Codici correttori: Un'introduzione (UNITEXT La Matematica per il 3+2)

Codici correttori Luca Giuzzi Codici correttori Un’introduzione 123 LUCA GIUZZI Dipartimento di Matematica Politec...

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Codici correttori

Luca Giuzzi

Codici correttori Un’introduzione

123

LUCA GIUZZI Dipartimento di Matematica Politecnico di Bari - Bari

ISBN 10 88-470-0539-6 Springer Milan Berlin Heidelberg New York ISBN 13 978-88-470-0539-6 Springer Milan Berlin Heidelberg New York Springer-Verlag fa parte di Springer Science+Business Media springer.com © Springer-Verlag Italia, Milano 2006

Quest’opera è protetta dalla legge sul diritto d’autore. Tutti i diritti, in particolare quelli relativi alla traduzione, alla ristampa, all’uso di figure e tabelle, alla citazione orale, alla trasmissione radiofonica o televisiva, alla riproduzione su microfilm o in database, alla diversa riproduzione in qualsiasi altra forma (stampa o elettronica) rimangono riservati anche nel caso di utilizzo parziale. Una riproduzione di quest’opera, oppure di parte di questa, è anche nel caso specifico solo ammessa nei limiti stabiliti dalla legge sul diritto d’autore, ed è soggetta all’autorizzazione dell’Editore. La violazione delle norme comporta sanzioni previste dalla legge. L’utilizzo di denominazioni generiche, nomi commerciali, marchi registrati, ecc., in quest’opera, anche in assenza di particolare indicazione, non consente di considerare tali denominazioni o marchi liberamente utilizzabili da chiunque ai sensi della legge sul marchio. Impianti forniti dall’autore Progetto grafico della copertina: Simona Colombo Stampa: Arti Grafiche Nidasio, Assago (Mi) Stampato in Italia

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n′ pi = 0

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C1 = C

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C k = {c ∗ d : c ∈ C k−1 , d ∈ C}.

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τ1 ∗ τ2 ∗ · · · ∗ τk = σ1 ∗ σ2 ∗ · · · ∗ σk ∈ C k ,

τ1 = σ1 , τ2 = σ2 , . . . , τk = σk .

C C = {0, 10, 100, 101} C 101010 ∈ C 3 101010 = 10 ∗ 101 ∗ 0 = 101 ∗ 0 ∗ 10.

= · ∗ − = .

= 000 ∗ 1 ∗0 ∗ 111 ∗000; ·

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i = 1!

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w w = σ1 ∗ σ2 = ϑ 1 ∗ ϑ 2 ,

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% C = {0, 10, 110, 1110, 1111} &

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{σ0 , σ1 , . . . , σr−1 }

A ∞ j=0

s A

||C||j s−j ≤ 1.

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C =

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r−1 i=0

!

s−|σi | ≤ 1.

k C C k k C !k !k r−1 X −|σ| X X −|σ| −|σi | s = s = s . i=0

σ∈C

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nmin nmax C ||C k ||l σi1 σi2 · · · σik ∈ C k l ! X

s−|σ| =

kn max X

l=knmin

σ∈C k

||C k ||l s−1 .

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A s

||C k ||l ≤ sl .

#

$ # kn max X

l=knmin

%

r−1 X i=0

||C k ||l s−1 ≤ knmax ,

s

−|σi |

!k

&

≤ knmax .

' % 1/k &

k → ∞

C = {σ0 , σ1 , . . . , σ9 } ⎧ 1 x = 1 ⎪ ⎪ ⎪ ⎨5 x = 2 ||C||x = ⎪ 4 x = 3 ⎪ ⎪ ⎩ 0

9 i=0

3−|σi | = 3−1 + 5 · 3−2 + 4 · 3−3 =

28 > 1. 27

M 71 1 2−|σ| = 2 · + 4 · 2−2 + 8 · 2−3 + 16 · 2−4 + 14 · 2−5 = > 1. 2 16 σ∈M

¾º¿ i ||M||i 0 0 1 2 2 4 . 3 8 4 16 5 14

Ì s φ : N → N

∞ X i=0

φ(i)s−i ≤ 1,

C

||C||x = φ(x)

r, s > 0 i = 0, 1, . . . , r − 1 ni C = {σ0 , σ1 , . . . , σr−1 } A |A| = s |σi | = ni r−1 X i=0

s−ni ≤ 1.

! " # $ A = {0, 1, . . . , s − 1}

ni

n0 ≤ n1 ≤ · · · ≤ nr−1 . % j 0 r − 1 & wj 0( $ j = 0' w0 = P j−1 nj −ni jP> 0' wj = i=0 s r−1 −ni %

) i=0 s ≤ 1'

wj =

j−1 X

snj −ni = sn j

i=0

j−1 X

s−ni < snj .

*

i=0

σj s & wj ' 0 + nj C = {σ0 , σ1 , . . . , σr−1 }

C " , - ' j, k 0 ≤ j < k ≤ r − 1

σj σk . ' wj = ⌊wk /snk −nj ⌋ /& Pk−1 nk −ni k−1 k−1 X n −n X n −n wk i=0 s = = s j i = wj + s j i ≥ wj + 1, nk −nj nk −nj s s i=0 i=j

¿¼

! ! " #

#

$ % & $ Ì' ! % ! # ( A = {0, 1, . . . , r − 1} i ∈ A pi #

p = (p0 , p1 , . . . , pr−1) ) * pi

!

0,1119 0,0073 0,0483 0,0398 0,1269 0,0116 0,0190

0,0141 0,0973 0,0000 0,0000 0,0614 0,0288 0,0706

0,0954 0,0255 0,0087 0,0604 0,0565 0,0568 0,0327

0,0225 0,0000 0,0002 0,0000 0,0044

Ì ) σi ! i ∈ A C = {σ0 , . . . , σr−1 } + , pi # σi

0.1300 0.1200 0.1100 0.1000 0.0900 0.0800 0.0700 0.0600 0.0500 0.0400 0.0300 0.0200 0.0100

0.0000

a b c d e f g h i j k l m n o p q r s tu v w x y z

0,0804 0,0154 0,0306 0,0399 0,1251 0,0230 0,0196

0,0549 0,0276 0,0016 0,0067 0,0414 0,0253 0,0709

0,0760 0,0200 0,0011 0,0612 0,0654 0,0925 0,0271

0,0099 0,0192 0,0019 0,0173 0,0009

C{σ0, σ1 , . . . , σr−1 } p = (p0 , p1 , . . . , pr−1) n=

r−1 i=0

pi |σi |.

ns (p)

p

s

Hs (p) ≤ ns (p) < Hs (p) + 1,

Hs

s

C/ns(p) − ǫ

¿¾

0.1300 0.1200

0.1100 0.1000

0.0900

0.0800

0.0700

0.0600

0.0500

0.0400

0.0300

0.0200

0.0100

0.0000

a bc d e f g h i jk lm n o p q r s t u v w x y z

A C

℄

p

! " # $ A % $ & A0 A1 $ '

% A1 ( % ) A0

* 0% A1 * 1% + A0 A1 A0 = A00 ∪ A01 ,

A1 = A10 ∪ A11 ,

, -

x ∈ Aσi σi .

/ A = {1, 2, 3, . . . , 7} ' ) + .$ {1, 2, 3} 0.60 $ {1, 2} 0.45 A0 = {1, 2},

A1 = {3, 4, 5, 6, 7}.

0

A 1 2 3 4 5 6 7

pi 0.25 0.20 0.15 . 0.15 0.10 0.10 0.05

Ì

A c∈C 1 00 01 2 3 100 . 4 101 5 110 6 1110 7 1111

A00 = {1}, A01 = {2},

A10 = {3, 4}, A11 = {5, 6, 7}.

1 00 2 01 A10 A100 = {3}, A101 = {4},

! A11 "

A110 = {5},

A111 = {6, 7}.

# A1110 = {6} A1111 = {7}

! A $ % &

¾º½º

¾º¾º

¾º¿º

∗

·

−

¿

¿º½

n n

!

"

# #

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$

A # " A

C n M M

n

% &

(n, M )

# "

A ' (n, M )

||C||

C n ||C|| : N → N

||C||(x) =

()*

(

M 0

x = n .

(7, 128)+

C = {(00000), (10110), (01011), (11101)}

(5, 4)

A, B

C n A k B

θ : B k → C C |B|k

¿º¾

n ! An " d V d : V × V → R a, b, c ∈ V # $ d(a, b) ≥ 0 d(a, b) = 0 a = b% & d(a, b) = d(b, a)% ' d(a, c) ≤ d(a, b) + d(b, c).

( x, y n A d(x, y) ) % d(x, y) = |{ i : xi = yi }|.

x = (10110) y = (11011) A = {0, 1}% d(x, y) = 3 x = (21012) y = (12001) A = {0, 1, 2} d(x, y) = 4

( x, y, z

n A $ d(x, y) = 0

x = y

x

y

¿º¿

¾º

d(x, y)

d(x, y) = d(y, x).

d(x, z) ≤ d(x, y) + d(y, z).

D1 = { i : xi = zi };

D2 = { i : xi = yi

xi = zi

zi = yi

"

A

xi = yi

yi = zi }.

D1 ⊆ D2

⊓ ⊔ C n w ∈ C w δ

!

#

Bδ (w) = { c ∈ C : d(c, w) ≤ δ}.

" $

"

%

C = R2 p = (p1 , p2 ) ∈ C &

2

B1 (p) = { (x, p2 ) : x ∈ R} ∪ { (p1 , y) : y ∈ R}.

B2 (p) = R2 . # '

¿º¿

( "

!

C

(n, M )

" "

m′

)

µ

k

λ : Ak → C

2 1 0 -1 -2 -2

-1

0

1

2

R2 ½º c′ m′ µ(m′ ) := µ(c′ ) = λ−1 (c′ ).

C m

e m m′ ∈ C d(m, m′ ) = e

E = {d(c, m) : c ∈ C} e = min E

Be (m) ∩ C x m′ = x

(e, m′ ) e

! "

M # $

C % &

n '

! " $

¿º¿

m

B2 B3

′

B1

½

C

m

e m m′ ∈ C d(m, m′ ) = e

e = 0

De = Be (m) ∩ C = ∅ |De | = 1

e De |De | > 1

e

e = e + 1

m′ ∈ C d(m, m′ ) = e

e

m′ C e m e

C 1

n n

De e 0 n C

!

!

"

k (n, M )

C # !

n "

! !

!

$

% (n, M )& A !

k ' R=

k . n

# ' M ≈ |A|k ( |A| = q R=

logq M ; n

! M ≤ q n R ≤ 1

% (n, M )& ! k ' r = n − k.

) $

!

*

# ' $ M = |A|k $

+ r = 1 − R. n

C (00) (01) (10) (11) (00) → (00000) (10) → (10110)

(01) → (01011) (11) → (11101).

R = 2/5 r = 3

e !

"

C (n, M )# $

C ! "

d = min{ d(x, y) : x, y ∈ C, x = y }.

% C d = 3 & (n, M )# C d

(n, M, d)#

' C = {c1 , c2 , c3 , c4 , c5 } A =

{a, b, c, d, e}

c1 = (abcde) c2 = (acbed) c3 = (eabdc) c4 = (dceda) c5 = (adebc).

i = j d(ci , cj ) = 4 d = 4

d < n

(n, M, d) δ=

d . n

δ < 1 d C

M

2

! " # $ (n, M ) C d

C r ≤ d−1 2 % m & m′ ∈ C d(m, m′ ) ≤ d−1 2

m′ '( )* #

¾

+ C

n d, + m

m′ ∈ C d(m′ , m) ≤

d−1 2 ,

e = d(m, m′ )

D = B(d−1)/2 (m), D ∩ C = {m′ } m′ e = d(m, m′ ), D ∩ C = ∅

d/2

m (d − 1)/2

2

(n, M )

Ì C (n, M ) d C

e = ⌊(d − 1)/2⌋ d − 1

C e = ⌊(d − 1)/2⌋

(n, M ) d

e = ⌊(d−1)/2⌋

d

1 ≤ t ≤ d − 1 m ⊓ ⊔ m′ ∈ C ! " e

#

$ #

% e

# (n, M, d) C

C i #

Bi (c) ∪ Bi (c′ ) = ∅,

c, c′ ∈ C C = {c1 , c2 , c3 , c4 } c1 = (00000),

c2 = (10110),

c3 = (01011),

c4 = (11101).

