q(l + {)t)

(1 + ~)t =

Thus

: ~(~(h + c{t))

belongs to

and

guarantee the existence of such an

and so

of

~(1) = 1

GF(q m)

On the other hand, by the first part of this proof

~(~(i

and

and

where

~(1 + ~) = 1 + w

by Lemma 5.6.

b + c{ t

{ = w + M

GF(qm)o

while

= b ~(~) + c ~(~ {t)

~(G ~t)

is of the form

belongs to a I + a2 t

tQ.

since

with

al; a 2

in

Combining this with (i) above completes the proof. Lemma 5.9 is the first major step in the Gordon, Mi!is~ Welch proof of Theorem

5o12 below.

To complete the proof some results about the implications of the

condition

(l + ~)t = al + a2 t

are required. every pair such that G F ( J +I)

If (5o17) holds for all w

PI' 02

of elements of

bl, b2,...,b u

(P I + 02 + ... +

GF(J+I),

GF(qN+I),

(PI + P2 )t = b! P~ + b2 P~" there exist

in

then it follows that for

there exist elements

By induction, given any

in

Ou)t

(5.17)

Q*

bl, b 2

pl, P2,.o.,pu

of

Q*, in

such that

t + t t = blP I b2o 2 + ..° + buO u o

(5.18)

Write (5.17) in the form

(I + ~)t = rw(l + s

with to

rw, s w Q

in

Q*.

Since

then neither does

w

t

(t,v) = i, and hence

t)

it follows that if rw, s

w

does not belong

are uniquely determined°

ll5

Lemma 5.10. elements

Let

N ~ 2

al, a 2

of

Q*

(i)

if

wt

T t ~ ~t

(ii)

if

G

and, for every

in

GF(J+i),

such that (5.17) holds.

is a primitive root of

For uniquely determined

let there exist

Then

are linearly independent over G F ( J +l)

which are not elements of Q, s

Proof.

w

Q~

and if

= 1 and

then s

s~/w = s~/T s /w.

= l,

then for all

w

(1 + w) t = 1 + wto

bl, b2, b3, Cl, c2, c3

of

Q*

blG~t + b2 Tt + b3~t = (~ + T + ~)t = Cl(W + T)t + c2~t

= clwt(1 + T/w) t + c2 ~t = c3(wt + sT/w Tt) + c2 ~t .

So

s iw = b2/b I

and by symmetry

S~l ~ = b3/b2, S~l w = b31b I

which establishes

part (i) of the len~ao Consider the

sw = i

assertion of (ii).

is not contained in any proper subfield of over Put

Qo

Hence

w = u,

i, t

and

I ~ u ~ J+l

2t

. I

Since

(t,v) = I

GF(J+I).

Hence

it follows that t

has degree

are linearly independent over and induct on

s 2 = s

s

Uo

Q

(as

G

t

N + i

N + i ~ 3).

By part (i) it follows that

= i

G

so

or

2.

Let

positive integers less than

uo

Since

as

sw = i

when

(t,v) = i.

i, t ,

ut

u = i

So

i

and

wt

or the elements

u ~ 3

and suppose that

sW = i

w

is not an element of

Q

are linearly independent over

i, G 2t, Gut ,

Let

i, ~ t ,

Gut

be independent

(j = i

and the induction hypothesis

S

=S

=S

neither is

s.=l

or

i, t , 2),

t w ,

Now the elements

are linearly independent over

if both these element sets are dependent then so is the set tradiction).

Q.

for all

G2t

Q;

for

(con-

then by part (i)

116

and so the first assertion of (ii) has been established. Since over

Qo

N + 1 > 3,

there is a

Then~ for suitable

~

such that

Cl, c2, c 3

in

l~ t

~t

are linearly independent

Q*.

(1 + ~ + {)t = Cl + c2 t + c3{t

with

Cl, c2, c3

uniquely determined.

by the linear independence of Qo

So

(i + ~)/~

is not in

i, t , Qo

Since ~t

s

= i, (i + w) t = rw(l + t ) ,

it follows that

a = r(l+~)/~.

and from this that

Theorem 5.11.

Hence r e = 1.

Let

v = (qN+l _ l)/(q - i) that for every such that

Proof. where

G

that and in

~

+ ar t

let

and let

be an integer relatively prime to

t

q

be a power of the prime

there exist non-zero elements Then

t

Without loss of generality

0 < t < v

and

is a fixed primitive root of

GF(qN+l).

Since

(t',v) = lo

Furthermore~ for any

w

p,

let Vo

al, a 2

Put in

p

GF(q)

modulo

v.

(i + G) t = r (i + sGGt), s~

t' = t + vc.

G F ( J +l)

Suppose

in

is congruent to a power of

e, 0 ~ c < q - 1.

cI = c3

as was to be shown.

N + i > 3,

G F ( J +l)

for some

+ a~t

By the same process it follows that

(1 + ~)t = 1 + t

(1 + ~)t = al + a2 t.

sG = v c

Q*

in

c I = c2° So

is not in

Thus

(i + w + ~)t = a(1 + ~)t + a~t = a r

where

(i + ~)t/~t

and

is in

Q*

it follows

Then 0 < t ' < J + l - 1

there exist

r' W

and

s' W

such that

(l+~)t'=r~(1+s'~t')o Note that

s' = io

Thus by part (ii) of Lemma 5o10

(i + w) t' = I + w

t !

(5o19)

117

for all

w

power of

of po

which are not elements of

q

N+l

- q

(5.19) by

roots°

Therefore

~u(1 + ~)u

yields

t' > q

= (l+~)~u

for all

~

of

G F ( J +I)

q

N+l

< 3q - 2

power of

p

the proof.

and

N + 1

N + I~ ~(x)

N+l

Let

t' +

u >

(Gordon, Mills and Welch)

~(y),

there corresponds a If

B

and

equivalent if and only if

Proof.

with

Start with respectivelY.

(t,v) = 1

Hence, since

B

B

and

Hence

(qN+l

2u k J + l

t' - l,

p

_ qo

So

- q)

N + 1 ~ 3, q ~ 2o

If

modulo

Thus

v,

t'

is a

which completes

q

be a power of a prime

N + 1 >m

~ 23

where

C

are

B

C

let

m

set

v,k,h-difference

set B

w,~,~-difference

sets then

B

and let with Hall

with Hall polynomial B

and

C

are

Co

and construct the Hall polynomials

is equivalent to

and

divides

~ = v/w,

w,~,~-difference

p

~

Ob(X), Oc(X )

then there exist integers

a,

such that

N + i > 3,

C

Let

is a cyclic shift of

and

is congruent to a power of B

~

To any

Ob(x) =- xao c(x t)

of

is not a

N+l u = q - 1 - t t.

Let

be given by (5°5) and (5.10),

be the polYnomial given by (5.7)°

B, C

- qo

Qo

is congruent to a power of

v, k, ~, w, ~, ~

O(x) = fl(x) @(X~)o

t

- 1 =

be a positive integer such that

polynomial

of

t'

Finally, the result promised above can be established.

Theorem 5.12. let

Suppose

+ (l+~)u

which is impossible since t

N+l

which do not belong to

q

or

Q.

Then (5o19) becomes a polynomial equation of degree at most

with at least Multiplying

GF(q N+I)~

(mod

x v - l) o

Lemma 5.4, Lemma 5.9 and Theorem 5.!1 establish that p

modulo

vo

Now every power of

p

t

is a multiplier

(applY Hall's Theorem 3.1 above to their complements).

So

ll8

*c(y t) -yU*c(y ) for some integer

Uo

Thus by Len~na 5°4

*'b(y) ~ y

and

B

(rood yw_ i)

is a cyclic shift of

¥c~y )

C

(mod

yw _ i)

as promised.

In the course of the above proof it was concluded that, for

Gb(X ) ~ xaGc(Xt )

only happens when

t

(mod

is congruent to a power of

same observation shows that only powers of sets the

~o

With

D(L,~)

powers of p

m = 1

p

p

x v - l)

modulo

v.

If

B = C

p

may be multipliers of

As mentioned earlier, Theorem 3ol shows that the

are always multipliers of these difference sets~ so:

Theorem 5.13.

(Gordon, Mills and Welch)

difference set or if

D

If

D

is a non-trivial Singer

is any non-trivial difference set derived from congruence

5°6, then the multipliers of

D

are precisely the powers of

p

modulo

Another by-product of Theorem 5o12 is the existence of parameters for which at least and let

J

j > 0

inequivalent difference sets exist°

denote the number of inequivalent

given by (5o10)o

Since

-1

2J

which is a cyclic shift of any other. v, k, Z-difference sets with

Theorem 5o14o

of

v, k, h

possible

v, k,

For, let sets with

w,~,~-difference

Hence there are at least given by (5.5) or (5.1)o

(Gordon, Mills, and Welch)

be positive integers with r

w,~,~-difference

vo

m _> 3~ w, ~,

is never a multiplier of a non-trivial difference set

(see Theorem 3~3) there are at least

m, M

this

may be multipliers of the difference

this shows that only the powers of

difference sets.

N+I>3,

m > 3.

Let

Let

q

inequivalent

In particular

be any prime power and let

N + 1 = ~u

prime numbers, not necessarily distinct.

2J

sets, none of

and let

M

be the product

Then there exist at least

2r

ll9

inequivalent

difference

sets with parameters

(5.5) and thus

(by taking

complements)

with parameters

N+I v_q

B.