& '

d = 3$ ( )

*

S #

A = {0, 1}

C

|S| = 32

5

1

Sc1 = {(00000), (10000), (01000), (00100), (00010), (00001)}

Sc2 = {(10110), (00110), (11110), (10010), (10100), (10111)} Sc3 = {(01011), (11011), (00011), (01111), (01001), (01010)}

Sc4 = {(11101), (01101), (10101), (11001), (11111), (11100)}.

5

S∗

24

32

S ∗ = {(11000), (01100), (10001), (00101), (01110), (00111), (10011), (11010)}. C

r = (00011)

d(c1 , r) = 2,

d(c2 , r) = 3,

d(c3 , r) = 1,

d(c4 , r) = 4.

r ∈ Sc3 r m′ = c3 s = (11000)

d(c1 , s) = 2, !

d(c2 , s) = 3, "

s %

d(c3 , s) = 3,

d(c4 , s) = 2.

# $

&

3

" %

C

3

w ∈ A

w

C

C

d(w, C) := min d(w, c). c∈C

C ⊆ A

n

ρ(C) = maxn d(x, C). x∈A

! ' (

n) m

Bρ (m) ∩ C

%

*

+

%

(n, M ) C A

e n A e C C

2e + 1

!

n M

d

C

A

" n # M ≤ |A| d ≤ n

Ì

A

C

(n, M, d)

M ≤ |A|n−d+1 . #

C

'

d−1

$%

M

C

&

d ( (n − d + 1, M, 1)) A *

⊓ ⊔

M ≤ |A|n−d+1

+ ,

-

|A| = q .

$%

n

q

logq M 1 ≤1−δ+ , n n

R=

δ ≤ 1 − R.

Ì

(n, M ) e

q = |A|

C ⊆ An

e n M (q − 1)i ≤ q n , i i=0

C

e M |Be (x)| ≤ q n .

Be (x) n x n

x n2 ! |Be (x)| =

e n (q − 1)i . i i=0

" #

⊓ ⊔

$ % & B r n ' q ( % B % ' Vq (n, r) & Vq (n, r) ' ) 0 < δ < 1/2

lim

n→∞

1 log2 V2 (n, nδ) = H(δ). n

V2 (n, nδ) '' V2 (n, nδ) =

nδ n . i i=0

" δ

n i+1 n i

=

n−i ≥ 1, i+1

*

i 0 nδ V2 (n, nδ)

n nδ

n . ≤ V2 (n, nδ) ≤ (nδ + 1) nδ

!

√ k! ≈ k (k+1)/2 e−k 2π.

" # $ # log2

n nδ

= −n(1 − δ) log2 (1 − δ) − nδ log2 δ + o(log n) = nH(δ).

% H(δ) ≤

1 1 log V2 (n, nδ) ≤ log2 nδ + 1 + H(δ). n n

n → ∞ log2 (nδ + 1)/n → 0

⊓ ⊔

" & '

( # q #) δ 0 1 lim

n→∞

1 logq Vq (n, nδ) = Hq (δ) + δ logq (q − 1). n

Hq (x) *# q +

Ì

C R δ C „ « δ R + H2 ≤ 1. 2

p = δ/2 (d − 1)/2 ≃ pn M V2 (n, pn) ≤ 2n ; 2 n R+

log V2 (n, pn) ≤ 1. n

⊔ ⊓

A n ≥ 1 ϕ :

An → An

An

l, m ∈ An

d(ϕ(l), ϕ(m)) = d(l, m).

An

C C

′

C

n A C ′

C

!

C

"

An

" "

#

A = {a, b, c}

C

(3, 3)$

(aaa), (abc), (cbb). %

C′

(aaa), (bac), (bcb),

C

&

C

' ′′

%

C

(baa), (bbc), (abb),

C

b

b

C ′′′

a

a

c

c

(baa), (cac), (ccb) "

C

(

)

D (aaa), (aab), (aca)

ϕ An

An Sn ! "

#

$ c = (c1 , c2 , . . . , cn ) " n c′ = (c′1 , c′2 , . . . , c′n ) c σ ∈ Sn " i 1 ≤ i ≤ n! c′i = cσ(i) . !

$

c′ = cσ C " n σ ∈ Sn I = {1, 2, . . . , n} C σ C σ = { cσ : c ∈ C}.

An

"

Sn

% & !

$

{1, 2, . . . , n} → Sym(A)

A

γ

n α :

A " n

A → A ' (

α, β

"

n

#

"

γ(x) = α(x)β(x),

x ∈ {1, 2, . . . , n}

) "

"

n!

Γn (A)

!

* # ! " αi α(i) + # Γn (A)!

θ ∈ Sym(A) ( θ i = j (θ, i)(j) = 1 i = j.

*

(θ, i)

θ i

Γni (A) := {(θ, i) : θ ∈ Sym(A)}

Sym(A) 1 Γn (A) Γni (A) ∩ Γnj (A) = {1}

i = j

α ∈ Γn (A) n Y α= (αi , i), (αi , i) ∈ Γni (A) i=1

(αi , i) ∈ Γni (αj , j) ∈ Γnj i = j

(αi , i)(αj , j) = (αj , j)(αi , i).

Γn (A) =

n Y

Γni (A)

i=1

n Sym(A)

! !

" c = (c1 , c2 , . . . , cn ) !! n

A α ∈ Γn (A) #

α c cα = ( α1 (c1 ), α2 (c2 ), . . . , αn (cn )).

C !! n ! α ∈ Γn (A) α C

Cα = {cα : c ∈ C}.

#! Γn (A)

An

Γ

n (A)

An

" c = (c1 , c2 , . . . , cn ) d = (d1 , d2 , . . . , dn ) An α !

i ! (ci di ) α ∈ Γn (A) α(c) = d ⊔ ⊓

$ ! % & ' C (n, M )(

A

Cα

C σ (n, M )(

Ξn (A) = Sn Γn (A)

Sym(An )

! An

) S

n Γn

! *

Ì

C D

ϑ:C→ D

ϑ:C → D

l, m ∈ C

σ ∈ Sn

d(lσ , mσ ) = |{ i : mσ i = lσi }| = |{ σ(i) : mi = li }| = |{ i : mi = li }| = d(l, m),

α ∈ Γn (A) 1 ≤ i ≤ n li = mi αi (li ) = αi (mi ) !

d(lα , mα ) = |{ i : αi (li ) = αi (mi )}| = |{ i : li = mi }| = d(l, m), ⊔ ⊓

! " # $

A

Ξn (A)

Ξn (A)

Sn

n

Γn (A)

Ξn (A) Sn Γn (A) Sn Γn (A) −1 α ∈ Γn (A) σ ∈ Sn σασ ∈ Γn (A) # c ∈ An

!

%

&

(σασ −1 )(c) = (σα)(cσ(1) , cσ(2) , . . . , cσ(n) = σ(α1 (cσ(1) ), α2 (cσ(2) ), . . . , αn (cσ(n) )) = (ασ −1 (1) (c1 ), ασ −1 (2) (c2 ), . . . , ασ −1 (n) (cn )). '

(σασ −1 )

⊔ ⊓

¿º½º

A = {a, b, c} C = {aab, abc, cbb}

C ′

C

ccc

¿º¾º M

¿º¿º C

(n, M, d)

A

C (t) = {c1 c2 . . . ct : ci ∈ C}

C (t)

(6, M, 4)

(8, 4, 5) C = {(1001 0100), (1000 1011), (0111 0011), (0110 1100)},

r = (0001 0011).

!"℄ $ %

& ' & (

) *

*

½ $ $ + *

1

A

A A

A

! " "

#$%℄ '

A = Fq Fq

q (

C

n Vn (Fq ) = Fnq n Fq (

C

) k K Vk (Fq ) k ≤ n Fq

M = qk

ϑ : Vk (Fq ) → Vn (Fq ) ϑ

* (n, M )+ C

[n, k]+ ,

Fq - K

k

Fq

. Vk (Fq ) M = q k / ϑ : Vk (Fq ) → Vn (Fq )

ϑ

k Vn (Fq ) sk =

(q n − 1)(q n−1 − 1) . . . (q n−k+1 − 1) ; (q k − 1)(q k−1 − 1) . . . (q − 1)

sk [n, k]+ 0 ! [n, k, d]

[n, k])

1 d

5 (

K = V2 (F2 ) {(0 0), (1 0), (0 1), (1 1)} C

F2 2

) K = V5 (F2 )

C

B = {(10111), (11110)}.

(

(00) → (00000) (10) → (10111)

(01) → (11110) (11) → (01001). C = {(00000), (10111), (01001), (11110)}.

C 2

Vk (Fq ) Vn (Fq ) Vk (Fq )

! Vn (Fq ) " # $% # $& ' n k ( ϑ

(

k Vn (Fq )

! " # $

% &

! ! %

!

$

& '

& '

& &$ % & %

! B Vn (Fq )

(

!

) F X # & FX f : X → F f, g ∈ FX λ, µ ∈ F (λf + µg)(x) := λf (x) + µg(x).

(λf + µg) ∈ FX

FX F X |X| = n X = {1, 2, . . . , n} E FX

( 1 i = j ei (j) = 0 i = j,

i ∈ X

i !

FX n = |X|

! T ⊆ X "

T F χT : X → F ( 1 j ∈ T χT (j) = 0 j ∈

T.

" χT # E

X ei (x). χT (x) = i∈T

!

Fq X

|X| = n $ [n, k]

k% FX q d(f (x), g(x)) = |{ x ∈ X : f (x) = g(x)}|.

f

∈ C C ≤ FX

w(f ) = |{ x ∈ X : f (x) = 0}|.

&

' ( ) *

( + (n, M )% , -

[n, k, d]

C F

q

(n, q k , d)

. X = {1, 2, . . . , n} Θ : C → Fnq

Θ : f → (f (1), f (2), . . . , f (n)).

. C Θ = {(f (1), f (2), . . . , f (n)) : f ∈ C} C Θ

Θ

n Fn q ( - C k Θ k ' C

q ( ' C (n, q )% . /

C C Θ -

* 0 Θ f, g ∈ C f = Θ(f ) = (f (1), f (2), . . . , f (n))

g = Θ(g) = (g(1), g(2), . . . , g(n));

'

d(Θ(f ), Θ(g)) = |{ x ∈ X : f (x) = g(x)}| = d(f (x), g(x)).

1

C (n, q k , d)%

⊔ ⊓

Θ(f ) f

E

Ì

q n d C [n, k, d] [n, k, d] C ′ C E C ′ X = {1, 2, . . . , n} ξ:

j

X Fn q → Fq P a = (a1 , a2 , . . . , an ) → a(x) = n i=1 ai ei (x).

ξ ! d(a, b) = d(a(x), b(x)).

" C ′ = ξ(C)

⊔ ⊓

" #!

$ %

&

• •

d ≤ n − k + 1

n (q − 1) ≤ q n−k . i i=0

e

i

[n, k, d] Fq

d = n − k + 1

C F5 B = {(3 4 1 0), (0 3 4 1)}. [4, 2] 25 ! "#

"$ % & 3 ' ( 3 = 4 − 2 + 1 C &

)*

(0 0 0 0) (1 0 3 4) (2 0 1 3) (4 0 2 1) (3 0 4 2)

(0 1 3 2) (1 1 1 1) (2 1 4 0) (4 1 0 3) (3 1 2 4)

(0 2 1 4) (1 2 4 3) (2 2 2 2) (4 2 3 0) (3 2 0 1)

(0 4 2 3) (1 4 0 2) (2 4 3 1) (4 4 4 4) (3 4 1 0)

(0 3 4 1) (1 3 2 0) (2 3 0 4) (4 3 1 2) (3 3 3 3)

Ì

x

w(x) 0

x ∈ Vn (F)

f ∈ C C

f = ξ −1 (f ) E FX

w(C)

[n, k] C

w(C) = min{ w(x) : x ∈ C, x = 0 }.

C

f , g ∈ C

d(f , g) = w(f − g).

f , g C ∆ = { i : fi = gi } ∆ = { i : fi − gi = 0}

d(f , g) = d(f − g, 0) = w(f − g).

d C

⊓ ⊔

d = w(C).

C n F

t Wt (C)

q

Wt (C) := { w ∈ C : w(w) = t}.

t ≤ n

Wt (C) C t c ∈ C

v1 = (1 0 0 0 1) v2 = (1 1 0 1 0) v3 = (1 1 1 0 1) [5, 3] C ⎞ ⎛ ⎞ ⎛ 10001 v1 G = ⎝v2 ⎠ = ⎝1 1 0 1 0⎠ . v3 11101

m = (m1 m2 m3 ) ∈ F32 c = mG = m1 v1 + m2 v2 + m3 v3 .

! " m = (1 0 1) ⎛ 1000 mG = 1 0 1 ⎝1 1 0 1 1110

⎞ 1 0⎠ = 0 1 1 0 0 . 1

# ! d = 2

$ [5, 3] u1 = (10001) u2 = (01010) u3 = (00111) 3 V5 (Z2 ) ! % & ⎞ ⎛ ⎞ 10001 u1 G = ⎝u2 ⎠ = ⎝0 1 0 1 0⎠ . u3 00111 ⎛

" d = 2

$ C k ' n V Vn (Fq )( V ) Vn (Fq )( ' E qk C k ) )

# ' & *

+

, [n, k] C ! k × n ' E Vn (Fq ) C

C

! " #

!

$ %&

' G C

G = (Ik A),

Ik k × k A k × (n − k)

'

G () %*

C [n, k]+ ' I = {i1, . . . , ik } k " v1 , . . . , vk (

c ci = v1 , ci = v2 , . . . , ci = vk 1

2

k

, I C

I C C ( I = {1, 2, . . . , k}

- m = (101) G′ () %* c = (10110)

m c

.

G′ , () %/

.

C [7, 3, 3]+ F2

⎛

⎞ 1101001 G = ⎝1 1 1 0 1 0 0 ⎠ . 0011010

I = {1, 2, 3}

C I = {1, 3, 4}

C

n−k

k

n−k

k

k

{1, 2, . . . , k}

!

" "

"

# $%%

&

C = {(0, 1), (0, 0)}

[2, 1]' # (2, 2)' %* C (2, 2)'

( ) C ′ = {(1, 1), (1, 0)}

) +

[n, k]'

, -

[7, 3]'

⎛ ⎞ 1000000 G = ⎝0 1 0 0 0 0 0⎠ , 0010000

C

C′

⎛ ⎞ 1001010 G′ = ⎝0 1 0 0 1 0 1⎠ . 0010110

C C ′

C ↔ C ′ ⎞ ⎛ 1000000 ⎜0 1 0 0 0 0 0 ⎟ ⎟ ⎜ ⎜0 0 1 0 0 0 0 ⎟ ⎟ ⎜ ⎟ T =⎜ ⎜1 0 0 1 0 0 0 ⎟ ; ⎜0 1 1 0 1 0 0 ⎟ ⎟ ⎜ ⎝1 0 1 0 0 1 0 ⎠ 0100001

T = T −1 C 1 C ′ 3

C C

ϕ : C → C ′

′

C C ′ G G′ C C ′ G′ G

! "

!

"

# [n, k]$ C Fq G = (Ik A) C [n, k]$ C ′ C % " k C # G = (gij ) C

½º

k = 1

G

−1 g11

1

k − 1

C

k ′

C

k − 1 C

′ ′′

G C

C

′′ G′′ = (gij ) C ′

σ

k × n ⎛ ⎞ ′′ ′′ 1 0 ... 0 g1,k . . . g1,n ′′ ′′ ⎜ 0 ⎟ 1 ... 0 g2,k . . . g2,n ⎜ ⎟ ⎜ ⎟ = ⎜ G ⎟ ⎜ ⎟ ′′ ′′ ⎝ 0 ... 1 gk−1,k . . . gk−1,n ⎠ gk,σ(1) gk,σ(2) . . . gk,σ(k−1) gk,σ(k) . . . gk,σ(n)

G

gk,σ(1)

C

gk,σ(2)

!

"

C ⊓ ⊔

# " $

C

≤ FX D ≤ FY ϕ : C → D f ∈ C w(ϕ(f )) ≤ w(f ).

! " f, g ∈ C

d(ϕ(f ), ϕ(g)) ≤ d(f, g). # $

C D

D

ϕ : C → ψ:D→ C

C D C D # %&'

C C ′ C ′ C

Ì ϕ : Vn → Vn

w ∈ Vn

ϕ(0) = 0 Vn ! " ϕ

v, w ∈ Vn # " $ %"

d(v, w) = w(v − w) = w(ϕ(v − w)) = w(ϕ(v) − ϕ(w) = d(ϕ(v), ϕ(w)).

& ϕ

⊔ ⊓

ϕ ' " ϕv (x) = x + v,

v Vn 0 Vn (" v = 0) " " * X " Y " σ : Y → X + F # " f ∈ FX σ ⋆ (f ) ∈ FY

XO σ

f

/ F ? σ ∗ (f )

Y

σ ⋆ (f ) " " FY " y ∈ Y" σ ⋆ (f )(y) = f (σ(y)). [n, k], # "

* X Y " F σ : X → Y * α : Y → F⋆

Y F Φσ,α σ α FX → FY

Φσ,α (f )(y) := f (σ(y))α(y).

' " + X = Y " α

j X F → FX α: f (x) → α(x)f (x), Φσ,α Φσ,α (f ) = f σα.

Ì Φσ,α FX FY

f, g ∈ FX

y ∈Y

Φσ,α (f )(y) + Φσ,α (g)(y) = f (σ(y))α(y) + g(σ(y))α(y) = (f (σ(y)) + g(σ(y)))α(y) = (f + g)(σ(y))α(y) = Φσ,α (f + g)(y).

λ ∈ F

λΦσ,α (f )(y) = λf (σ(y))α(y) = (λf )(σ(y))α(y) = Φσ,α (λf )(y). ⊔ ⊓

x ∈ X

α, β : X → F

α·β

X → F

(α · β)(x) = α(x)β(x).

α β

FX

1 ∈ F " # 0 ∈ F ⋆ "! X → F = F \ {0} ! $ ·% ⋆ & FX ⋆ X → F n = |X|

# β : X → {1, 2, . . . n} ⋆ ⋆

FX F{1,2,...,n}

Γn (Fq ) Fq n '( ) γ ∈ Γn (Fq )

X f ∈ F

x∈X

!

f γ (x) = f (β −1 γ(β(x))).

#

* '

F⋆X

(

!

F⋆X

x ∈ X

F

⋆

x∈X

αy y

!

+

F⋆X =

Y

α ∈ F⋆q

|X| = n

)

(α, x)

x=y x = y.

Υx = { (α, x) : α ∈ F⋆ } Υx

x∈X

β

$ %

(α, x)(y) =

Γn

F⋆

Ì

Σ

FX

Φσ,α Φϑ,β Σ

(Φσ,α Φϑ,β )(f )(x) = Φσ,α (f (ϑ(x)β(x))) = f (ϑ(σ(x))β(σ(x)))α(x) = f (ϑ(σ(x)))β(σ(x))α(x) = Φϑσ,βσ·α (f )(x).

ϑσ ∈ Sym(X) β ∈ F⋆X βσ ∈ F⋆X

x ∈ X β(σ(x)) ∈ F⋆

α, β ∈ F⋆X x ∈ X α(x)β(x) ∈ F⋆ F⋆X ! Σ

1 : X → F ! X " F 1X X Φ1X ,1 Φσ,α = Φσ1X ,α1X 1 = Φσ,α ,

#

%$Φσ,α Φ1X ,1 = Φ1X σ,1σα = Φσ,α ,

#&%

Σ ' Φσ,α ∈ Σ α : X → F α(x) = α(σ −1 (x))−1 .

Φσ −1 ,α Φσ,α = Φσσ−1 ,ασα = Φ1X ,(ασ−1 σ)−1 α = Φ1X ,1 ;

Σ ( Φσ,α Φθ,β Φη,γ Σ Φη,γ (Φσ,α Φθ,β ) = Φη,γ (Φθσ,βσ·α) = Φ(θσ)η,(βση·αη)·γ = Φ(ση),αη·γ Φθ,β = (Φη,γ Φσ,α )Φθ,β ,

" ( !

⊔ ⊓

Σ

Σ

⋆

Sym(X) FX

FX

)

Υ :

j

Sym(X) → Σ σ → Φσ −1 ,1

! $ Υ (1) = Φ1X ,1 & Υ (σ −1 )Υ (σ) = Φσ,1 Φσ −1 ,1 = Φ1X ,1 , Υ (σ −1 ) = (Υ (σ))−1 * Υ (σ)Υ (η) = Φσ −1 ,1 Φη−1 ,1 = Φη−1 σ −1 ,1 = Υ (ση) Υ (σ) = Φ1X ,1 σ −1 " Sym(X)

σ = 1

Λ:

j

F⋆X → Σ α → Φ1X ,α

Λ(1) = Φ1X ,1 ! " Λ(α)Λ(α−1 ) = Φ1X ,α Φ1X ,α−1 = Φ1X ,α1X ·α−1 = Φ1X ,αα−1 = Φ1X ,1 Λ(α−1 ) = (Λ(α))−1 ! # Λ(α)Λ(β) = Φ1X ,α Φ1X ,β = Φ1X ,α1X ·β = Φ1X ,αβ = Λ(αβ)! Φ1X ,α = Φ1X ,1 α $ F⋆X Φ1X ,β

Λ

F⋆X % σ −1 ∈ Sym(X) α ∈ F⋆X Λ(α)Υ (σ −1 ) = Φ1X ,α Φσ,1 = Φσ,1σ·α = Φσ,α ,

Σ Sym(X) F⋆X &

α ∈ F⋆X σ ∈ Sym(X) Υ (σ)−1 Λ(α)Υ (σ) ∈ Λ(F⋆X ).

'(( Υ (σ)−1 Λ(α)Υ (σ) = Φσ,1 Φ1X ,α Φσ −1 ,1 = Φσ,1 Φσ −1 ,ασ = Φ1X ,σ−1 ασ .

&

Φ1X ,β Λ(F⋆X ) Σ Σ ) ⊔ ⊓ Vn *

+

( , θ : Vn → Vn Vn

σ ∈ Sn α1 , . . . , αn ∈ F⋆ x = (x1 , x2 , . . . , xn ) ∈ Vn θ : (x1 , x2 , . . . , xn ) → (α1 xσ(1) , α2 xσ(2) , . . . , αn xσ(n) ).

-

.

/ n In

0 M

P D

0 &

$

% C C ′

θ : C → C ′ % # !

% " 1

0 #

Ì [n, k] C C ′ Fq

G G′ ′ D

G = GD,

C

C′

G′ = GP,

P

C C ′ n

ϕ : C → C′

C C ′ θ : C → C ′

C

G=

„

« 100 . 011

C

θ : (c1 , c2 , c2 ) → (c2 , c1 , c1 ),

C !

"

# $ % &

C D

' % (

) )

* ! ' % C D C D + , #

& # & # '

& - .

/

0 1

$

C D n

C D Fq % FX q X = {1, 2, . . . , n} 2

Φσ,α : C → D 2 3 +4 5

Φσ,α = Λ(α)Υ (σ),

α ∈ F⋆q X σ ∈ Sym(X)

Λ(α) Υ (σ) FX

f ∈ FX !

Λ(α)(f )(x) = α(x)f (x) " 0 f (x) = 0

α(x) = 0 #

α(x) " x$

w(Λ(α)(f )) = w(f );

Λ(α) " % # Υ (σ)

w(f ) = |{ x ∈ X : f (x) = 0}| = |{ σ(x) ∈ X : f (σ(x)) = 0}| =

|{ x :∈ X : f (σ(x)) = 0}| = w(Υ (σ)(f ));

⊔ ⊓

&

" Σ

FX " ' FX

Σ FX Wt (FX ) t 0 n = |X|

# ( )) Wt (FX ) " Σ FX

Σ " Wt (FX ) * t

0 ≤ t ≤ n

f, g ∈ Wt (FX ) # ωf = { x ∈ X : f (x) = 0},

ωg = { x ∈ X : g(x) = 0}.

#

|ωf | = |ωg | = t β : ωg → ωf ! + f (β(x)) = 0 g(x) = 0 ( β '

σ X f (σ(x)) g(x) *

j g(x)/f (σ(x)) g(x) = 0 α(x) =

g(x) = 0 1 , α(x) ∈ F⋆X -% j ﬀ 0

g(x) = 0 Φσ,α (f )(x) = α(x)f (σ(x)) = = g(x), g(x)f (σ(x))/f (σ(x)) g(x) = 0

⊔ ⊓

- FX

FX

Σ

Ψ FX Ψ FX Ψ

E Ψ ! ! i ∈ X j ∈ X

λ ∈ F⋆

Ψ (ei ) = λi ei′ .