Nthpower

Residue

A difference or of the

-1 q-l'

Difference

set which

is some reason for distinguishing

restricted

to divisors

residues

and the

that the

f

of non-zero venient

5.15.

form a difference

Proof. residue

(N, v - l )

coincide°

rl, oo.,r f

for the number theory required difference

When

is a prime,

set with parameters

residue

srl, sr2, sr

r,

say

[Nagell

residue

r I - r 2 =- ro

If

residues

Thus one of s

also.

equally often as a difference

and vice versa.

r 2 - r I - -r)

residue with the same number of representations v , k = 4t - l, 2t - l;

The set

~1,2~4~

modulo

7

thus

The best known (1933):

residues modulo

v

is a quadratic r I - r 2, r 2 - r 1 residue of

Thus eve~# equation

v

set w i t h

the fact

(1951) is a con-

is any quadratic

residue of

(e.g.,

N th power

sets of Paley

c, -c

st. - s r . -= sr l j

these congruences

sets,

v, k, h~ n = 4t - l, 2t - l, t - l, t.

with an equation

Reversing

set.

form a subgroup of the group

corresponds

is represented

the

v = N f + l,

the quadratic

1 < c < v - lo

are quadratic

v,

Attention m a y be

in this section].

sets are the quadratic

v = 4t - 1

some prime

difference

= d >l

For primes

w i l l be used extensively

for a prime of this form,

then

the two types°]

Recall from number theory that exactly one of

is a quadratic v,

v

N th power residue

since if

N th power residues

residues modulo

N th power residue

Theorem

d th power residues

distinct

reference

v - l,

N th powers modulo

N TM power residue difference

is called an

when there

of

1 q-l"

Sets and C~clotomy

zero are called modified

N

N-I h=q

is composed of all the

N th powers and zero,

[Those containing

N q-1 q-l'

k

yields

also.

r. -r. -=r

Hence every quadratic of quadratic

residues.

every quadratic

non-

Hence this is a difference

h = t - lo

is the first non-trivial

quadratic

residue

120

difference set.

Note that the parameters imply that every quadratic residue set

is a Hadamard difference set (see section IV.B. for a discussion of this special type). Another class of residue difference sets was discovered by Chowla (1944):

Theorem 5o16o

The biquadratic residues of primes

a difference set with parameters

v = 4x 2 + l, x

v,k,% = 4x 2 + l, x 2, (x2 - 1)/4.

The first non-trivial biquadratic residue difference set is for v,k,~ = 37,9,2 Other

and

odd, form

x = 3,

here

D = {1,7,9,10,12,16,26,33,34}.

N th power residue difference sets have been discovered and a general

theory for them has been developed by ~mma Lehmer (1953)o

In order to explain her

results a little of the theory of cyclotomywill have to be introduced.

No more

cyclotomy than is necessary for an understanding of the difference set results of sections V.B., V.Co, and V.Do is developed here.

[A complete introduction to

cyclotomy from the classical point of view is given ab initio in Dickson (1935 ABC); T. Storer's booklet "Cyclotomy and Difference Sets" (1967 A) gives a different development also ab initio and discusses most of the difference set results of these sections.] Let

v = Nf + 1

An integer

R

number R

f

g

be a fixed primitive root of

is said to belong to the index class

exists an integer sists of

be an odd prime and let

x

such that

distinct numbers

(~,m)N

R ~ gNX+~ (mod v).

~.

That is,

R + 1

(~,m)N

with respect to

g

if there

Thus, the index class

g~,g N+~, .o. ,gN(f-1)+~

counts the number of times

belongs to index class

~

modulo

v.

Vo

~

con-

The cyclotomic

belongs to index class is the number of solutions

m

when x, y

of the congruence

g

where the integers

Nx+~

x, y

that there are at most

+ 1 ~

gNy+m

(mod

are chosen from N2

numbers depend not only on

0,1,...,f- 1.

v)

This congruence shows

distinct cyclotomic numbers of order v, N, ~, m

(5°20)

but also on which of the

N

and that these ~(v- l)

121

primitive roots

g

of

v

is chosen.

The following elementary cyclotomic facts are all that is needed for an understanding of the results of this section (note that when

f

is odd,

N

is

necessarily even):

(~,m)N = (~',m')N

when

~ ~ ~'

and

m ~ m'

(mod

(m,~)N

N)

f

even

f

odd

(5.22)

(~,m)N = (N - ~, m - ~)N = I

(m + N/2,

l

N-I Z

(~,m)N = f - n~,

where

m=0

~ + N/2) N

II

~ ~ 0 (mod N)

f

even

L

~ ~ N/2 (mod N)

f

odd

n~ = < 1 0

then

(~,m)~ s

is prime to

Proof. nition.

is based on the primitive root

(5°23)

otherwise

(5.24)

(~,m)~ = (s~,sm)N

where

(5.21)

g' ~ g

s

modulo

v;

necessarily

v - lo

Equation (5.21), of course, is an immediate consequence of the defi-

The first part of (5.22) follows from the definition after congruence

5.20 is multiplied through by the inverse of its first term

[i.e., by

gN(f-x)-~].

Similarly, the second part of (5.22) follows after congruence 5°20 is multiplied through by

-1,

that is by

-i -= g(Nf)/2 = ~

gN(f/2)

f

even

(5.25)

L gN(f-1)/2

+N/2

f

o~d .

The sum in equation (5.23) is simply the number of successors of members of index class

~ which belong to any index class at all.

Since

-1

is the only element

122

whose successor does not belong to an index class, equation correctness of (5.23).

Equation

to

sx

v - 1

implies the

(5.25) implies the

(5.24) follows from the definition,

ranges with

x

since

s

prime

over a complete set of residues modulo

fo In terms of these cyclotomic n~nbers

(~,m)N

and sufficient conditions for the existence of

it is possible to give necessary N th power residue difference sets;

in fact:

Theorem 5.17o

(Lehmer,

1953)

N th power residues of a prime even,

f

Necessary and sufficient conditions,

v = Nf + 1

form a difference set, are that

N

is

is odd and that

(~,o) N :

(f - I ) / N

for

The parameters of such difference sets are

~ ~ o , I .... , ½ N -

v,k, ~ = v,f,

(5.26)

l o

(f- I)/No

Necessary and

sufficient conditions, that the

N th power residues and zero for a prime

form a difference set, are that

N

be even,

1 + (0,0)N = (~,0)N = (f + I)/N

The parameters of such difference sets are [Note that equations

f

for

~ = 1,2 .... , ½ N - 1 .

(~,0)N

N th power residues

then there are exactly

h

solutions

r i - rj ~ 7

and hence the congruence

(5.27)

v,k,Z = v, f + i, (f + I)/No

(5.22) and (5°24) show that these existence conditions

individual cyclotomic ntnnbers

If the

v = Nf + 1

odd and that

(5°26) and (5.27) are independent of the primitive root

Proof.

that the

g,

even though the

are not°]

rl,.oo,r f ri, rj

are a difference set modulo

v3

to the congruence

(mod

v)

(5.28)

123

riril

-= Trj I + i

for all

7 ~ 0

(c,0)N = ~; as

thus

v.

But for

(i,0)N = ~

h = (f - 1)/N

is an f

modulo

for

follows from

(mod

T

in index class

0 < i < N - 1

v,k = v,f.

If

N th power residue, hence a multiplier,

is odd and as

and

f

all

i.

Nf

is necessarily even,

is odd, equation Hence

That is, the v,k,~ = v,f, When

0

(5°22) provides

v

c

(5.29)

this implies that

and in particular f

so is

(5.26) holds

is even~ then by (5.25),

which contradicts N.

(f - I)/N

form a difference

-1

Theorem 3.3~

Conversely,

since

0)N : (f-

(i,0)N = (i + 2 '

(5°29) and (5.28) have exactly N th powers modulo

v)

N

Thus is even

I)/N

solutions for all

for

7.

set with parameters

(f - I)/N. is added to the set of

N th power residues the only effect is that

differences

r. - 0 = r. 1

and

l

have to be counted alsoo

As before,

So,

is even) and

each

f

is odd (hence ri

(N/2, 0)N

and

-r i

In particular,

v,k = v, f + lo

odd and that equation (5.22)] it follows, difference

f

l

even implies that -1

(5.27) holds.

equation

Since

that the

v,k = v, f + i;

(5.27)holds,

[1,2,4},

(ii)

v = 4x 2 + i, x

IN = 4, Chowla odd;

e.go, the

and (f+l)/N

is even, here

f

is

[by equation

and zero form a

N,

i.eo, those for which

with the following results:

sets exist which consist of (i)

the quadratic residues of primes

~=

thus

h = (f + I)/N.

Theorem 5.17 has been applied for several values of

Difference

N

N (i + ~ , 0)N = (i~0)N

hence

N/2;

(0,0)N

since

N th power residues

the eyciotomic numbers have been computed,

is a multiplier°

belongs to index class

On the other hand, assume that

as before,

set with

Theorem 5.18.

-i

is represented once more than the numbers

indicate.

follows from

N

0 - r. = -r.

l

v = 4t - i;

e.g., the

(1944)] the biquadratic v,k,Z = 39,9,2

set

IN = 2, Paley

v,k,h = 7,3,1

(1933) ]

set

residues of primes [1,7,9,10,12,16,26,33,34],

124

(iii)

[N = 4,

Lehmer

(1953) attributes

residues and zero for primes [0,1,3,9),

(iv)

[N =8,

v = 4x 2 + 9, x

Lehmer

1 = 64b 2 + 9, k = a 2, ~ = b 2 [1,2,4,8,16,32,37,55,64), (v)

IN = 8,

Lehmer

these to M. Hall, Jr. ] the biquadratic odd;

e.g., the

(1953) ] the octic residues of primes

with

a, b

odd;

e.g., the

the next such prime

v

is

v,k,h = 26041,3256,407

odd,

b

(by these methods)

even;

v < 34,352,398,777.

it can be shown that n__oomodified quadratic

sets exist and further that no non-trivial

residue difference

sets exist when

(1966), of

v.

(x)

(viii)

N = 16,

The case where

2

(vl)

N = 6,

N = 12, Whiteman

Whiteman

Lehmer

is not an octic residue is open;

(xi)

(1967),

Baumert and Fredricksen

(submitted for publication) The case where

5

for the case when

is not a biquadratic

The calculations

(ix)

(1957) for the case when

set here then (xii) 5

residue or modified

(1953),

(1960A),

that if the residues form a difference N = 18,

v = 8a 2 +

e.g., the

residue difference

(1960 B),

set

140,411,704,393 ,

set, there are no more such primes

On the other hand

set

v = 8a 2 +

v,k,h = 73,9,1

(1953)] the octic residues and zero for primes

49 = 64b 2 + 441, k = a 2 + 7, ~ = b 2 + 7, a

Whiteman

v,k,?~ = 13,4,1

2

(vii)

N = 10,

N = 14, Muskat is an oetic residue however,

it is known

v > 1,336,337. N = 20,

Muskat and Whlteman

is a biquadratic

residue of

v.

residue remains unsolved.

involved in determining

the cyclotomic numbers of these

orders are far too extensive to present here, the numbers themselves may be found for

N = 2 - 6

in Dickson

papers cited above for By way of example,

N = 8

in Lehmer

though~

if

thus

N = 2

and

(0,0)2 = (1,1)2 = (1,0)2 = ( f -

v = 2f + l, f

quadratic residue difference

f

odd, equation

1)/2

and

(5.22)yields

In the case

is more difficult;

(0,1)2 = (f + 1)/2.