α : X → FX α(i) = λi σ ∈ Sym(X) " ! i ∈ X i′ Φσ,λ ! x ∈ X Φσ,α (ei )(x) = α(x)ei (σ(x)) = λi (x)eσ(i)(x) = λi ei′ (x);

Φσ,α = Ψ # ! #

⊔ ⊓

$

" FX

Ì

C D FX

Ψ : C → D Ψe X e F Ψ|C = Ψ

% & '()*℄ '((℄

C D Fq D Φ : C → D Ψ : C → D Ξ : C → # ! Φ "

D ! ( , Ψ C →

- . Ψe FX - Ψe ( , ( ! / , ! $ 0 ! / ⊔ ⊓

B

V

V

B : V × V → F

α ∈ F B(x + y, z) = B(x, z) + B(y, z) B(x, y + z) = B(x, y) + B(x, z) B(αx, y) = B(x, αy) = αB(x, y) ! "

t #

j V →F Bt : x → B(t, x)

V

F x, y, z ∈ V

Bt :

V % &V v ∈ V 0 ∈ F

$

j

V →F x → B(x, t)

$ #

B

V → F

rad V B = { t ∈ V : Bt = &V }.

rad V B = { t ∈ V : B t = &V }.

rad V = { t ∈ V : B(t, x) = 0, ∀x ∈ V },

rad V = { t ∈ V : B(x, t) = 0, ∀x ∈ V }. '

$ #

( )

*

V F rad V B rad V B V

F dim rad V B = dim rad B rad V B = {0} rad V B = {0}

B

B

V x, y ∈ V

$ #

+

rad V B = rad V B

B(x, y) = B(y, x),

$

B

' $ # ,

$ #

B

rad V B = rad V B = {0}. !

#

V

V F V ·, · : V × V → F

V

V ≃ Rn n R j V × V → PR ·, · : x, y → n i=1 xi yi x, y, z ∈ V !

" n n X X x, y = x i yi = yi xi = y, x; i=1

#

x + y, z =

$ α ∈ R αx, y =

n X i=1

n X

i=1

(xi + yi )zi =

x i zi +

n X

xi (αyi ) = α

n X i=1

i=1

i=1

(αxi )yi =

n X

n X i=1

i=1

yi zi = x, z + y, z;

xi yi = αx, y = x, αy;

B = {e1 , . . . , en } V y ∈ V !

yi = y, ei .

y yi = 0 i% & y = 0 Rn ' ! x ∈ Rn \ {0} x, x > 0.

B V F x ∈ V x = 0 ! B(x, x) = 0 ( B X ) FX

X f, g = f (x)g(x). x∈X

* f g E !

n X f , g = fi gi . i=1

+ &

E

·, · :

Vn (Fq ) × Vn (Fq ) → Fq n (f , g) → i=1 fi gi .

C ⊥

C

Fq

Vn (Fq ) C ⊥ = { x ∈ Vn (Fq ) : x, y = 0

[n, k]

y ∈ C }.

C [n, k] F C ⊥ [n, n−k] F

C ⊥ Vn (Fq )

k C n − k ⊓ ⊔

!

C ⊥ C " C ⊥ = C

C

c ∈ C ⊥ ∩ C

C

C ⊥

G = (Ik A) C H = (−AT In−k ) C ⊥

H n − k ! # H C ⊥ $ H C ⊥ %

" m m

c = mG.

H GH T = 0,

cH T = mGH T = 0,

c #

H T

C ⊥ ⊓ ⊔

C [7, 4] Z3 ⎛ 0 ⎜0 G=⎜ ⎝2 2

00 01 10 00

G′ = (I4 A)

C 3 2 7 1 4 6

00 ⎜0 0 ⎜ ⎜0 1 ⎜ P =⎜ ⎜0 0 ⎜1 0 ⎜ ⎝0 0 00

⎛

100 ⎜0 1 0 ′ ⎜ G =⎝ 001 000

00 00 02 12

D⊥

G

00 10 00 00 00 01 00

⎞ 0 0⎟ ⎟. 0⎠ 1

G

⎛

211 102 202 100

H′

10 00 00 00 00 00 01

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 0⎠ 0

⎞ 21 1 2⎟ ⎟ = (I4 A), 2 2⎠ 10

⎛

02 ⎜0 1 ⎜ A=⎝ 22 21

⎞ 1 2⎟ ⎟. 2⎠ 0

⎛

5 3 2 6 1 7 4

PT

⎛ 0 ⎜0 ⎜ ⎜0 ⎜ =⎜ ⎜0 ⎜1 ⎜ ⎝0 0

000 010 100 000 000 000 001

10 00 00 01 00 00 00

GH T = 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟, 0⎟ ⎟ 1⎠ 0

C ⊥ ⎛ ⎞ 1100001 H = ⎝0 1 2 1 1 0 2⎠ . 0110210

5

⎞ 0011100 H ′ = (−AT I3 ) = ⎝1 2 1 2 0 1 0⎠ ; 2110001

D

H

n − k = 3

C [n, k] F

C H C ⊥

q

H

[n, k] C

v

Vn (Fq ) C vH T = 0

H

[n, k]

C F

x ∈ Vn (Fq )

s ∈ Vn−k (Fq )

s = xH T .

0 ! "

#" "

C

C d − 1 " H

H

Hi

[n, k, d]

i

$$ r % i1, i2 , . . . , ir+1 " ci1 Hi1 + ci2 Hi2 + · · · + cir+1 Hir+1 = 0,

ci

& " c " " ij ci C " r + 1' d ≤ r + 1 ⊓ ⊔

j

j

( "

H

[n, k] C

r − 1 H

d C r

r − 1

H

r = 0 0 r d ≥ r ⊓ ⊔

Ì

C Fq [n, k, d] T n − k Fn−k q ⊥ C C

n − k

H C !! w(C) ≥ n − k + 1 " w(C) = n−k +1 #

H $ 0 n−k+1

n − k

% C &' [n, k, d] (

H C C ⊥ ) v ∈ C ⊥ * # H d − 1 = n − k

H v

* d − 1 0 v n − (d − 1) + 1 = k + 1 C ⊥ + ⊓ ⊔ #

[n, k, d]

C ! k

! ! C " ! k

, # $ #

C -

. $ ! C # ( #.

! $ #

C $ # . /

0 1

C A DC (Z) =

1 d(c,d) Z . |C| c,d∈C

C

AC (Z) :=

C

Z w(c) .

c∈C

C AC (Z) = DC (Z).

C n Ai C

i !

" AC (Z) =

n

Ai Z i .

i=0

#

C $ % &

" (Ai )ni=0

WC (X, Y ) =

n

Ai X i Y n−i ,

i=0

n '

$()* +,-℄ +/℄ '

* 0 C [n, d, k]) 12 Fq C

" Ai =

i−d n i − 1 i−j−d (−1)j . (q − 1) q i j j=0

Ì

q C

WC ⊥ (X, Y ) =

C⊥

1 WC (Y − X, Y + (q − 1)X). |C|

℄ ! ℄ " # $ ω ∈ Fq n%

& '( ! ω n = 1( ω j = 1 0 < j < n ) Tr : Fq → Fp

Fq Fp V = Fn q

& f : V → F ! & fˆ : V → F X Tru,v ω f (v). fˆ(u) = v∈V

) * ( " # ! ) * *+ * &

C

X

u∈C

X

u∈C

Fq

fˆ(u) = |C|

X

f (v).

v∈C ⊥

,

X X Tru,v X X Tru,v ω f (v) = f (v) ω . fˆ(u) = u∈C v∈V

v∈V

u∈C

$ v ∈ C ⊥ ( Tru, v = 0- ( . |C| $ v ∈ C ⊥ ( Fp * " ν (

ν(1 + ω + · · · + ω p−1 ) = 0. / *

⊔ ⊓

0 * 1 2 $ R & X, Y

C- Vn n Fq ( *

C C ⊥ 3 * v ∈ Vn (

f (v) = X w(v) Y n−w(v) .

" # f !

fˆ(u) =

X

X w(v) Y n−w(v) ω Tr(u,v) .

v∈Vn

δ : Fq → {0, 1}

δ(0) = 0 δ(a) = 1 a = 0 P x ∈ Vn w(x) = δ(xi ) fˆ X X δ(v1 )+···+δ(vn ) Y (1−δ(v1 ))+···+(1−δ(vn )) ω Tr(u1 v1 +···+un vn ) ; v1 ,v2 ,...,vn ∈F

fˆ(u) =

n X Y

X δ(vi ) Y 1−δ(vi ) ω Tr(ui vi ) .

i=1 vi ∈F

ui = 0 Y + (q − 1)X ui = 0 Y − X + q/p(1 + ω + · · · + ω p−1 )X = Y − X.

!

fˆ(u) = (Y − X)w(u) (Y + (q − 1)X)n−w(u) ,

" #$

⊔ ⊓

% AC (Z) [n, k]&

C AC⊥ (Z) C ⊥

„ « (1 + (q − 1)Z)n 1−Z AC AC⊥ (Z) = ' #( . |C| 1 + (q − 1)Z ) * +

* ,

- .

/ 01 ℄

m, n, s 0 ≤ m ≤ n s ≥ 2 3,m&

s& x ! m X j x (−1) Km (x; n, s) = n − xm − j(s − 1)m−j , j j=0 4 ! x x(x − 1) · · · (x − j + 1) , = j! j

x ∈ R

Km (x; n, s) m x i 0 ≤ i ≤ n

Km (i; n, s) z m (1 + (s − 1)z)n−i (1 − z)i .

Km (x; n, s) x m xm Km (x; n, s) ! m m X 1j (−1)m−j (−1)m X m (s − 1)m−j = (−1)j (s − 1)m−j = j! (m − j)! m! j j=0 j=0

(−s)m

= 0, m!

z m

! ! ! ! m X n−i i m−j j (s − 1) (−1) . m−j j j=0

Km(i; n, s) K0 (i; n, s) = 1 K1 (i; n, s) = (n − i)(s `− 1)´ − i = (s − 1)n − si Km (0; n, s) = (s − 1)n mn 1 ≤ m ≤ n 1 ≤ i ≤ n

⊔ ⊓

Km (i; n, s) = Km (i − 1; n, s) − Km−1 (i − 1; n, s) − (s − 1)Km−1 (i; n, s).

((1 + (s − 1)z)n−i (1 − z)i )(1 + (s − 1)z) =

((1 + (s − 1)z)n−(i−1) (1 − z)i−1 )(1 − z).

z m !

Km (i; n, s) + (s − 1)Km−1 (i; n, s) == Km (i − 1; n, s) − Km−1 (i − 1; n, s),

C Fs P n n P

i i AC (z) = n c z

A (z) = i D i=0 i=0 bi z

D = C⊥

⊔ ⊓

n 1 X Km (i; n, s)ai = bm . |C| i=0

"#$ AD (z) =

n 1 X ai (1 + (s − 1)z)n−i (1 − z)i , |C| i=1

AD (z) =

n 1 X ai K(i; n, s)z i , |C| i=1

⊔ ⊓

n X i=0

ai Km (i; n, s) ≥ 0.

DC (z) !

" #

℄

! n = (q

r

− 1)/(q − 1) Hr (q) r Fq [n, n − r]" Pr r × n # $ # 1" Frq (q r − 1)/(q − 1) % & r Fq $

# ' ( & ( k ≥ 2 Pr 2) C1 C2 # C1 + C2

Pr ( * ++ # &

r≥2

#

$

1

d=3

2

, & Pr Fr2

r Fq

(

x ∈ Hr (q) |B1 (x)| = 1 + n(q − 1) = q r .

q n−k 1 Hr |Hr |q k = q n

Fnq - # ⊓ ⊔

(

Hr

r = 3

F2

⎛

100101 P3 = ⎝0 1 0 1 1 0 001011

[7, 4] H3 (2)

⎞ 1 1⎠ . 1

#

d = 3

!" 4 $ 24 = 16

A0 = 1& # A1 = A2 = 0

& $ (1111111)

A7 = 1 # 4 % x

p&

(1111111) − x

7 − p ( A5 = A1 = 0 A6 = A2 = 0 )

A3 = A4 = 7

H3 (2) $

%

3

'

1 + 7x3 + 7x4 + x7 .