So

set [ i.e., an alternate proof

and (5.27) prohibits

sets.

which may be

hence the quadratic residues of

odd, form a difference

of Theorem 5.15 has been given],

the cyclotomic numbers

and in the

(5.23) provides two linear relations,

(5.26) of Theorem 5.17 provides no restrictions, every prime

(1955 B),

N = 10,12,14,16~ 18, 20 o

(0,0)2 = (l~l)2 = (1,0)2 solved for

(1935 A),

the existence

N = 4, f

however,

they are

of modified

odd, the computation of

125

(0,0) 4 = (2,2) 4

:

(2,0)4

= (v - 7 + 2y)/16

(O,l) 4 :

:

(3,2)4

= (v + 1 + 2y - 8x)/16

(1,2) 4 = (0,3) 4 = (3,1)4

= (v + 1 + 2y + 8x)/16

(1,3) 4

(0,2)4

and the rest are

(v - 3 - 2y)/16,

where

Thus (5.26) of Theorem 5.17 requires

= (v + l

- 6y)/16

v = y2 + 4x 2

and

y ~ 1

v - 7 + 2y = v - 3 - 2y = v - 5

[That is, a proof of Theorem 5.16 has been given.]

or

f = 2j + l, x

Y = -3,

then

thus

v : 9 + 4x2.

v = 8j + 5 : 9 + 4x2.

Since

Thus

or

4. y = 1.

The existence of modified

biquadratic residue difference sets requires, by (5.27), that 2y = v + 3

modulo

v + 2y + 9 = v - 3 -

v = 4f + l, f

odd, let

8j - 4x 2 = 4, 2j - x 2 = l;

hence

is odd, which establishes part (iii) of Theorem 5.18. A completely analoguous theory of cyclotomy can be developed for prime powers

v = p (for

i

= Nf + 1

[see, for example, Hall (1965) or Storer (1967 A)].

N = 2,4,6,8)

in the same conditions for

residue difference sets as before. are not cyclic and, in fact, for

However N = 4,8

Theorem 5.18 are only satisfied for

N th power and modified

(for

i > l)

residues plus for

00

N th power

these difference sets

the quadratic conditions imposed by

i = l,

i.eo,

v

is prime.

quadratic residues, this was established by A. Uo Lebesque rediscovered by Hall (1965).

This results

[For the bi-

(1850) and recently

Hall's paper also contains a proof for the biquadratic

Storer (1967 A, Theorem 20) states this result without proof

N : 8.] The multiplier group for

N th power residue difference sets has been de-

termined:

Theorem 5.19.

(Lehmer, 1953)

multipliers of a non-trivial

Proof.

Clearly the

another multiplier,

then

The N th power residues themselves are the only

N th power residue difference set.

N th power residues are multipliers. tD ~ D + s

modulo

v,

where

t

Suppose

t

was

belongs to an index

126

class

j,

with

j ~ 0 (mod N)o

trl,.o.,trf;

so

0

D, r

is not in

s ~ 0 u

modulo

+ s ~ tr

r s u

has exactly

f

In fact, the index class

-1

solutions,

y

v.

(mod v)

+ i ~ tr s y

If

(5.26) when 0

D

belongs to

belong to index class

-1

(mod

(i,0)N = 0,

i.

Then, if

v)

(N - i,j - i)N = (i,J)N = f.

which contradicts the existence

is non-trivialo D,

Thus, by (5.25),

then s

tD ~ D + s

Do

tr u

-

-1

is a multiplier also, contradiction.

for some

consists of

implies

to

s ~ 0

s

i.eo, the cyelotomic number

But this implies, b y (5.23), that condition

Let

j

u;

(with

belongs to index class thus

t

and

-1

s ~ 0) N/2.

implies

0 - s

belongs

On the other hand,

belong to the same index class°

As mentioned in sections I.Eo and III.Ao above, every divisor of is a multiplier for all known cyclic difference sets.

Hence

n = k - h

This is not readily apparent

for the residue difference sets of Theorem 5.18, but was established by ~mna Lehmer (1955 A).

First consider the quadratic residue difference sets for prime

for these it is necessary to use the law of quadratic reciprocity Nagell

(1951)]o

whether

(= +l)

As usual, the Legendre symbol or not

Theorem 5.20.

(=-l)

p

(p/q) = ± 1

All divisors of

v,

are quadratic residues.

(mod 8),

q.

are multipliers

v = 4t - lo

By Theorem 5.19, it is sufficient to show that all divisors of

quadratic residues modulo n

will be used to indicate

n (= k - h)

for the quadratic residue difference sets for primes

Proof.

[see, for example,

is a quadratic residue of the odd prime

(Lehmer, 1955 A)

v = 4t - l;

but then

2

an odd prime divisor of

Now

which, of course, n = (v + 1)/4

(and thus of

are

follows if the prime divisors of and

2

divides

n

only if

is a quadratic residue by the reciprocity lawo n

n

v + 1 = qm)o

Let

v ~ -1 q

be

By quadratic reciprocity

127

i.e.,

q

is a quadratic residue of

v.

More generally there is:

Theorem 5.21. cyclic

C.

Nthpower

(Lehmer, 1955 A)

All divisors of

n

are multipliers for the

residue difference sets of Theorem 5.18.

More Cyclotomic Difference Sets Denote each index class

i

by

Ci,

then if

v = 31

and

N = 6

these

classes are:

Letting

CO + C I + C 3

CO + C I + C3

CO:

i,

2,

4,

8,

16

CI:

3,

6,

12,

17,

24

C2:

5,

9,

10,

18,

20

C3:

15,

23,

27,

29 ,

30

C4:

7,

14,

19,

25,

28

C5:

ii,

13,

21,

22,

26

denote the union of these classes,

forms a difference set with parameters

it can be verified that

v,k,Z = 31,15,7.

This is a

special case of Theorem 5.23 given below°

Note that if a union of index classes of

the

v = Nf + i

N TM power residues of some odd prime

it is immediate that all the N th power residue if Nthpower for

f

f

N th power residues are multipliers.

is even (in fact, all solutions of

xf ~ i

Since

-I

modulo

is an v

are

residues) such a union of index classes can only form a difference set odd (see Theorem 3.3 above).

Lemma 5.22.

is odd and

N

Thus:

A union of index classes

power residues of some odd prime

(with or without

v = Nf + i

Ci+ s + Cj+ s + ... + Cm+ s

added) of the

N th

can form a difference set only if

C. + C. + o-o + C m G m for any

s;

f

forms a difference set then

this second difference set can be

obtained from the first by multiplying b y any element of to the first.

0

is even°

If the union of index classes so does

form a difference set, then

Cs

and thus is equivalent

(This fact considerably shortens any exhaustive examination of index

128

class unions.) Let

N = 2,

then with

k < v/2,

as usual, only

CO

need be considered and

this leads to the quadratic residue difference sets of paragraph V.B. N = 4

(with

k < v/2)

only

C0, C 0 + O, C O + C 2

and

CO + C 1

For

need be considered.

The first three of these correspond to the biquadratic residues, the biquadratic residues plus

0

and the quadratic residues;

which of course constitute difference

sets under the conditions given in Theorem 5.18 above.

The cyclotomic numbers

discussed in section V.B. will be used to rule out the remaining possibility. In general,

congruence 5.20 shows that the cyclotomic number

(~,m)N

is the

number of solutions of

- ~ -= 1

for

~

in

C~

and

~

in

Cm.

Thus

(mod

(~-

s, m -

v)

is the number of solutions

S)N

of

g

Nx+~-s

g ~+m-sNv =- i

(rood v)

and therefore of

g

for fixed

d

in

C . s

Nx+~

-

gNy+m

(rood v)

md

Thus, the number of solutions of

- B ~ d

for fixed

d

index classes d

in

C

s

is

in C

C

s

with

. z l' "°'Cz h'

in

CI, G

(mod

in

Cm

is

v)

(~ - s, m - S)N.

(5.3O)

So, for

the nt~nber of solutions of congruence 5.30 for fixed

129

h Js =

h

~

I

i=l

j=l

Thus a difference set corresponds to

(zi - s, zj

C

+ ... + C zI

J0 = J1 . . . . . so only

JN-1 = %°

J0 . . . . .

Since

f

J(N/2)-l = %

Now consider

CO + C 1

(5.31)

S)N

if and only if zh

is odd,

Ji = Ji+N/2

(by equation (5.22))

need be checked.

for

N = 4,

then

Jo = (0'0)4 + (1'°)4 + (°'1)4 + (i'1)4 = (4v - 12 - 8x)/!6 Jl = (-i''i)4 + (0''1)4 + (-1'0)4 + (0'0)4 = (4v-12 + 8x)/16

as the values of x = 0;

but

CO + C1

N/2

given in section V.B. show.

v ( = y 2 + 4x 2)

is

0 - d

Ci

thus

in the union for which

Theorem 5.23. v = 6f + 1

Js i ~ s

Since

f

is odd,

and

C3

should be increased by modulo

belongs to class 1

for each index

N/2. N = 6

v - 3

modulo

4

or

(2)

for an appropriate choice of primitive root 4x 2 + 27.

and

established:

the index classes g

of

v

whenever

g

puts the residue

3

in

C1. )

N = l0 the same technique was applied by Hayashi (1965) yielding:

A set of residues forming a non-trivial difference set modulo

v = 10f + 1

v

The only possibilities are equivalent to one of the

(The appropriate primitive root

Theorem 5.24. a prime

-1

which includes the sextic residues as multipliers may consist

is representable as

For

for which

A set of residues forming a non-trivial difference set modulo

of (1) the qudratic residues for

above.

v

only if

N = 4.

Hall (1956) applied this technique to the case

CO, C 1

J0 = J1

then, so there is no prime

must be counted also.

(congruence 5.25 above);

a prime

Thus

0 is to be added to the index class union~ differences of the type

and

class

y2

form a difference set with

When d - 0

(~,m)4

which includes the

l0 th

power residues as multipliers may

13o consist of classes

(1) the quadratic

CO

and

Yamamoto

CI

index classes

His test has the advantage

5.25.

Let

residues° Let or

g = ii

0

N.

when

added)

be prime with

fth power residues

be the set of all residues

or

i

and define D

or

(2)

the index

v = 31o s

such

forms a difference

The initial form of his result

E

Then the set

4

D = E

of

is a difference

of

N v,

D = E

even and

v)

with

f

odd°

exactly

such that

0

of the

is:

containing

a (mod

N th

set°

that it does not require the prior determination

Let

d = 0 I.

root

modulo

a test for deciding whether a union of

v = Nf + i

be a subset of the set of

v - 3

(with or without

numbers of order

Theorem

for

for the primitive

(1967) developed

power residue

cyclotomic

residues

a

f

Let s

B such

is in

added according

as

Bo d = 0

set if and only if

s(sf + 2d - l) ~ 0

(mod

( jf

if)

N)

(5.32)

(rood v)

(5.33)

i=O

for of

j = 2,4, ~o.,N - 2,

where

Kr

is the sum of the

r

th

powers of the elements

Bo After modifying

N = 4,6,8,10,12

Theorem

congruence

5.26.

A set of residues

v = Nf + i

residues

as multipliers,

residues

(with or without

for

v = 4x 2 + 27

forming a non-trivial

N = 4~6,8,10

condition,

non-trivial

residue

or

12,

0),

the octic residues

or the special

31, 6, I Yamamoto

and, b y way of example, or modified

residue

difference

which includes

m a y consist of the quadratic

In a further attack on the problem, different

applied this test for

w i t h the result:

a prime

sets for

5.33, Yamamoto

residues,

the

set modulo

N th power

the biquadratic

(with or without set of Theorem (1969) develops

0),

the Hall

5.24. a slightly

uses it to show that there exist no

difference

sets for

N = 6,10,14.

[l.e.,

131

he reestablishes parts

(vi), (vii),

(ix) of Theorem 5.18, section V.Bo above.]

In addition to the above facts it is known (Baumert and Fredricksen, that there are

6

index class unions of the

which lead to inequivalent

v,k,h = 127,63,31

18 TM power residues for difference sets.

For

v = 127 N = 6

(1965) developed the requisite prime power cyclotomy and generalized his difference sets to prime powers

v.

However,

W. H. Mills], the only prime powers

4x 2 + 27

sets,

v = 4x 2 + 27

as is shown there

Hall

v = 4x 2 + 27

[proof due to

are in fact primes.

These difference

of Hall, were examined from the point of view of the multi-

plier problem and it was found that indeed every divisor of these sets.

1967)

n

was a multiplier for

[Emma Lehmer (1955 A) gives a proof of this fact and attributes an

earlier proof by means of cubic reciprocity to Hall.]

D.

Generalized Cyclotomy and Difference Sets In 1958 Stanton and Sprott (1958) published a generalization of the following

result:

Theorem 5.27 . and

p + 2

Let

g

are both prime.

be a primitive root of both

p

and

p + 2,

where

p

Then the numbers

2 1, g,g2,...,g(p - 3 ) / 2

0, p + 2, 2(p + 2),...,

form a difference set with parameters

(p- 1)(p + 2)

v,k,h = p(p + 2), (v-i)/2,

(v-3)/4,

i.e.,

a Hadamard difference set. These difference sets (the so-called twin prime sets) were in fact already known, although in slightly different guise.

They had been independently discovered

not only by Stanton and Sprott but also by Kesava Menon (1962), Brauer (1953), Chowla (1945), perhaps first by Gruner (1939) and probably a few others, as they seem to belong to that special class of mathematical objects which are prone to independent rediscovery. Motivated by Theorem 5.27, Whiteman the

k = d + p

numbers

(1962) investigated the problem of when

132

l,g, g2

consistute

for

g

a difference

, .... g

k=

a con~non primitive

root

defined b y

condition

odd primes

Whiteman

p

and

q. Since

Whiteman

showed:

let

x

constitute v

N = 2

(Theorem

and dN

akin to a primitive

and

d

Theorem) is

it follows that set to exist;

cyclotomy

this

for

root must be established.

root of both primes

dN = (p - 1)(q - 1).

and

p

Then there exists an

integers

(s = O,1,...,d

a reduced residue

(5.35)

5.27).

a generalized

be a fixed common primitive

such that the

gSxi

to

g

N = (p - l, q - l)

integer

for such a difference

this problem b y developing

In order to do this something

q;

(p - l, q - l) = N

k = d + p = (v - 1)/N,

condition

v = pq.

Let

~ = (v- I - N ) / N 2

Here

sufficient when

approached

Len~na 5.28.

(5.34)

(easily provided by the Chinese Remainder

is a necessary

is3 in fact,

O,q,2q, .... (p- l)q

(v- l)/N,

(p - l)(q - i) = dN.

q = (N - 1)p + 2

;

set with parameters

v = ~,

of the distinct

d-i

- l;

system modulo

i = %l,...,N

v = pq.

- l)

(5.36)

[That is, all residues prime

are of this form.]

Proof.

Let

x, y

be a pair of integers

satisfying

the simultaneous

con-

gruences

x =- g (mod p),

y - i (mod p)

x -= i (rood q),

y m g (mod q) .

(5.37)

The existence

and uniqueness

of such

x~ y

are guaranteed by the Chinese

133

Remainder g

Theorem.

modulo

v

Note that

i = j.

s # t,

gSxi m gtxJ,

T > 0, ~ > 0. ~;

Note further that the order of

p - l, q - l;

N

gSxi m gtxJ

divides

T.

i.e., it is

contrary to the lemma's assertion.

Since

g

can be written

By (5.37) this implies that so

of

v.

while that fact that the order of

So the assumption

divides

modulo

is the least common multiple

Now assume that shows that

xy m g

p - 1

is

Then (5.37)

rules out the case

x T m g~ m (xy) ~

divides

0 < ~ < N

d

do

~ - T

with

and that

this is a contradiction

q - 1 and the

lemma has been established° [Note that

x

is not unique,

In Whiteman's numbers

generalized

for

y

obviously

cyclotomy the index class

and the generalized i

consists of the

(i,J)N

v)

d

s, t

s i

(5.38)

is the number of members of index

which are followed by members of index class

g x

j.

That is,

(i,J)N

is

of the congruence

+ i -= gtxJ

(mod

v)

(5.39)

0 < s, t < d - io Certain elementary properties

difference

set applications

q - 1 = Nf', d = Nff' f, f'

(mod

c7clotomic number

the number of solutions

where

i

(s = 0,1~°.o,d - l)

a m gSxi

class

serves equally as well.]

below.

of this generalized N

is, of course,

cyclotomy are needed for the even and

for some relatively prime integers

f,f'

p - 1 = Nf, (in particular

are not both even).

x

N

=

g~

-i ---

(mod v)

I

gd/2

for some (mod v)

~ ~ l, 0 _< ~ < d - 1 when

ff'

(5.40)

is odd (5.41)

gW xN/2 (mod v)

when

ff'

is even

i34

where

v

is some fixed integer,

(i,J)N = (i',J')N

0 < v < d - i.

when

r

(J

+

N/2, i

(5.42)

(rood N)

i ~ i', j ~ j'

+

N/2) N

ff'

even

(5.43)

(i,J)N = (N - i, j - i)N = J

<

ff'

(J,i) N

odd

N-I (i,J)N =

(~ - 2)(q - 2 ) N

i

+ 5. !

ff' ff'

even odd

(5.4~)

j=O

where

li

i ~ N/2 (mod N) i ~ 0 (mod N)

1

otherwise

N-I f

(j,i) =

(~ - 2)(q - 2) -

+

N

(5.45)

~. l

j=0

where

i

i -: o (N)

0

otherwise

ei

Proof. This implies If

~ = i,

If

x N ~ g~

xN-i =ogS then

x

for any

~,

then by (5.36)

which contradicts Len~na 5.28.

is a primitive root of

v = pq;

.

x N ~ gSxi So

with

x N = g~

i > O.

for some

~.

this contradicts elementary

135

number theory since

p, q

are distinct odd primes.

So (5.40) has been

established. Suppose modulo

p

Let and

ff'

d/2

i =

Select q;

modulo

q; s

s

p

and

as in (5.37). gd/2 m -1

gS m -1

s ~ 0, d/2 q,

gd/2 ~ xd/2yd/2 ~ -1

v.

v.

Then

(p-l)/2

x s m -1

f, f'

modulo

p

as well as by (q-l)/2.

is the only candidate.

since one of

i -= g0x0 - g2Sx2i

Thus

Then

mod

modulo

is divisible by

and , as

modulo both

0 < i < No

so

x, y hence

be even and suppose

divides

gd/2 ~ -1 with

is odd.

as well as modulo

yS _ -1

Thus

ff'

is even.

But So

s

-I =- g x

1

and it follows from Lemma 5°28 that

N/2. Equation (5.42) is an immediate consequence of the definition and the relation

(5.40) above.

The first part of (5.43) follows on multiplying

the inverse of its first term and using

(5.39) through by

(5.40) to give

I + g-S-~xN-i ~ g t-SxJ-i

The second part of (5.43) follows from multiplying

(5.39) by

-i

as given in

(5.41).

si g x

The summation if (5.44) counts, for fixed

i,

+ 1

That is~ all those

which are in any index class at allo

which are prime to by

v,

let

N

v.

Of the

d

such numbers,

be the number divisible by

p

let

the numbers of the form

N

v

+ 1

be the number divisible

but not by

q

and let

P number divisible by

si g x

N

be the q

q

but not by

p.

Then, clearl~v,

N-1 (i,J)N = d - ~

- ~

- Nq.

(5.46)

j=0

By (5.41)

N v = 5.. 1

Since

the non-zero residues of

g p

is a primitive root modulo are assumed

(q - 1)/N

p

times by

it follows that all gS

and hence by

136

s i

g x N

q

as

s

ranges from

= (p - 1)/N - 5.. l Equation

relations

0

to

Equation

d - i.

Hence

= (q - I)/N - 8. l

p

(5.44) thus follows

(5.45) follows w h e n

from 5.46.