Hr (q)

Sr (q)

#

r

#

(q r − 1)/(q − 1)

* C

C

0

#

[7, 3]

C = S3 (2) + # ,!

f (X, Y ) = Y 7 + 7X 3 Y 4 + 7X 4 Y 3 + X 7 .

-./ !

g(X, Y ) =

C

C&

1 f (Y − X, Y + X) = Y 7 + 7X 4 Y 3 , 24

$

d = 4

0 $ --

0 C = S (q) q r−1

r

G Sr (q)

q r vi c C

0 I ⊆ {1, 2, . . . , r} c=

αi (v0 i , v1 i , . . . , vqr −1 i ),

i∈I

αi ∈F⋆q c

v Frq i∈I αi vi = 0 q r − 1 q r−1 − 1 q r − 1 r − v ∈ Fq : − = q r−1 . w(c) = αi vi = 0 = q−1 q−1 q−1 i∈I

ASr (q) (z) = 1 + (q r − 1)z q

r−1

⊓ ⊔

.

! "#$

% % Hr (2) A(z) =

1 [(1 + z)n + n(1 − z 2 )(n−1)/2 (1 + z)]. 2r

& r = 3 F = F3 ' [13, 10] % 3 (

⎛

⎞ 1001011201211 P3 = ⎝ 0 1 0 1 1 0 1 1 2 0 1 2 1 ⎠ . 0010111012112

H3 (3) ) ! "* +#$℄

& C [n, k]

r

e

c ∈ C r c − r C

1

e = r − c )

0 -

.

i 0 1 2 3 4 5 6

Ai 1 0 0 104 468 1404 4056

i 7 8 9 10 11 12 13

Ai 8424 11934 13442 11232 5616 2080 288

Ì H3(3) 0 n 0

1

Hr (2)

n − r

H3 (2) ⎛ ⎞ 1101100 H = ⎝1 0 1 1 0 1 0⎠ . 0111001

⎛

⎞ 1010101 H ′ = ⎝0 1 1 0 0 1 1⎠ . 0001111

! i i " Hr (2) [2r − 1, 2r − 1 − r] H ′ 1, 2, 4, . . . , 2r−1 #

# $ % & '

n [n, k] C Fq

q k

Vn (Fq ) Vn (F) t = q n /q k = q n−k C0 = C C1 Ct−1

C [7, 4] H3 (2)

C V7 (F2 ) 23 = 8

(0 0 0 0 0 0 0) (1 0 0 0 0 0 0) (0 1 0 0 0 0 0) (0 0 1 0 0 0 0) . (0 0 0 1 0 0 0) (0 0 0 0 1 0 0) (0 0 0 0 0 1 0) (0 0 0 0 0 0 1)

Ci 0 ≤ i ≤ t − 1

Ii ! I0 = 0 C

C = {I0 , c1 , . . . , cqk −1 }.

¾ C S q n−k ×q k Vn (Fq ) (i+1, j +1)

Ii + cj . " # $

Vn (F) S % C

C [5, 2]& ' 10101 G= . 01110

! C ' ()

C [n, k] Fq cj 0 ≤ j ≤ q k − 1 S C

Ii 0 ≤ i ≤ qn−k − 1 (i + 1, j + 1) Ii + cj

d(Ii + cj , cj ) ≤ d(Ii + cj , ch )

i j h 0 ≤ i ≤ qn−k − 1 0 ≤ j, k ≤ qk − 1 2

Ii

Ì

d(Ii + cj , cj ) = w(Ii ) d(Ii + cj , ch ) = w(Ii + (cj − ch )). cj − ch ∈ C

Ii

Ci

Ii + (cj − ch ) ∈ Ci

⊓ ⊔

w

w ∈ Vn (Fq )

w

S

w

i

!

c

"

Ci

(1, i)

## $

##

#

Ì C d Fq

x

w(x) ≤ ⌊

d−1 ⌋. 2

x Vn (Fq )/C y

x w(y) ≤ w(x)

"

z=y−x∈C

x

w(z) < d

%

z=0

y = x ⊓ ⊔

& '(!

) *

& # e # + * r Ij

r

)

C⊥ r r+C e

Ii

C

˜ r

r

r

! " #$ %

Fn → Fn−k

Ì

H [n, k] C Fq x, y ∈ Vn (Fq ) Vn (Fq )/C

xH T = yH T . &

x, y

z z ∈ C '( x − y = 0 ⊓ ⊔

) ) * +,

- (. ,, * ## )

Ij

r

C

n H r

r ∈ C ˜

d(r, ˜r) < ⌊ d−1 2 ⌋

Vn (F )/C r s = rH T e = Ii s

˜r → r − e r ˜ ! " e # $% ! &

' C [6, 3](

⎛ 11010 G = ⎝0 1 1 0 1 10100

⎞ 0 0⎠ , 1

⎛

⎞ 100101 H = ⎝0 1 0 1 1 0⎠ . 001011

d = 3 ) S C $$ * $+, w = 1 #

V6 (Z2 )/C # ! # 0 * # - . $+ ' r = (100011) & rH T = (010)

(010) ! I5 # r − I5 = (100011) − (010000) = (110011).

C

[70, 50]!

C

"

70

250 = 270 /250 #

2

20

70 · 220 + 20 · 220 = 90 · 220 ,

11

$

% & ' (!)

Ë½

* +

T s,

*

"

i = 1 T w = T (s) w = 0

+

•

-

•

-

s T (s)

w = 0

s = rH T

c = r

T ((r + ei )H T ) ≤ T (rH T ),

r → r + ei i! i = i + 1

ei

%

*

Ii 000000 000001 000010 000100 001000 010000 100000 001100

110100 110101 110110 110000 111100 100100 010100 111000

011010 011011 011000 011110 010010 001010 111010 010110

101001 101000 101011 101101 100001 111001 001001 100101

101110 101111 101100 101010 100110 111110 001110 100010

110011 110010 110001 110111 111011 100011 010011 111111

011101 011100 011111 011001 010101 001101 111101 010001

000111 000110 000101 000011 001111 010111 100111 001011

Ì

0

Ii 000000 000001 000010 000100 001000 010000 100000 001100

s 000 101 011 110 001 010 100 111

Ì i > n

r ! " # $ % &

' ( n &

) % % C * $ * (

+

s w(Ij ) 000 0 101 1 011 1 110 1 001 1 010 1 100 1 111 2

ÍÒ

!

C

r

c ∈ C r " #

$

% C P (C)

u

C

& Pu (C) ' r ∈ C r = c

C p

Pu (C) = WC (p, 1 − p) − (1 − p)n ,

WC (X, Y ) C % 1 ≤ i ≤ n e i ' P (e, i) = pi (1 − p)n−i ,

i n − i

e ' C ( i Ai Ai ' i C Pu (C) =

n i=1

Ai P (e, i) =

n i=1

Ai pi (1 − p)n−i = WC (p, 1 − p) − (1 − p)n . ⊓ ⊔

C Li i

r = m + e

e

Pc (C)

Pc (C) =

n i=0

Li pi (1 − p)n−i .

Pe (C) Pe (C) = 1 − Pc (C).

C !" # AC (Z) = 1 + 4Z 3 + 3Z 4 .

$ L0 = 1 L1 = 6 L2 = 1 % &"

1/10

Pu (C) =

3159 106

≈ 0.0032

Pd (C) =

110717 125000

≈ 0.8857

Pe (C) =

14283 125000

≈ 0.1142

' p $ √ √ 5 51/4 5+3 5 √ ≈ 0.5992. − p= + 4 4 2 2

( p Pu (C) ≈ 0.087

) !* +, p > 0.2644 -

.

r (1 − p)6 / r 1

p(1 − p)5 ( 0 C 11 1 − (1 − p)6 − p(1 − p)5

1

Pe (C)

Pd (C) Pu (C)

0 0

1

p

1/2

p > 0.1297

C 3 2 3

1 − (1 − p)6 − p(1 − p)5 − p2 (1 − p)4 . #

1/2

C

%&

1/2

p > 0.1502 Pu (C)

!" $

!'

() $

Ì C n d Pu (C)

C p n n i Pu (C) ≤ *" p (1 − p)n−i . i i=d

1

1 2

Pu (C)

0 0

1

p

Pu (C)

n i i ⊓ ⊔

n i Pu (C)

!" # $ %

1/2

p > 0.4215&

!

p > 0.1502

'

( ) *++, -

) *+.,

n, k, d

(

¿

n

3

s

C

d

Ωn,d

9 A0 = 1 > = Ai = 0 1 ≤ i ≤ d − 1 , = (A0 , A1 , . . . , An ) : A > > ; : Pin≥ 0 d ≤ i ≤ n i=0 Ai Km (i; n, s) ≥ 0 1 ≤ m ≤ n 8 > <

|C| ≤ max

(

n X i=0

AC (x) =

An : A = (A0 , A1 , . . . , An ) ∈ Ωn,d Pn

i=0

)

.

ai xi C

|C| =

n X

ai .

i=0

a0 = 1 C 1 d − 1 n X i=0

ai Km (i; n, s) ≥ 0,

m 1 n ! a = (a0 , a1 , . . . , an ) ∈ Ω

" C ⊔ ⊓ # $

" C %

ÍÒ

d

M

C C2 C

2 1

M 2

{x,y}∈(C 2)

1

d(x, y) ≤ M 2

{x,y}∈(C 2)

M d = d M = d. 2 2 1

C

d

Ì ËÙÔÔ Fq n M d

d≤

nM (q − 1) . (M − 1)q

C [n, k]

Fq

Λ C

Λ=

w(C).

c∈C

i 1 ≤ i ≤ n

i

qk−1 (q − 1) Di C i 0 1 C |Di | = q k−1

Di C/Di q − 1 nqk (q − 1) Λ qk − 1 d(qk − 1)

Λ

nq k−1 (q − 1) ≥ Λ ≥ d(q k − 1).

d d≤

qk

nq k−1 (q − 1) nM (q − 1) Λ ≤ = . −1 (q k − 1) (M − 1)q ⊓ ⊔

d > n q−1 q −1 q−1 . M ≤d d−n q

!"#$

%

(n, M, d) ! $ Fq & '

(

' )

n d Fq ) * + n, d Aq (n, d) = max{ M :

(n, M, d) q }.

% (n, M, d) C

|C| = Aq (n, d) ' d > n q−1 q −1 q−1 . Aq (n, d) ≤ d d − n q

,

Aq (n, d)

(n, d)

!

" # !

Ì A2 (n, 2l − 1) = A2 (n + 1, 2l).

C

C

#

2l

C

2l

% # #

n

C′

C

#

C′

n + 1

n+1

2l − 1

! &

# #

2l − 1 C

$ #

n

⊔ ⊓

C C ′ C

C ′

! " C [n, k, d]# Fq G $ G G=

1 1 ... 1 0 0 ... 0 G1 G2

.

% G2 C G C ′

[n, k, d] C

d [n − d, k − 1]

" G C G2 C ′ % g G !

C ′ n−d G2 k−1 "

! & C k

c G ' ! & C d d c ' % c − c1 g C d(

c = c1 g ) G k ⊓ ⊔

½¼¼

q = 2 C C ′

d c′ C ′ C

w ≤ d/2

d 0 C ′ d − d/2 = d/2

Ì [n, k, d] Fq

n≥

k−1 i=0

⌈d/q i ⌉.

G C d

!

· · 0). (111 · · · 1 0 · d

n−d

" G

⌈d/q⌉ !

d #

$% & C G [n− d, k − 1, d′ ]' d′ ≥ ⌈d/q⌉ ( ⊓ ⊔

)

* [n, k]' + ,

! * - . $ / ! 0 * 12

-

n, k, d

q

n−k

>

d−2 i=0

n−1 (q − 1)i = Vq (n − 1, d − 2); i

[n, k] Fq d

3 C *

- H H (n − k) × n H1 H (n − k)'

Fq

H2 Fn−k

q H1 j − 1

d − 1 d − 2

j − 1 d−2 i=0

j−1 (q − 1)i i

Hj Fqn−k

d − 2

j − 1 H n − k

d − 1 H !! d ⊓ ⊔

" #

$$ d = 3 k = n − r

%

`P ` ´´

n − k ≥ logq ei=0 (q − 1)i ni n − k ≥ d − 1 d ≤ nq k−1 (q − 1)/(q k − 1) P i n ≥ k−1 ⌈d/q ⌉ i=0 Ì

[n, k, d]&

C '

( ⎛

11 ⎜1 1 H =⎜ ⎝1 0 01

11 00 10 10

000 110 101 100

⎞ 0 0⎟ ⎟. 0⎠ 1

'

r = (0001 0101 1011) [12, 4, 6]&

C ⎛

10 ⎜0 1 G=⎜ ⎝0 0 00

000 001 101 010

11 01 11 00

11 10 01 11

⎞ 010 1 1 0⎟ ⎟. 1 1 1⎠ 111

k [12, k, 5]& )

[n, k] C k G

! " # $ % & # %

'

0

( T V

n (Fq )

(a0 a1 . . . an−2 an−1 ) ∈ T

(an−1 a0 a1 . . . an−2 ) ∈ T.