(5.43) is applied to

(5.44).

Let the index class

i

Lemma 5.29 .

be a fixed integer divisible by

Let

r

be denoted by

y

Thus all the

C.. 1

in class

C1

and

p

q

or

but not by

v.

of the congruence

y - z -= r

with

and similarly

above have been established°

Then the number of solutions

is

N

z

(mod

in class

CO

v)

(5.47)

is independent

of the value of

r

and

(p - 1)(q - l)/N 2.

Proof. modulo

v

Let

p

divide

as in Lemma

some fixed integer

r

5°28.

u

and let Then

such that

g, x

x ~ 1

generate

(mod

v),

0 < u < p - 2.

the reduced residue

and

gUx ~ i

system

(rood p)

In order for (5.47),

i.e.,

for in

order for

t

g x-

to be solvable,

(mod Hence,

q),

r

ranges of

s.

t - s = u

(q - 1)/N

q - 1

Since

g

these are congruent modulo

for fixed

(5.47).

that

5.48 for any

a, ga,...,gq-2a.

any fixed

by

(rood v)

(5.48)

(mod

p - I).

values of

Thus for each

t It = u + s + m ( p - 1 ) ,

for which the right side of (5.48) is divisible by

congruence

are

-r

there are precisely

0 < m < (q - I)/N]

ferences

S

it is thus necessary

s (0 < s < d - l)

and consider

g

m,

as

precisely

s

ranges

once.

Thus a fixed

consecutive

is a primitive

root of

v

to

p, 2p,..., (q - l)p

from

sO

to

For each value of r

values of

is represented

By symmetry the same result

sO + q - 2 m

is true when

s;

the dif-

and

a ~ 0

in some order.

d/(q - l)

(p - l)(q - I)/N 2 q

Fix

(5.48) represents

there are

exactly

q

p.

but not

v

such times

divides

r

m

137

and the lemma is proved. Using these cyclotomic facts:

Theorem 5.30° of

p - 1 (=Nf)

let

Let

(Whiteman~ 1962) and

q - 1 (=Nf')

(p - 1)(q - l) = dN

and let

N

denote the greatest common divisor

where

g

p

and

q

are distinct odd primes,

be a primitive root of both

p

and

q.

Then

the numbers

!,g3g2

,..o,g

d-i

form a difference set with parameters if and only if

ff'

;

0,q, 2q,.o.,(p - l)q

(5.49)

v = pq, k = (v - I)/N,

h = (v - 1 - N)/N 2

is even and the following two conditions are satisfied:

q = (N - 1)p + 2

(i,o)~ : (N - i)

[Here

[(p - 1)/N] 2

n = k - h = (Nv - v + 1 ) / ~

(5.50)

(i : 0,1 ..... ~ - l ) .

= (p - f)2

by 5.50 and so

(5.51)

(n,v) = 1

for all

difference sets of this type.] The statement that the numbers

(5.49) form a difference set is equivalent to

the statement that for every fixed integer precisely

h

y, z

not divisible by

v,

there are

solutions to the congruence

y-

with

re

chosen from 5.49 .

and the numbers

z~r

(mod v)

The numbers

0,q,..o 2 (p - 1)q

(5.52)

l,g,..o,g d-i

form the index class

will be said to be members of the class

CO

Q.

Before proving Theorem 5.30 a couple lemmas are required.

Le~na 5.31.

Let

r

be a fixed integer not divisible b y

of solutions of congruence 5.52 with (p - 1 ) / m

y

in

CO

and

z

in

q. Q

Then the number

is equal to

138

Proof. (p - 1)/N Since

As in the proof of Lemma disjoint

g

subsets~

is a primitive

these subsets

Lemma

5.32.

Let

each of which

root of

such that

gS . r

r

q~

When

of

g

t

- g

gm(p-l)+s excluded q - i

in

r

v)

is divisible

is that

_ gS ~ r (mod

consecutive

g

v)

y

by

divides

integers,

establishes

Proof of Theorem

t m ~ (mod

with

by

and q

p

integers.

in each of

z

but not by

qo

both in class

but not by

p

Then

CO

is

the number of

condition for the solvability

p - 1).

1 < m < (q - 1)/N.

r.]

Fix

m

Thus this congruence [The value

m = 0

in this range and let

then as before precisely

(p - 1)(q - 1 - N)/N 2

p, q

5.52 with

s

into

(q - 1)(p - i - N)/N 2.

m r (mod

for then

consecutive

- i

and the lemma follows.

As in the proof of Lermna 5.29 a necessary

s

there are

q

be a fixed integer divisible

(p - 1)(q - 1 - N)/N 2.

Proof.

q - 1

0,1,...,d

is exactly one number

is divisible b y

of congruence

is

contains

there

the number of solutions

such solutions

5.29 divide the integers

solutions

s

one solution

overall,

becomes

is

vary over

appears.

as was to be proved.

So S~umuetry

the remainder of the lemmao

5.30.

Suppose that the numbers

set with the prescribed parameters.

Then

(5.49) constitute

(5.50) is an immediate

a difference

consequence

of

k=d+p. Let in

CO

r

be relatively prime to

corresponds

Thus if

r

belongs

(i,0)N.

Lemma

then there are Lemma CO .

v.

with a solution of

~r + 1 -~ zy

to

- l)

C i (i = % . . . , N

5o31 shows that if (p - 1)/N

y

solutions

5.31 also shows that there are Since

solutions

(r#v) = l, y to

Every solution of

and

z

belongs to

(rood v),

here

to

CO

and if

z

By considering

-r

(p - 1)/N

solutions w i t h

y

cannot both be in

N

and

Q

~

is

to

Q

instead, in

Q~

z

in

and so the total number of

v-l-~ N2

z

~z =- 1 (rood v).

belongs

(5.52).

2(p-l)

y

the number of such solutions

(5.52) is

(i,o)~ +

(5.52) with

139

from which

(5.51) follows°

Suppose

ff'

were odd.

the number of solutions of

Consider the cyciotomic number r + 1 ~ s (mod

v)

when

r, s

Corresponding to any such solution is another solution since

-s, -r

also belong to

from the first unless (i,i)N

is even except when

So

ff'

(by 5.41).

r ~ -s ~ -r - 1 (mod

(i,i)N = (N - i,0)N, odd.

C. l

(v - 1)/2

(i,i)N;

it counts

belong to

-s + 1 ~ -r

C i. (mod

v);

This second solution is distinct v),

i.e., unless v = 2r + 1.

belongs to

this implies that equation

C i.

Thus

Since, by (5.43),

(5.51) cannot hold when

ff'

is

is even and the necessity part of the proof is complete.

Now suppose that conditions

(5.50) and (5.51) hold;

it is to be shown that

congruence 5.52 hasIa uniform number of solutions independent of the value of r (r ~ 0

modulo

v)

when

y, z

are chosen from among the numbers

(5.49)°

There

are three cases: (i) y

in

Q

Let

p

and also

divide

r.

(p - 1)/N

(p - l)(q - I - N)/N 2

By Lemma 5.31 there are solutions with

additional solutions when

(5.50) the total number of solutions is (ii)

Let

q

divide

(q - l)(p - I - N)/N 2 solutions.

z

r.

When

solutions.

If

in

(p - I)/N

Q.

y, z

Lemma 5.32 shows are both in

Let

r

CO .

Using

(v - i - N)/N 2.

y, z y, z

both belong to

CO

both belong to

Q

So, by (5.50) the total number of solutions is

(iii)

solutions with

be relatively prime to

v.

Lermma 5.32 counts there are

p

(v - 1 - N)/N 2.

In the necessity part of the proof

it was shown that the number of solutions in this case is

(i,0)N + 2(p - I)/N

when

(v - 1 - N)/N 2.

r

belongs to

So the numbers h = (v - 1 - N)/N 2

C.. 1

By (5.50),

(5.51) this is again

(5.49) do indeed form a difference set modulo and

k = d + p = (v - I)/N.

v = pq

with

That is, Theorem 5.30 has been

established. As an example of this theorem consider the case and

ff'

is necessarily even.

Equation

so (5.51) is automatically satisfied.

N = 2o

(5.43) shows that

Thus the numbers

By (5.50)

q = p + 2

(0,0)2 = (1,0)2 = (1,1)2 ,

(5.49) form a difference

14o

set whenever

q : p + 2.

That is, a proof has been given for Theorem 5.27 above.

Of course, more direct proofs of this result exist immediately below Theorem 5.27). 0, l, 2, 4, 5, 8, l0

With

constitute

P = 3, q = 5, g = 2

a difference

In applying Theorem 5.30 to the case numbers

(i,0)4

he shows,

for

are needed° ff'

(see the references

Whiteman

then, the numbers

set with parameters

N = 4

given

v,k,h = 15,7,3.

formulas for the cyclotomic

(1962) derives these numbers.

In particular

even, that

8(0,o) 4 : -a + ~

+ 3,

8(0,1)4 = -a - 4b + 2M - 1

8(o,2) 4 : 3a + ~

- l,

8(0,3)4 = -a + 4b + 224 - 1

(5.53)

8(1,o)4 = a + 2M + 1

the remaining

(i,J)4's

being equal to one of these as (5.43) shows°

M = [(p - 2)(q - 2) - 1)/4]

v:

a

2

and

a, b

+ 4b 2,

are determined

a-= 1

Whiteman observes that the common primitive two classes a power of

G, G'. g

Here if

g'.

a '2 + 4b '2,

unequal.

(0,0)4

a 2 + 4b 2

and

(0,0)~

G'.

Theorem 5.31.

(Whiteman)

Let

1, q -

and let

roots of

both) of the sets

G'

p, q

g

of

v

p

and

q

g, ~ gr

G;

g'

respectively

the 1964),

are always

g, g'

gives

be distinct primes such that

d = (p - l)(q - 1)/4.

with

is

as in (5.54)o

(Carlitz and Whiteman,

and

G

are all powers of any

(5.51) this implies that at most one of N = 4.

primitive

can be divided into

then every primitive root in

Furthermore

computed for

set for

1) : 4

p, q

(%54)

is used with the primitive roots of

with those of

In view of equation

rise to a difference

(p-

G

4) o

There are two distinct representations

One such representation

the numbers

belongs to

and similarly the primitive roots of

particular one

other,

g

from a representation

(mod

roots of

Here

(rood v)

Let

g, g'

for any

r.

be distinct common Then one

(but not

141

l~g,g2 ,...,g d-i ;

O,q,2q, .... ( p -

l,g,,g,2,...,g,d-i

is a difference only if

set with parameters

q = 3P + 2

The generalized (Whiteman,

and

k

(5.55)

0,q, 2q,..., (p - l)q

(5.56)

v = pq, k = (v - 1)/4,

h = (v - 5)/16

if and

is an odd square°

cyclotomic numbers

1962 ) and for

l)q

N = 6,8

(i,J)N

(Bergquist,

have been computed for 1963 ) .

N = 2,4

They may be found

(ff'

even only) together with a detailed discussion of the material of this section in Storer's booklet concerned,

(1967 A)0

whenever

The ntmubers

q = p + 2

numbers form a difference g

whenever

q = 3P + 2

(5.49) form a difference

(Whiteman,

sets are

N = 4

particular

For

N = 4

and

k

is an odd square.

For

v = pq

(Whiteman,

and the next example has

1962) these root

(Bergquist,

1963 )

the

set occurs with

v = 6,575,588,101.

(1967 B) developed further cyclotomies succeeded in constructing

N = 6,8

for

set.

the first example of such a difference

v = pq = 901 = 17.53 Storer

1962).

set modulo

set for a suitable choice of the common primitive

these numbers never form a difference For

as far as difference

there is:

Theorem 5.32. N = 2

By way of summary,

along these lines and in

v,k, }~ = 133,33,8

difference

set of Hall

(1956) by these methods. Concerning the multiplier group of these difference

sets and the multiplier

conjecture there are the following facts:

Theorem 5.33.

The multiplier

consists of the residues

Proof. index class (mod

v)

13g,°..,g

group of any difference d-1

and so

(j ~ 0

modulo

s ~ 0 (mod

N) v).

D

given by (5.49)

and no others.

These numbers are clearly multipliers. j

set

Suppose

was also a multiplier. Suppose

q

t

belonging to

Then

does not divide

td -= D + s s.

Since

0

142

belongs

to

power of (mod

D

so does

g.

q),

Thus

-s

and since

(by 5.41)

tg I - s m 0

same index class.

s

belongs

for some

Thus

-1

i.

that

primitive over

C O + s m Cj (mod

root modulo

CO .

Thus

and establishes

Theorem difference

r,

g

D

Proof.

When

prime to

v,

of Theorem

with

p)

and if

t t

s.

s = q~,

every non-zero

+ qZ ~ 0 (mod

For

N = 2

given by

N = 2

it follows N/2.

that

-s

Again since

as w e l l as

-1

is a s ~ 0

belong to the

is, contradiction. Then, with

t

(s,p) = i.

Since

g

occurs

as

residue modulo

for some

all divisors

i~

p

which

in

Cj, tD m D + s

contradicts

is a g

i

ranges

C O + s ~ C. J

v

'

l,g,~o.,g

n

are multipliers

d-i

for any

are precisely

the Jacobi symbol

to show that every divisor

law for the Jacobi

5°20 above.

of

(5.49).

the elements

for which

Thus it is sufficient the reciprocity

s

divides

v)

q)

the theorem.

5.54.

set

i

p~

So

q

(mod

to index class

is a multiplier

Now consider the case where implies

s ~ 0

d

of

those residues

[see Nageil, n

symbol the proof proceeds

has

1951] , v

= +i.

is

+i. Using

exactly like the proof

VI.

MISCELLANY

In this chapter a few facts, some important, which was not convenient to mention earlier, are gathered.

A.

Multiple inequivalent Difference Sets Singer (1938) showed that non-trivial difference sets with parameters

@+l_ V

exist whenever

q

1

k:

q-i

@{- 1

'

is a prime power and

of these difference sets.

~

qml

-------~ q '

N > 2;

1

(6.1)

q-i

see section V.A. for a discussion

Singer noted that when

h = i

there seemed to be only

one equivalence class of difference sets with these parameters and he conjectured that this was in fact the case.

This conjecture of Singer is still open.

(1956, p. 984) has verified Singer's conjecture for 27,32.

Hall

q = 2,3,4,5,7,8,9,11,13,16,25 ,

Beyond this work of Hall apparently very little has been done on this

problem. Strangely enough the largest known class of multiple inequivalent difference sets also have parameters given by (6.1). composite, thus

h > i.

$ > 0,

by (6.1) for which there exist at least k ~ i00, k < v/2,

as follows:

2

121, 40, 13

and

is necessarily

for

2

for

127, 63, 31.

For while those parameters

~

there exist values of

v, k, k

given

pairwise inequivalent difference sets.

the known multiple inequivalent difference sets are

for 31, 15, 7; 6

N + i

In fact (see section V.Ao for details) Gordon, Mills and

Welch (1962) have shown that for any

For

Here, though,

43, 21, i0;

2

for

63, 31, 15;

4

for

[There may well be others in this range.

v, k, h (k ~ i00)

which have associated cyclic

difference sets have been determined (see section VI.B) not much beyond the work of Hall (1956, limited to

k ~ 50)

is known about multiple inequivalent difference

i44

sets in this range.] Two block designs with the same parameters are considered equivalent if there exists two permutations (generally different), one acting on the objects and the other acting on the blocks, which take the one design into the other.

On the other

hand, equivalence of difference sets (see section I.B) is a more restrictive relation.

Thus, it could happen that two cyclic difference sets were inequivalent

while their associated block designs were not. this behavior.

There are no known examples of

In particular, it does not happen for the parameter sets

mentioned just above.

(k ~ 100)

In these cases, inequivalence of the associated block

designs follows from the distribution of sizes of the intersections of triples of blocks.

For example, the distribution of intersection size for all block triples

containing a particular fixed block is 420 of size of the

v,k,~ = 31,15,7

difference sets.

set has such block triples intersections of size

4.

3

and

15 of size

Whereas the other 90 of size

2,

195

7

31, 15, 7 of size

for one difference

3

and

150

Since the distributions of block intersection sizes must be the same

for equivalent block designs, it is clear that these designs are not equivalent. The question of whether the multiple inequivalent difference sets of Gordon, Mills and Welch necessarily lead to inequivalent block designs is open. [While no examples of inequivalent cyclic difference sets generating the same block design are known, there are examples of block designs which are generated by more than one difference set. these examples is non-cyclic.

Of course, at least one of the difference sets in

In fact, Bruck (1955 , p. 475) has shown that the

block design associated with a cyclic difference set with parameters n

2

+ n + i, n + i, i

n ~ i

B.

modulo

3

(these are only known to exist for prime power

v,k, h = n)

for

can also be generated by a non-Abelian group difference set.]

Searches In 1956 Marshall Hall (1956) published the results of his search for difference

sets having

k < 50.

v,k,h(k ~ 50),

With but

12

exceptions, he decided, for each parameter set

whether or not a difference set existed.

When the difference set

145

was not a member of one of the families of Chapter he listed the residues modulo

v.

V

(and in some other cases)

For many of the smaller parameter sets he decided

whether or not multiple inequivalent difference sets existed (see section VI.A. above ). Hall's twelve undecided cases were all resolved negatively.

In fact many of

the existence tests presented in section II.E. above were developed specifically for the purpose of deciding Hall's twelve cases. Baumert (1969) extended Hall's search to exactly

74

parameter sets

sets existed.

These

difference sets;

74

Utilizing these powerful tests

k < i00o

v,k, h(k _~ I00)

He found that there were

for which non-trivial cyclic difference

parameter sets have associated with them

85

known

there may be more, since Baumert made no attempt to find multiple

inequivalent difference sets beyond those already known. below contains all

85

known difference sets for which

Table 6.1 of section VI.Do k < i00.

As reported in more detail in section IV.A., Hall (1947) checked that, for n _~ I00,

all planar

(i.eo,

h = i)

difference sets had

n

a prime power.

This

conjecture, which is still outstanding, was checked further by Evans and Mann (1951) up to

n < 1600.

Dembowski (1968, p. 209) states that V. Ho Keiser (un-

published) has checked it to

n _~ 3600.

Hadamard difference sets

(i.e., those with

have been searched for, through

v < i000.

v,k,h = 4t - i, 2t - i, t - i)

The results of these searches (Golomb,

Thoene, Baumert) are that, except for possible additional multiple inequivalent difference sets, the only unknown Hademard difference sets possible, have

v = 399, 495, 627, 651, 783

or

975°

v < 100%

(See section IV.B. for a discussion of

the known Hadamard difference sets. ) The difference sets associated with circulant Hadamard matrices (see section IVoC. ) have been sought.

These have parameters

v = 4N 2, n = N 2 .

Turyn (1968)

surveys the results in this area, most' of which are due to him, and shows that, except for

v = 4,

none such exist with

v < 12,100.

lh6

C.

Some Examples The general tenor of these examples seems to be that, at least from an alge-

braic number theoretic point of view, difference sets are no better behaved than they absolutely have to be. Consider the quadratic residue difference set with parameters 25.

Here

modulo

n = 26,

103

is

are minimal.

the order of

17,

i.e.,

2

modulo

103

251 e l, 1317 ~ 1

Furthermore, no power of

13

is modulo

51

and the order of 103,

K({103)

defined by

the automorphism 8 ~i03 -+ ~i03'

T,

P's

hand, Theorem 5.20 shows that set.

13

~i03 -+ ~i03'

defined by

fixes both

2

or and

4

modulo

13

in the

103.

(13) : PiP2P3PiP2P 3

where the bar denotes complex conjugation. ~,

2

as:

(2) : ~ ,

automorphism

13

where the exponents

is congruent to

Thus Theorem 2.19 gives the prime ideal factorizations of cyclotomic field

v,k, ~ = 103,51,

and 2

Q's, 13

248 -= 13

it follows that the

fixes all these prime ideals;

2 4103,

~i03

and

Since

since

fixes only the

Q's.

1316 ~ 8 (mod 103).

But

whereas T 3,

i.eo,

On the other

are both multipliers of this difference

So, without loss of generality, the prime ideal decomposition of the ideal

(G(~I03))

is

(e(~lO3)) = QP~P33

where the action of the multiplier according to the multiplier

PI ~ P2 ~ P3 ~ PI" 2,

2

permutes the prime ideal divisors of

That is, while the ideal

(0(~I03))

13

is fixed by

its individual prime ideal divisors are not.