) Vn (Fq ) * {0} ) + v = (v0 v1 . . . vn−1 ) ∈ V σ ∈ Sn , (1 2 . . . n − 1) ( w ∈ V v i ∈ {0, 1, . . . n − 1} w = (vσi (0) vσi (1) . . . vσi (n−1) ) = σ i (v).

w #

w = (vi vi+1 . . . vn−1 v0 . . . vi−1 ).

n w w = (vi vi+1 . . . vi+n−1 ).

Ì

T

V

T

s = (s0 s1 . . . sn−1 ) ∈ T i ∈ {0, 1, . . . n − 1}

σ0 (s) = s ∈ T σ1 (s) ∈ T σi (s) ∈ T

σi+1 (s) = σ(σi (s)) ∈ T σj j = 0, . . . n − 1 s T ⊓⊔ ! n Fq " Fq

" Vn (Fq )

# C " $

# c = (c0 c1 · · · cn−1 ) ∈ C,

"

c(x) = c0 + c1 x + · · · + cn−1 xn−1

% Vn (Fq ) n $

x Fq n − 1 # Fq [x]

x

& Fq % " Rn (Fq ) = {f (x) ∈ Fq [x] : deg f (x) ≤ n − 1},

Fq [x] n − 1 ' Rn (Fq ) " n Fq E = {1, x, x2 , . . . , xn−1 }.

Rn (Fq ) Vn (F) " ϕ : Vn (Fq ) → Rn (Fq ) ϕ : (a0 , a1 , . . . , an−1 ) → a0 + a1 x + . . . an−1 xn−1 .

Rn (Fq )

(xn − 1) Fq [x] (xn − 1)

Fq [x]

F

Rn (F)

n ! 2n − 2 " ! #

F

n $

!

% f (x) ∈ Fq [x] r Fq [x]/(f (x))

Fq [x]

f (x)

r Fq &

Fq [x]/(f (x))

r − 1' (

#

F = {1 + (f (x)), x + (f (x)), x2 + (f (x)), . . . , xr−1 + (f (x))}. % Fq [x]n

Fq [x]n = Fq [x]/(xn − 1).

π : Fq [x] → Fq [x]n

π : f (x) → f (x) + (xn − 1),

Rn (Fq ) (

& ##

(

ξ = πϕ : Vn (Fq ) → Fq [x]n

ϕ

Vn (Fq ) .

ξ

/ Rn (Fq ) π|Rn Fq [x]n

)

C

Fq [x]n &

v = (v0 v1 . . . vn−1 ) ∈ C

ξ v(x) + (xn − 1) v(x) = v0 + v1 x + · · · + vn−1 xn−1 .

&

σ(v) = (v1 v2 . . . vn−1 v0 )

xv(x) + (xn − 1) = x (v(x) + (xn − 1)) Fq [x]n %

ξ −1 # V

*

a, b (

ab = ξ −1 (ξ(a)ξ(b)).

V Fq [x]

I

Fq [x]n

I ′ = { f (x) ∈ Fq [x] : πf (x) ∈ I}.

I′

Fq [x]

π(I ′ ) = I

I ′ π(I ′ ) = I t(x) ∈ Fq [x] \ I ′ π(t(x)) ∈ I I ′ ! Fq [x]

I

f (x), g(x) ∈ I ′ " π(f (x) + g(x)) = π(f (x)) + π(g(x)) ∈ I f (x) + g(x) ∈ I ′ # I ′ $ h(x) ∈ Fq [x] h(x)f (x) % π(h(x)f (x)) = π(h(x))π(g(x)) ∈ I h(x)f (x) ∈ I ′ & −1f (x) = −f (x) ∈ I ′ $ I ′ Fq [x] ⊔ ⊓

' π

# Fq [x]n Fq [x] (xn − 1)

T

T

ξ

I

Vn (Fq ) Fq [x]n

Fq [x]n

& T V xI = I Fq [x]n $ f (x) = a0 + a1 x + . . . ∈ Fq [x] ' f (x)T (

a0 I + a1 xI + a2 x(xI) + . . . = a0 I + a1 xI + a2 xI + . . . = a0 I + a1 I + a2 I + . . . = I.

I Fq [x]n )

I Fq [x]n " xI = I * ξ −1 ⊔ ⊓

T V

v(x) v

v ∈ Fq [x]n

C

Fq [x] I 1! " I Fq [x] g(x) ∈ I

I = { g(x)h(x) : h(x) ∈ Fq [x]}.

C

g(x) ∈ C g(x) C g(x)

n

Ì

C = {0}

dim C = n − r

g(x)

n

Fq

r

C

C = { g(x)q(x) : q(x) ∈ Fq [x]; deg q(x) < n − r}.

g(x)

xn − 1

Fq [x]

# $ g(x) C $

% Fq [x] #

$ g(x) C $

"

& C

$

Fq ' C = {g(x), g(x)x, . . . , g(x)xn−r−1}.

( ) % * + %,

xn − 1 = h(x)g(x) + s(x),

deg s(x) < deg g(x) . (xn − 1)

s(x) = (−h(x))g(x) ∈ C / s(x) = 0 && s(x) ∈ C && r $ s(x) = 0 ⊔ ⊓ 0

Fq [x]n

C Fq [x]

g(x) ∈ C C

C Fq Fq [x]

D = Fq [x]/(g(x)) g(x) [n, k] C n − k

Fq C ⊥ v(x) ∈ Rn (Fq ) v(x) D v ∈ C

n

V g(x) ∈ Fq [x] f (x) = xn − 1

n (Fq )

7

V7 (Z2 ) f (x) = x7 − 1 Z2

x7 − 1 = (x + 1)(x3 + x2 + 1)(x3 + x + 1);

f (x) g1 (x) = 1 g2 (x) = x + 1 g3 (x) = x3 + x2 + 1 g4 (x) = x3 + x + 1 g5 (x) = (x + 1)(x3 + x2 + 1) g6 (x) = (x + 1)(x3 + x + 1) g7 (x) = (x3 + x2 + 1)(x3 + x + 1) g8 (x) = f (x).

V7 (Z2 ) 8 g6 (x)

S ={(0000000), (1011100), (0101110), (0010111) (1001011), (1100101), (1110010), (0111001)}.

g7 (x) S = {(0000000), (1111111)}.

[15, 9] !

g(x) = (1 + x + x2 )(1 + x + x4 ) x15 − 1 g(x) 9 V15 (Z2 )

½

k k = n − r k

! " #

k $ C [n, k]% Fq g(x) C G = (R Ik )

$ f (x) deg

0

f (x) f (x) $

deg f (x) − deg0 f (x) > 1 f (x)

G

& i = 0, 1, . . . , k − 1 xn−k+i g(x)

xn−k+i = qi (x)g(x) + ri (x),

deg ri (x) < deg g(x) = n − k ri (x) = 0' ( $ pi (x) = xn−k+i − ri (x) = qi (x)g(x) ∈ C;

) $

deg pi (x) − deg0 pi (x) ≥ i;

* + p(x) = xn−k+i − ri (x) ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

1

0

r−1

−r0 (x) −r1 (x) −rk−1 (x)

r

n−1

⎞ 1 0 ... 0 ⎟ ⎟ 0 1 ⎟ , ⎟ ⎠ 0 0 ... 0 1

G = (R Ik ) R −ri (x) 0 ≤ i ≤ k − 1'

½½¼

G C G C

C

[7, 4]

1 + x + x3

x3 = (1)(x3 + x + 1) + (1 + x) x4 = (x)(x3 + x + 1) + (x + x2 ) x5 = (x2 + 1)(x3 + x + 1) + (1 + x + x2 ) x6 = (x3 + x + 1)(x3 + x + 1) + (1 + x2 ).

⎛

11 ⎜0 1 G=⎜ ⎝1 1 10

01 10 10 10

⎞ 000 1 0 0⎟ ⎟ = (R I4 ), 0 1 0⎠ 001

⎛ 1 ⎜0 R=⎜ ⎝1 1

⎞ 10 1 1⎟ ⎟. 1 1⎠ 01

R # $ 1 + x x + x2 1 + x + x2 1 + x2 % & m = (1011) ' G c = mG = (100 1011) ! "

(

)

g(x)

*

"

hR (x)

k i h(x) = i=0 ai x deg h(x) = k h(x)

hR (x) =

k

ak−i xi .

i=0

deg h(x) = k xk h(1/x) +

C

C R

hR (x)

,

)

C

C R

-

C

(c0 c1 . . . cn−1 ) ∈ C (cn−1 cn−2 . . . c1 c0 ) ∈ C R g(x) C

g0−1 gR (x)

C g(x) C h(x) g(x)h(x) = xn − 1.

Fq [x]n

(g(x) + (xn − 1)) (h(x) + (xn − 1)) = 0 + (xn − 1).

h(x)

g(x) ∈ Fq [x] n− k f (x) = xn − 1 C [n, k] h(x) = (xn −1)/g(x) C C = { c(x) ∈ Fq [x] : deg c(x) ≤ n, c(x)h(x) = 0

(mod xn − 1)}.

c(x) ∈ C q(x) deg q(x) < n

c(x) = q(x)g(x) ! c(x)h(x) = q(x)g(x)h(x) = q(x)(xn − 1) = 0

(mod xn − 1).

" # $ c(x) ∈ Fq [x] deg c(x) < n c(x)h(x) = p(x)(xn − 1).

$

c(x)h(x) = p(x)(xn − 1) = p(x)g(x)h(x); (c(x) − p(x)g(x))h(x) = 0

Fq [x]

g(x)h(x) = xn − 1 % h(x) = 0 !

c(x) = p(x)g(x) c(x) ∈ C ⊓ ⊔ c(x) C

c(x)h(x) % xn − 1 & % '

( $ ( h(x)

g(x) [n, n − r] C Fq C ⊥ hR (x) h(x) C

½½¾

v w Vn (Fq ) σ (0 1 2 · · · n−1) i = 0, 1, . . . n−1 v, w = σ i (v), σ i (w).

σ i (v), σ i (w) =

n−1 X j=0 n−i X

j=0 n X

vj+i wj+i +

i−1 X

vj wj =

j=0

vj wj +

j=i n−1 X j=0

vj+i wj+i =

i−1 X

vj wj =

j=0

vj wj = v, w.

⊔ ⊓

g, hR = 0,

g hR n

g(x) hR (x) g(x) =

r X

gi xi ;

hR (x) =

i=0

n−r X

hn−r−i xi .

i=0

n−r X

gi hn−r−i ,

i=0

! hR n − r

(xn − 1) = g(x)h(x) =

n X t X

gi ht−i ,

t=0 i=0

g, hR " n − r g(x)h(x) = xn − 1 # 0 < r < n

0 $ g(x) hR (x) % D hR (x) m(x) ∈ C n(x) ∈ D & ' a(x) b(x) m(x) = a(x)g(x),

n(x) = b(x)hR (x).