Consider the planar difference set whose parameters are consists of the residues

i, 6, 7, 9, 19, 38, 42, 49

notation of section III.D.,

modulo

v,k, h = 57,8,1. 57.

Using the

It

i~7

@[57](x) : x + X 6 + x 7 + x 9 + x !9 + x 38 + x 42 + x 49

e[19](x ) = 2 + x + x 4 + x 6 + x 7 + x 9 + x II

and the prime ideal divisors of

are (see Theorem 2.19):

(@(~19))

2

A : (7, ~ 9 + 6~I 9 + 3~19 + 6) 2 + 5~i 9 + 6)

C = (7, ~ 9

while

A, B, C

ideals over

divide

K({57).

which lie above

A

(@(~i~)). Let

and

al, a 2 B

2 + 4~i 9 + 4~i 9 + 6)

Each of these six ideals splits into two prime and

bl, b 2

respectively.

be the prime ideals of

Then

(e(~7)) ~ ala2hlb2ClC 2

where

2 + 2~57 + 6) a I = (7, ~ 57 + ~57

~2 = (7, ~ 7

2 + 6~57 + 5~57 + 6)

b I = (7, ~7 + 3 ~57 2 + 6~57 + 6) 2

~2 = (7, ~ 7

+ 2~57 + 2~57

c I = (7, ~ 7

+ ~57 + 6)

c 2 = (7, ~ 7

+ 4~57 + 6).

+ 6)

K({57)

148

Thus, the tempting assumption that the fact that

A

divides the ideal

(e(~19))

implies that one of

2 2 al, a2, ala 2

(e(~57))

Besides certain theoretical implications, this complicates the

is wrong.

necessarily divides the corresponding ideal

application of the constructive existence test of section III.D. Consider the planar difference set v, k, h K(~73 )

are

73, 9, l;

n = 8.

[1,2,4,8,16,32,37,55,64)

whose parameters

Now one of the prime ideal divisors of

(2) in

is

6 2 A = (2, ~ 3 + ~73 + ~ 3 + ~73 + I)

and

A~

divides

(e(~73)).

Thus, the simplifying assumption, that one and only

one of a complex conjugate pair of prime ideal divisors of

n

divides

(e(~))

is

not correct in general°

D.

A Table of Difference Sets Table 6.1 below contains all

85

known difference sets for which

k ~ i00.

As pointed out earlier, there well may be others; for the question of the existence of multiple inequivalent difference sets has not been solved for all of the parameter sets

v,k, ~ (k ~ i00)

has been shown that these

74

having associated difference sets.

74

However, it

parameter sets are the only ones that need be

considered. Each difference set is identified by are the prime divisors of

n)

v, k, h, n, pl , P2' P3

(where the

Pi

and by a class indicator which indicates the family

or sub-family to which it belongs.

These are:

SN

-

Hyperplanes in projective

N

L

-

Quadratic Residues (Theorem 5.15)

B

-

Biquadratic Residues (Theorem 5.16)

BO

-

Biquadratic Residues and

0

-

0ctic Residues (Theorem 5.18)

H

-

Hall's Sets (Theorem 5.23)

O

space (Theorem 5.1)

(Theorem 5.±8)

149

TP GMW

-

Twin Prime Sets (Theor~n 5.27) Gordon, Mills, Welch (Section V.A)

W

-

Whiteman Set (Theorem 5.31)

*

-

not one of the above .

Where multiple inequivalent sets are known, they are distinguished by etc., which is added to the parameter v,k,k = 31,15~7 respectively.

v.

Thus the two inequivalent

difference sets are designated

31A, 15, 7

and

31B~ 15, 7

The table is sorted in order of increasing values of

k.

A, B, C

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10 46 90 121 160 198 25! 291

13 92 169 22~ 296 395 ~8~ 571 636 729 835

9 ~3 88 1~0 157 196 ~50 290

12 89 168 ~23 295 39~ ~77 570 63~ 7~ 8~8 898

I~ 93 173 225 300 397 ~92 57~ 639 735 837

12 47 9! 122 161 206 253 ~95 16 106 17~ 226 307 ~OO ~9~ 577 6~1 737 840

1~ 48 9~ 1~3 16~ 209 25~ 799 22 108 182 729 308 ~01 502 579 656 7~2 8~3

16 ~9 95 t2~ 165 ~11 ~56 300 ~5 !09 183 731 313 &O~ 503 581 658 7~6 8~5

19 55 97 126 167 ~3 760 302 ~I I~0 192 232 318 ~05 516 583 661 752 8~8

22 56 100 129 168 220 ~61 303

8233 8296 8360 8560 87~0 8817 8930 5 7 8 9 I0 13 14 16 43 @5 ~6 47 @9 50 51 79 80 81 86 89 90 91 117 121 122 123 12~ 1~5 126 161 162 165 169 172 175 177

W 43 !I~ ~9~ 233 319 ~07 523 585 66~ 760 8~9

~5 117 196 235 325 4i9 526 587 665 766 85~

~7 53 12~ !~5 198 20~ 244 ~5~ 326 3@5 @22 ~23 529 530 596 599 688 689 767 77~ 853 859

59 126 203 256 350 ~2~ 531 602 691 778 862

60 133 205 259 352 ~33 533 611 69~ 780 863

1071 ~8~3 s~6~ 72~ 9011 9098 9126 92~@ 937@ 9@09 L 18 20 23 25 26 28 52 5~ 56 57 58 61 92 9~ 98 100 102 103 ~28 130 131 ~32 139 1@0 178 180 182 18~ 187 188 TP 25 26 77 ~9 30 31 57 64 65 66 71 75 lOi I0~ 105 106 107 108 133 13~ 137 138 1~0 1~l 169 171 172 173 ~78 181 72~ 225 227 ~ 8 229 231 ~6~ 263 26~ 265 266 269 30~ 305 308 310 31~ 315

CLASS ~2

.1 ~ 137 360 568 ~11 6~7 67o 69~ ? ~ 833 8~9 963 1070 ~26~ 1378 I~0~ 1503 198~ 1989 206~ ~163 ~2~s ~301 ~308 ~670 27~8 2793 2~02 2826 3169 3186 3211 3527 3782 39~9 ~128 4257 ~536 ~59, ,~2s ~7~s ~ 1 8 52o9 s~ls 5~5~ ~s~o ~ 9 0 ~ 8 ~ ~03~ 61~3 61a~ 62~6 6 3 ~ 639~ 6 ~ 6 6~66 6~96 ~ 7 ~ 6687 69~I 7221

78@8 7862 79@8 ~091 99 49 50 ~ *I ~ 4 5 33 35 36 40 65 66 70 7~ 11~ 11~ 115 116 155 157 15~ 160 323 161 80 81 3 *0 1 3 ~ 38 ~0 41 ~2 79 81 83 87 11~ 115 116 118 1~7 1 ~ 149 152 191 192 193 195 ~1 243 2~7 2~9 ~78 279 2 ~ 285 321 901 225 56 169 13 *0 i 5 9 71 79 80 81 152 156 159 167 2t~ 215 219 222 277 286 292 293 371 379 382 387 &61 ~65 ~67 @69 5~5 550 559 56~ 621 622 6~5 630 711 713 7~0 72~ 813 82~ 876 8~7 878 881 8~6 897

V 9507

65 139 208 26~ 355 ~#5 536 617 695 786 865

35 76 109 1~3 18~ 23~ 270 316

29 62 I0~ 1@4 193

1073 2~96 5~ ~66~ 9@40

69 i~ 209 265 363 ~7 5@0 618 706 795 870

96 77 111 144 188 237 271 3!7

31 63 106 1~5 196

1107 3000 ~58, ~60~

70 ~47 212 274 369 ~60 543 6~0 708 801 87~

37 78 113 1~6 190 239 273 3~8

32 6@ ~11 151

11~ ~008 ~8 7~

oo

REFERENCES

Generally speaking a survey such as this is expected to contain a fairly exhaustive bibliography.

However, since the book of Dembowski (1968) contains

an excellent 49 page bibliography covering this subject, the listing here may be, and is, restricted to items explicitly mentioned in this report.

Alanen, J. D., and D. E. Knuth 1964

"Tables of Finite Fields", Sankhy~ 26, 305-328.

Albert, A. A~ 1953

"Rational Normal Matrices Satisfying the Incidence Equation", Proc. Amer. Math° Soe. 4, 554-559.

Albert, A. A., and R. Sandler 1968

An Introduction to Finite Projective Planes, Holt, Rinehart and Winston, New York

Barker, R. H. 1953

"Group Synchronizing of Binary Digital Systems", in Cormutuuication Theory, W. Jackson, Editor, London, 273-287.

Baumert, L. Do 1969

"Difference Sets", SIAM J. Appl. Math. 17, 826-833.

Baumert, L. D., and H~ Fredricksen 1967

"The Cyclotomic Numbers of Order Eighteen with Applications to Difference Sets", Math. Comp. 21, 204-219.

Belevitch, V. 1968

"Conference Networks and Hadamard Matrices", Ann. Soc. Sci. Bruxelles Set. 1 82, 13-32.

Bergquist, J. W. 1963

"Difference Sets and Congruences Modulo a Product of Primes", Dissertation,

University of Southern California.

160

Brauer, A. !953

"On a New Class of Hadamard Determinants",

Math. Z. 58, 219-225.

Brualdi, R.Ao 1965

"A Note on Multipliers of Difference Sets", J. Res. Nat. Bur. Standards Sect. B 69B , 87-89.

Bruck, R. Ho 1955

"Difference Sets in a Finite Group", Trans. Amer. Math° Soc. 78, 464-481.

Bruck, R. H., and H. J. Ryser 1949

"The Nonexistence of Certain Finite Projective Planes", Canad. Jo Math. l, 88-93.

Carlitz, L., and A. L. Whiteman 1964

"The Number of Solutions of Some Congruences Modulo a Product of Primes", Trans. Amer. Math. SOc. ll2, 536-552.

Chowla, S. 1944

"A Property of Biquadratic Residues", Proc. Nat. Acad. Sci. India, Sect. A 14, 45-46.

1945

"Contributions to the Theory of the Construction of Balanced Incomplete Block Designs", Math. Student 12, 82-85.

Chowla, S., and H. J. Ryser 1950

"Combinatorial Problems",

Canad. J. Math. 2, 95-99.

Dembowski, P. 1968

Finite Geometries,

Springer-Verlag,

New York.

Dickson, Lo E. 1935A

"Cyelotomy, Higher Congruences and Waring's Problem",

Amer. J. Math.

57, 391-424 and 463-474. 1935B

"Cyclotomy and Trinomial Congruences",

Trans. Amer. Math. Soc. 37,

363-380. 1935C

"Cyclotomy When E is Composite",

Trans. Amero Math. Soc. 38, 187-200.

Evans, T. A., and H. B. Mann 1951

"On Simple Difference Sets", Sankhy~ ll, 357-364.

161

Gillies, D. B. 1964

"Three New Mersenne Primes and a Statistical Theory", Math. Comp. 18, 93-95.

Goethals, J. M., and J. J. Seidel 1967

"0rthogonal Matrices with Zero Diagonal", Canado J. Math. 19, 1001-1010.

Goldhaber, J. K. 1960

"Integral P-Adic Normal Matrices Satisfying the Incidence Equation", Canad. J. Math. 12, 126-133o

Goldstein, R. M. 1964

"Venus Characteristics by Earth-Based Radar", Astronom. J. 69, 12-18.

Golomb, S. W. et al. 1964

Digital Communications with Space Applications, Prentice-Hall, Englewood Cliffs, New Jersey.

Gordon, B., W. H. Mills and L. R. Welch 1962

"Some New Difference Sets", Canad. Jo Math. 14, 614-625.

Gruner, W. 1939

"Einlagerung des Regularen N-Simplex in ~en N-Dimensionalen Wurfel", Comment. Math. Helv. 12, 149-152.

Halberstam, H., and R. R. Laxton 1964

"Perfect Difference Sets", Proc. Glasgow Math. Assoc. 6, 177-184.

Hall, M. Jr. 1947 1956 1965

"Cyclic Projective Planes", Duke Math. J. 14, 1079-1090. " A Survey of Difference Sets", Proc. Amer. Math. Soc. 7, 975-986. "Characters and Cyclotomy", Proc. Sym. in Pure Math., Amer. Math. Soc.

8, 3t-43. 1967

Combinatorial Theory, Blaisdell, Waltham, Massachusetts.

Hall, M. Jr., and H. J. Ryser 1951

"cyclic Incidence Matrices", Canad. J. Math. 3, 495-502.

1954

"Normal Completions of Incidence Matrices", Amer. J. Math. 76, 581-589.

Hayashi, H. S. 1965

"Computer Investigation of Difference Sets", Math. Comp. 19, 73-78.

162

H~Ighes, D. R. 1957

"Regular Co!lineation Groups", Proc. Amer. Math. Soc. 8~ 165-168.

Johnsen, E. C. 1964

"The Inverse Multiplier for Abelian Group D~fference Sets", Canad. J. Math. 16, 787-796 .

1965

"Matrix Rational Completions Satisfying Generalized Incidence Equations", Canado Jo Math. 17, 1-12.

1966A

"Integral Solutions to the Incidence Equation for Finite Projective Plane Cases of Orders

1966B

N -= 2 (mod 4)", Pacific J0 Math. 17, 97-120.

"Skew-Hadamard Abelian Group Difference Sets", J. Algebra, 588-402.

Jones, B. W. 1950

The Arithmetic Theory of Quadratic Forms, Carus Math. Mono. i0, Wiley, New York.

Kesav8 Menon, P. 1962

"Certain Hadamard Designs", Proc. Amer. Math. Soc. 15, 524-531.

Lebesgue, A. U. 1850

"Sur L'!mpossibilite,

en Nombres Entiers, de L'Equation

xm = y

2

I. + 1 ,

Nouve!!es Ann. Math. 9, 178-181. Lehmer, Eo

1953

"On Residue Difference Sets", Canad. J. Math° 5, 425-432.

i955A

"Period Equations Applied to Difference Sets", Proc. Amer. Math. Soc. 6, 453-442.

1955B

"On the Number of Solutions of

u k + D -= w 2 (rood p)", Pacific J. Math.

5, 103-118. Magnus, W. 1937

"Uber die Anzahl der in Einem Geschlecht Enthaltenen Klassen yon Positiv Definiten Q~adratischen Formen", Math. Ann. 114, 465-475.

Mann, H. B. 1952

"Some Theorems on Difference Sets", Canad. J. Math. 4, 222-226.

1955

Introduction to Algebraic Number Theory, Ohio State Univ. Press, Columbus, Ohio.

163

Mann, H. B. 1964

"Balanced Incomplete Block Designs and Abelian Difference Sets", lllinois J. Math. 8, 252-261.

1965

Addition Theorems, The Addition Theorems of Group Theory and Number Theory, Interscience,

1967

New York.

"Recent Advances in Difference Sets", Amer. Math. Monthly 74, 229-235.

Mann, H. B., and S. K. Zaremba !969

"On Multipliers of Difference Sets", Illinois J. Math. 13, 378-382.

McFarland, R., and H. B. Mann 1965

"On Multipliers of Difference Sets", Canad. J. Math. 17, 541-542.

Muskat, J. B. 1966

"The Cyolotomic Numbers of Order Fourteen", Acta Arith. Ii, 263-279.

Nagell, T. 1951

Introduction to Number Theory, Wiley, New York.

Neumann, H. 1955

"On Some Finite Non-Desarguesi~1 Planes", Arch. Math. 6, 36-40.

Newman, M. 1963

"Multipliers of Difference Sets", Canad. J. Math. 15, 121-124.

0'Meara, 0. T. 1963

Introduction to Quadratic Forms, Springer-Verlag, New York.

Ostrom, T. G. 1953

"Concerning Difference Sets", Canad. J. Math. 5, 421-424.

Paley, R. E. A. Co 1933

"On Orthogonal Matrices", Jo Math. and Phys. 12, 311-320.

Rankin, R. A. 1964 Reuschle, 1875

"Difference Sets", Acta Arith. 9, 161-168. C. G. Tafeln Complexer Primzahlen,

Welche aus Wurzeln der Einheit Gebildet

Sind, Berlin. Roth, R. 1964

"Collineation Groups of Finite Projective Planes", Math. Z. 83, 409-421.

164

Ryser, H. J. 1950

"A Note on a Combinatorial Problem", Proc. Amer. Math. Soc. l, 422-424.

1952

"Matrices with Integer Elements in Combinatorial Investigations",

Amer.

J. Math. 74, 769-773. 1963

Combinatorial Mathematics,

Schutzenberger, 1949

Carus Math° Mono. 14, Wiley, New York.

M. P.

"A Non-Existence Theorem for an Infinite Family of Symmetrical Block Designs", Ann. Eugenics 14, 286-287.

Shrikhande, 1950

S. S. "The Impossibility of Certain Symmetrical Balanced Incomplete Block Designs", Ann. Math. Statist. 21, 106-111.

Singer, J.

1938

"A Theorem in Finite Projective Geometry and Some Applications to Number Theory", Trans. Amer. Math. Soc. 43~ 377-385.

Skolem, Th., S. Chowla and D. J. Lewis 1959

"The Diophantine Equation

2 n+2 - 7 = x 2

and Related Problems", Proc.

Amer. Math. Soc. 10, 663-669. Spence, E. 1967

"A New Class of Hadamard Matrices",

Glasgow Math. J. 8, 59-62.

Stanton, R. G.~ and D. A. Sprott 1958

"A Family of Difference Sets", Canad. J. Math. 10, 73-77.

Storer, J., and R. Turyn 1961

"On Binary Sequences", Proc. Amer. Math. Soc. 12, 394-399.

Storer~ To 1967A

Cyclotomy and Difference Sets, Markham,

Chicago

1967B

"Cyclotomies and Difference Sets Modulo a Product of Two Distinct Odd Primes", Michigan Math. Jo 14, 117-127.

Turyn, R. 1960

"Optimum Codes Study", Final Report, Sylvania Electric Products~ Inc.

1961

"Finite Binary Sequences 't, Final Report, Products, Inc.

(Chapter VI), Sylvania Electric

165

Turyn, Ro 1964

"The Multiplier Theorem for Difference Sets", Canad. J. Math. 16, 386-

388. 1965

"Character Sums and Difference Sets", Pacific J. Math. 15, 319-346.

1968

"Sequences with Small Correlation",

in Error Correcting Codes, Ho B.

Mann, Editor, Wiley, New York, 195-228. 1970

"An Infinite Class of Williamson Matrices", to appear.

Van Der Waerden, B. L. 1949

Modern Algebra vol. i, Ungar, New York.

Wallis, J. 1969 A

"A Class of Hadamard Matrices", J. Combinatorial Theory 6, 41-44.

1969B

"A Note of a Class of Hadamard Matrices", J. Combinatorial Theory 6,

222-223. 1970

"Combinatorial Matrices",

Dissertation,

La Trobe University.

Watson, G. L. 1960

Integral Quadratic Forms, Cambridge University Press, Cambridge.

Whiteman, A. L. 1957

"The Cyclotomic Numbers of Order Sixteen", Trans. Amer. Math. Soc. 86, 401-413.

1960A

"The Cyclotomic Numbers of Order Twelve", Acta Arith. 6, 53-76.

1960B

"The Cyclotomic Numbers of Order Ten", Proc. Sym. in Appl. Math. i0, Amer. Math. Soc., 95-111.

1962

"A Family of Difference Sets", Illinois J. Math. 6, 107-121.

1970

"An Infinite Family of Kadamard Matrices of Williamson Type", to appear.

Yamamoto, K. 1963

"Decomposition Fields of Difference Sets", Pacific J. Math. 13, 337-352.

1967

"On Jacobi Sums and Difference Sets", J. Combinatorial Theory 3, 146-

181. 1969

"On the Application of Half-Norms to Cyclic Difference Sets", in Combinatorial Mathematics and its Applications, R. Co Bose and T. A. Dowling, Editors, The University of North Carolina Press, Chapel Hill.

166

Yates~ D. L. 1967

"Another Proof of a Theorem on Difference Sets", Proc. Cambridge Philos. Soc. 63, 595-596 .