( m n

m(x) n(x) )

m, n =

n X i=0

D

ai bi σ i (g), σ i (hR ) =

n X i=0

ai bi g, hR = 0,

C

D r C ⊥ deg hR (x) = deg h(x) = n − r D r hR (x) ⊥ ⊔ ⊓

C

G [n, k]

C g(x)

G = (R Ik ),

! " # $ % a

i = 0, 1, . . . , k − 1 ri (x) x

xn−k+i = qi (x)g(x) + ri (x). &

% a = (a0 a1 . . . ak )

a(x) =

k−1

ai xi .

i=0

'

G

( k ) ( n − k ) a(x)

)

C

a0 xn−k a1 xn−k+1

ak xn . q(x)

xn−k a(x) = q(x)g(x) + t(x),

* +

q(x)g(x) = −t(x) + xn−k a(x),

t(x) = 0 deg t(x) < deg g(x) = n − k [ −a0 r0 (x) a0 0 . . . 0 ]+

−a1 r1 (x) 0 a1 0 ]+ [

]+ [ −ak−1 rk−1 (x) 0 . . . 0 ak−1 ]= [ [

−t(x)

a0 a1 . . . ak−1 ] .

a(x) −t(x) t(x) = k−1 i=0 ai ri (x) t(x) ! xn−k a(x) " g(x)

" # [n, k]$

% [n, k]$ C " g(x)& % a = (a0 a1 . . . ak−1 )

% s = (s0 s1 . . . sn−k−1 ) (a s) ∈

C

' " g ˜ = (g0 g1 . . . gn−k−1 )

( sj = 0 0 ≤ j ≤ n − k − 1

( i = 1

" )

' ak−i = sn−k−1 sj = sj−1 j n − k − 1 1 s0 = 0

* ' ak−i = sn−k−1 sj = sj−1 + gj j n − k − 1 1 s0 = g 0

' i > k '+

" "" " ! " G = (R Ik )

ri (x)

C

m = (1011) g ˜ = (110) (a0 a1 a2 a3 ) = (1011) (s0 s1 s2 ) = (000)

i i 0 1 2 3 4

s ak−i 000 110 1 101 1 100 0 100 1

m = (1011)

(100)

c = (1001011).

! g(x) = 1 + x4 + x6 + x7 + x8 " [15, 7]#

m = (1011 011) g˜ = (1000 1011)

i 0 1 2 3 4 5 6 7

s ak−i 0000 0000 1000 1011 1 0100 0101 1 1010 1001 0 0101 0100 1 1010 0001 1 1101 1011 0 0110 1101 1

m = (1011 011) $ c = (0110 1101 1011 011).

%

[n, k]#

C Fq " g(x) " & $ " G

G = (R|Ik ) "

C H = In−k | − RT . '() ri (x)

⎛

⎞

1 0 ... 0

⎜ ⎜0 1 º H=⎜ ⎜ ⎝ 0 0 ... 0 1

r0 (x)

...

r1 (x)

⎟ ⎟ ⎟ ⎠

rk−1 (x) ⎟ .

h(x) C r s = rH T s r H

hR (x)

s(x)

Fq [x]n−k

!"

s

h(x)

C

H

Ì [n, k]

C Fq g(x) H

v ∈ Fnq s H v(x) s(x)

s(x)

v(x) g(x) # $ % q(x) deg t(x) < deg g(x) = n − k v(x) = q(x)g(x) + t(x)

t(x) v ′ (x) ∈ C + v(x)

v ′ (x) = q ′ (x)g(x) + t(x); t(x) & C+v(x) si xi ⎞ ⎛ k−1 rj,i ⎠ , si = vi ⎝1 +

s(x) =

n−k i=0

j=0

xi rj (x) ( rj (x) n−k+i

x g(x) t(x) n−1 n−k−1 vj xj . t(x) = vj rj−(n−k) (x) +

rj,i

'

j=0

j=n−k

'

ti

i

x

t(x)

ti = vi +

k−1

vi rj,i ,

j=0

t(x) = s(x)

⊓ ⊔

!

C

[7, 4]"

# $ %

⎛

11 ⎜0 1 G = (R I4 ) = ⎜ ⎝0 0 00

01 10 11 01

00 10 01 10

⎞ 0 0⎟ ⎟. 0⎠ 1

& ' (

⎛

⎞ 1001011 H = (I3 | − RT ) = ⎝0 1 0 1 1 1 0⎠ . 0010111 !

s = rH T = (001)

r = (101 1011))

r

%

(

r*

r(x) = 1 + x + x2 + x3 + x5 + x6 . +

r(x)

g(x)

r(x) = (x3 + x2 + x + 1)g(x) + x2 . %

r

(

s(x) = x2

% ,

C

v v

C

[n, k]

deg s(x) = n − k − 1

xr(x)

Fq s(x) =

g(x) r(x) P n−k−1 si xi i=0 deg s(x) < n − k − 1 xr(x) xs(x)

xs(x) − sn−k−1 g(x)

s(x) r(x) g(x) r(x) = g(x)q(x)+s(x)! xr(x) = xg(x)q(x) + xs(x) 2 "# deg xs(x) < deg g(x)! xr(x) g(x) $ xq(x) $ xs(x)! xr(x) xs(x)!

r(x) = v(x) + e(x)

v(x)

n

+

e(x) s(x)

g(x)

e(x)

s(x) = r(x) (mod g(x))

deg xs(x) = deg g(x) (xq(x) + ⊔ ⊓ sn−k−1 ) xs(x) − sn−k−1 g(x)

! "

r w = (1 1 0 1 1 0 1)

w(x) = xr(x) w

#

wH T = (1 1 0) = t.

$

xs(x) − 1 · g(x) = x3 − (x3 + x + 1) = 1 + x.

% r

w

v = (v0 v1 · · · vn−1 ) vi , vj v j = i + 1 i = n − 1 j = 0

k ≤ n vi , vi , . . . vi v k

k r

r e r(x) e(x) e(x) w(e) ≤ t ! 0

1

k

e

k

e

σi

i

0 ≤ i ≤ n−1 e

i

σi

eσi

n−k

i r i si (x) xi r(x)

xi e(x) w(e) ≤ t w(sj ) ≤ t ! i

w(si (x)) ≤ t

xi e(x)

"#

%$deg si (x) < t. & si (x)

xi r(x)

g(x)

g(x) | xi (r(x) − e(x));

'

xi e(x) = si (x) "#(%

(mod xn − 1).

"#(%

' '

xi e(x) = (si , 0).

e(x) = xn−i (si , 0)

) *

xn−i (si , 0)

n − i

(si , 0)

+

, #- sj (x) sj−1 (x)

. [n, k]/ ' C g(x)

d

. r = r(x)

c(x) ∈ C r(x)

½¾¼

Ë½

t = ⌊(d − 1)/2⌋

i = 0

s0 (x) r(x)

r(x) = q(x)g(x) + s(x).

w(si (x)) ≤ t e(x) = xn−i (si , 0) c(x) = r(x) − e(x) c(x) i = i + 1 i = n si (x) deg si−1 (x) < n − k − 1 si (x) = xsi−1 (x) deg si−1 (x) = n − k − 1 si (x) = xsi−1 (x) − g(x) ! "

! # # $$ [n, k]% # &

si (x) ' ( )*

+ g(x) = 1 + x2 + x3 [7, 4]% #

$ d = 3 , [7, 3]% - c c(x) = a(x)g(x) $ a = (111) a(x) = 1 + x + x2 . c(x) = 1 + x + x5 #

c ##

$ r = c + e r(x) = 1 + x + x5 + x6 / e s(x) r(x) r(x) = (x3 + 1)g(x) + (x + x2 ) s(x) = x + x2 .

0 w(s(x)) > 1 s1 (x) xr(x). deg s(x) = 2 = n − k − 1 1 s(x) x g(x) s1 (x) = 1

w(s1 (x)) ≤ 1 $ e(x) = x7−1 (s1 , 0) = x6 (1000000) = x6 ,

C [15, 7, 5]% # g(x) = 1 + x4 +

x + x7 + x8 2% 2 $ 2 $ $$ 6

7

1 2

r = (1100 1110 1100 010).

s(x) r(x) r(x) = (x5 + x4 + x2 + x)g(x) + (1 + x2 + x5 + x7 ) s(x) = 1 + x2 + x5 + x7 .

si (x) xi r(x) i w(si (x)) ≤ 2 ! "# $ i 0 1 2 3 4 5 6 7

si (x) 1010 0101 1101 1001 1110 0111 1111 1000 0111 1100 0011 1110 0001 1111 1000 0100

Ì

w(s7 (x)) ≤ 2 e = x15−7 (s7 , x) = x8 (1000 0100 0000 0000) = (0000 0000 1000 010).

r → r − e = c = (1100 1110 0100 000).

C [15, 5, 7]%

! g(x) = 1+x+x2 + x + x + x + x & b ≤ 3 5 ˆ e = (10000 10000 10000)

3

ˆ e

' (

eˆ(x) 1 + x5 + r1 (x) r1 (x) x10 g(x) &

r si (x) xi r(x) 0 ≤ i ≤ 14 w(si ) ≤ 3 w(si − r1 ) ≤ 2 ' x15−i (si − r1 , (10000)) 4

5

8

10

½¾¾

r = (11110 10100 11101).

si (x) i = 0

i si (x) 0 01100 00100.

w(s0 ) ≤ 3

e = x15−0 (s0 , 0) = (01100 00100 00000);

c = r − e = (10010 10000 11101).

r = (11100 01111 00100),

si (x) xi r(x) i = 0, 1, 2, . . .

si (x) − r1 (x) r1 (x)

r1 (x) = 1 + x + x2 + x4 + x5 + x8 i 0 1 2 3 4

si (x) 00110 10001 11110 11010 01111 01101 11010 00100 01101 00010

si (x) − r1 (x) 11011 00011 00011 01000 10010 11111 00111 10110 10000 10000

Ì

w(s4 − r1 ) ≤ 2 e = x11 (10000 10000 10000) = (01000 01000 01000);

c = r − e = (10100 00111 01100).

C 10 g(x) = x4 + x3 + x2 + x + 1

[12, 3, 6]

C g(x) = x9 + x8 + x5 + x4 + x + 1.

m = (0 1 1) C

h(x) C r = (0 1 1 0 0 1 0 1 0 1 1 0)

g(x) = x2 + x + 1

[9, 7, 2]

C2 g(x) F2 F4 [9, 8, 2]

ÍÒ [n, k] q g(x) ∈ Fq [x] n−k n − k

i = 0, . . . , n − k − 1 gi xi g(x) = xn−k +

n−k−1

gi xi ;

i=0

αi g(x)

! " g(x) =

n−k i=1

(x − αi );

# $ n − k (γi , g(γi )) γi Fq %

& '( ) * &

" +

( γ ∈ Fq n ,"& γ $ , xn − 1 = 0. -./

Fq

xn − 1 = f1 (x)f2 (x) · · · ft (x) ∈ Fq [x].

i = 1, . . . , t γi fi(x) Fq fi(x) ∈ Fq [x] fi(x) γi ! fi (x)

γi " fi (x) c(x) ∈ Fq [x] c(γi) = 0 C fi(x) # C = {c(x) ∈ Fq [x] : c(γi ) = 0, deg c(x) ≤ n}.

$

C %

g(x) = g1 (x)g2 (x) · · · gw (x)

Fq i = 1, . . . , w βi gi (x) Fq ! C = {c(x) : g(x)|c(x), deg c(x) ≤ n} =

w

{c(x) : c(βi ) = 0, deg c(x) ≤ n};

i=1

C # C = {c(x) : c(β1 ) = c(β2 ) = . . . = c(βw ) = 0, deg c(x) ≤ n}.

&

! ' ( n, m, w ) $ α1 , α2 , . . . , αw ∈ Fq & ! ) Fq Fmq * 1 ≤ i ≤ w Hi m × n

m

m

1, αi , α2i , . . . , (αi )n−1 .

+ , w wm × n Fq %

H ( c = (c0 c1 . . . cn−1 ) c(x) =

n−1 i=0

ci xi .

H

cH T

= 0 cH T = 0

c(αi ) = 0 i = 1, 2, . . . , w Fm q

⎧ c0 + c1 α1 + c2 α21 + · · · + cn−1 αn−1 =0 ⎪ 1 ⎪ ⎪ ⎨ c0 + c1 α2 + c2 α22 + · · · + cn−1 αn−1 =0 2

⎪ ⎪ ⎪ ⎩ n−1 c0 + c1 αw + c2 α2w + · · · + cn−1 αw = 0.

H

H

H

!

α1 , α2 , . . . , αn−k

Fq

C

C = { c(x) ∈ Fq [x] : c(α1 ) = c(α2 ) = . . . = c(αn−k ) = 0}. "

⎞ α21 · · · α1n−1 α11 α01 ⎜ α02 α12 α22 · · · α2n−1 ⎟ ⎟ ⎜ H=⎜ ⎟ ⎠ ⎝ n−1 α0n−k α1n−k α2n−k · · · αn−k ⎛

C ⊥

Fq H (n − k) × n Fq

C

#

$

%

Ì

n = (q m − 1)/(q − 1) β ∈ Fqm n gcd(m, q − 1) = 1

C = {c(x) : c(β) = 0, deg c(x) ≤ n}

Hm (q) [n, n − m] Fq

"

n = (q − 1)(q m−2 + 2q m−3 + · · · + m + 1) + m, gcd(n, q − 1) = gcd(m, q − 1) = 1 β i(q−1) = 1 i = 1, 2, . . . , n−1 & β i ∈ Fq i = 1, 2, . . . , n− 1 H 1, β, β 2 , . . . , β n−1

Fm q Fq # H ! % [n, n − m] ⊓ ⊔

C g(x) !

" # $% C n Fq & Fq [x]n = Fq [x]/(xn − 1)'

g(x) ( & Fq [x]n

h(x) )*

! &

k Fq ! * ! + RC = Fq [x]n /(h(x)) ≃ C

C ! ! 1 ∈ Fq [x]! & ! ! C = Fq [x]n RC ! ! !

Ì

C n Fq

g(x) h(x) g(x) h(x) c(x) ∈ C RC , - g(x) h(x) *

! ! . / 0. 1! a(x), b(x) ∈ Fq [x] a(x)g(x) + b(x)h(x) = 1.

, c(x) = a(x)g(x) = 1 − b(x)h(x).

, # $2! c(x) ∈ C ) ! & p(x)g(x) ∈ C ! c(x)p(x)g(x) = p(x)g(x) − b(x)h(x)p(x)g(x) = p(x)g(x)

(mod xn − 1),

c(x) RC 3 c(x) ∈ Rn (Fq ) ⊔ ⊓ ) deg c(x) > deg g(x) 4 ! -

p(x) c(x)p(x) = p(x) (mod h)(x)! c(x) 0 1 C 5 gcd(q, n) = 1! * f (x) = xn − 1 nxn−1 = 0 x = 0' f (x) ' ! *

Fq ) ! g(x) h(x) & *

! # % 3 & * gcd(n, q) = 1 3 6 !

) c(x) ∈ C !

RC !

C

c(x)2 = c(x) c(x)

C

RC v(x) ∈ C

Fq [x]n v(x)c(x) c(x) RC Fq [x]n !

g(x) Fq [x] C "

Fq [x]

(c(x)) ⊂ (g(x))

g(x) ∈ (c(x)),

π(c(x)) = π(g(x)) = J J Fq [x]n

Ì

C1 C2 c1 (x) c2 (x) C1 ∩ C2 c1 (x)c2 (x) Fq 2 C1 + C2 a + b a ∈ C1 b ∈ C2 c1 (x) + c2 (x) + c1 (x)c2 (x).

# $ % & C1 C2 c1 (x) + c2 (x) + c1 (x)c2 (x) = c1 (x) + (1 + c1 (x))c2 (x)

C1 + C2 ! C1 + C2 % p(x) = a(x)c1 + b(x)c2 " ' Fq 2 (( p(x)(c1 (x) + c2 (x) + c1 (x)c2 (x)) = a(x)c21 (x) + a(x)c1 (x)c2 (x) + a(x)c21 (x)c2 (x)+ b(x)c2 c1 (x) + b(x)c22 (x) + b(x)c1 (x)c22 (x) = a(x)c1 (x) + 2a(x)c1 (x)c2 (x) + 2b(x)c1 (x)c2 (x) + b(x)c2 (x) = p(x);

⊔ ⊓

Fq C1 C2 g1 (x) g2 (x) C1 ∩ C2 g1 (x)g2 (x) xn − 1

½¿¼

q = pt p k α ∈ Fq n Fq α ∈ Fp α = 1 0 < k < n ! n

αn − 1 = 0.

α

n" #

αi 0 ≤ i ≤ n

m" # m|n ! $ i d n αi

d" # % Ci q n i m αim

d" # ! & Ci = {i, iq, iq 2 , . . . , iq m−1 },

m '

iqm = i (mod n)

p = 2 c(x) ( C c(x) xi

c(x)

x2i ! c(x) =

n−1

i

2

ci x = c(x) =

n−1

ci x2i .

i=0

i=0

) # t−1

xi + x2i + · · · + x2

i

{i, i2, · · · , i2t−1} = Ci

* (

# xn − 1 %

p $

c(x)p = c(x),

+

Ci

Fq

fi (x) =

t∈Ci

(x − αt ),

α n αtn = 1 fi (x) (xn − 1)

f (x) (x

− 1)

i fi (x)

Mi− hi (x) = (xn − 1)/fi (x)

n

i

Mi+

Mi−

(Mi+ )⊥

=

Mi−

Mi+

Mi+ Fq [x]n Mi−

!!

Mi− [n, k] q g(x) ∈ Mi− c(x) ∈ Mi− j 0 ≤ j ≤ q k − 1

c(x) = t(x)j .

Mi− RM − = Fq [x]n /Mi+ = Fq [x]n /(fi (x)), i

fi (x) Mi− ! fi (x) " a(x), b(x) ∈ Mi− a(x)b(x) = 0

fi (x) a(x) b(x)

a(x) = 0 b(x) = 0 # Mi−

# $ % Mi− K = Fqk k = deg fi $ %&

K⋆ K

φ ∈ K⋆ K⋆ = { φi : i = 0, 1, . . . , q k − 1}.

%

t(x)

'

φ ⊔ ⊓

" t(x) #

½¿¾

Mi−

g(x) dim Mi− = k c(x) = 0 j

c(x) = xj g(x)

n = 2k − 1

c(x) ∈ Mi−

(mod xn − 1)

n n = 2k −1

g(x) g(x) Mi− ⊓ ⊔

Mi−

C [n, k] Fq

C

C n(q − 1)q k−1

i 0 n − 1 t ∈ Fq

q k−1 c ∈ C ci = t (q − 1)q k−1 i

n

⊔ ⊓ !

[2k − 1, k] k−1 k−1 k 2 n2 (2 − 1) = = 2k−1 . 2k − 1 2k − 1

Mi−

i θi ! " # F2

(x7 − 1) = (x − 1)(x3 + x + 1)(x3 + x2 + 1).

# g(x) = (x − 1)(x3 + x + 1) = x4 + x3 + x2 + 1 g(x) [7, 3] C $ (x2 + 1)g(x) = x6 + x5 + x3 + 1

C

Fq

2 xn − 1

Ì θi θPi (x)θ j (x) = 0 i = j ti=1 θi (x) = 1

t 1 + θi (x) + θi (x) + · · · + θi (x) fi (x)fi (x) · · · fi (x) 1

1

r

2

2

r

Mi ∩ Mj = {0} i = j ! " # $ " M1− + M2− + · · · + Mt− = Fq [x]n ,

%& Mi1 + Mi2 + · · · + Mir fi1 (x)fi2 (x) · · · fir (x)' ! ⊔ ⊓

ω F16 x4 + x + 1

F4 15

ω x2 + x + ζ, ζ F4 ζ 2 + ζ + 1 = 0

ω 4

C

n DC = {αi1 , αi2 , . . . , αil } n

c(x) ∈ C ⇐⇒ ∀ξ ∈ DC : c(ξ) = 0

C DC

C

D = {αi1 , αi2 , . . . , αil } n β i D = {β i , β i+1 , . . . , β i+l−1 }.

D = {α

, αi2 , . . . , αil } M (D)

k ⎞ . . . α(n−1)ii . . . α(n−1)i2 ⎟ ⎟ . ⎟ ⎠

i1

l × n 1, α , α ik

2ik

(n−1)ik

,...,α

⎛

1 αi1 α2ii ⎜1 αi2 α2i2 ⎜ M (D) = ⎜ ⎝

1 αil α2il . . . α(n−1)il

D

D

M (D)

D l

M (D)

l

l

l M (D)

j1 , j2 , . . . , jl

Y j β (j1 +j2 +...+jl )i (β r − β js ). r>s

β ⊔ ⊓

β n

i1 , i2 , · · · ik i1 < i2 < · · · < ik = i1 + t − 1 ≤ n.

t

M (β i1 , β i2 , · · · , β ik ) k

! A, B ⊆ Fqm

AB = { ξη : ξ ∈ A, η ∈ B}.

∗ A B M (A) ∗ M (B) M (AB) C1 C2 n1 n2 n1 ≤ n2 C1 Fq [x]n2 = F[x]/(xn2 − 1).

C1 c1 ∈ C1 c2 ∈ C2 c1 (x) c2 (x) d c1 (x)c2 (x)

(mod (xn2 − 1))

Fq [x]n−2

A B A ∗ B

ab a A b B

A B n ! A ∗ B n "

#

A B n

A ∗ B

rank (A) + rank (B) ≤ n.

λi

j = 1, . . . , n

j

B B′

! !

λj

B′

! !

A

!

rank (A) + rank (B) = rank (A) + rank (B ′ ) ≤ n, ⊔ ⊓

"

# $ " % !

c = (c0 c1 · · · cn−1 )

A

Supp c ci = 0

$ " $ # &

! !

i

" %

"

A

c∈C

%

C

"

I

%

AI

w(c) = |Supp (c)|

'

I

( $

# !#

Ì

A B

C

F

F

A∗B

rank (AI ) + rank (BI ) ≤ |I|,

I

c ∈ C

⊔ ⊓

AI BI |I|

d

!"℄ $ % " & d

∗

δ AI BI |I| I {1, 2, . . . , n}

|I| < δ d d ≥ δ '

(

Ì

dA

A

B

|B| + dA − 2 d ≥ |B| + dA − 1

B

n

AB

) rank (M (A)I ) =

j

|I|, ≥ dA − 1

|I| < dA |I| ≥ dA .

$ ' ! M (B) j |I| < dA 1, rank (M (B)I ) = |I| − dA + 2 dA ≤ |I| ≤ |B| + dA − 2. * % " rank (M (A)I ) + rank (M (B)I ) > |I|

|I| ≤ |B| + dA − 2;

I + , |B| + dA − 1 &

C - AB . C |B| + da − 1 ⊔ ⊓

C C

Ì

C n

R

R 1

c ∈ C 4

1 ∈ R R γ ∈ R

γ −1 ∈ R γ c(x) = xi1 + xi2 + · · · + xik

c ∈ C w(c) = k 1 ∈ R k n ξ c(ξ)c(ξ −1 ) = 0 F[x]n ξ i−j = ξ l−m ξ j−i = ξ m−l

c(ξ)c(ξ −1 ) 4 4 k 1 k(k − 1) = 0 (mod 4) ⊓ ⊔

k

n p (mod n) q = pk [n, k]

Fp

Ì

p

β ∈ Fq

n

V = { v(ξ) = (Tr(ξ) Tr(ξβ) . . . Tr(ξβ Tr(ξ) [n, k]

ξ

Fp

n−1

pk =

)) : ξ ∈ Fq },

Fp

V

!

! #

v(ξ)

v(x)

V

!

$&

Fq → Fp

$

v(ξβ −1 ) ! Vn (Fp )

Fp "

Tr(ξβ i ) v(ξβ −1 ) ∈ V

%

v(ξ)

V

β

Fq

$ !

t(x) = t0 + t1 x + · · · + tk xk

Fp [x]

k X

k

v(ξ) = (v0 v1 . . . vn−1 )

$

vi ti = Tr(ξt(β)) = Tr(0) = 0.

i=0

t(x)

'

"

(

h(x)

( V xk t(x−1 ) $& t(x) ! $

) *

h(x) = tR (x) = xk t(x−1 ).

h(x)

V k [n, k] ⊔ ⊓

C [15, 11, 3] g(x) = x4 + x + 1.

C

[63, 57] [31, 20, 6] C g(x) = x11 + x10 + x9 + x6 + x5 + 1.

i

i + 1 i − 1

!

"# #$# % &'()'*℄, !

i

- .

/

0

'

1 2

-

" 03 4

b

e

p b

! "

#$ "

% #

p &

& ' ( e

) # e

p e *

e p |p| l # # p e )

& p i (p, i)

+ p i n

e

" , e = (01000 00110) e -! ) ,

100 00011 11001 10010 00001

1 8 6 5 7 10

Ì 3

e w

w , ) -. #

e (p, i) e p 0 ! "

i + |p| + 1 i − 1 # (p, i)

100 00011 11001 10010 00001

1 (8, 0) 6 (2, 3, 4, 5) 7 −

Ì

e (p, i) (p′ , i′ ) e = (e0 e1 · · · en−1 ).

Z Z ′ p = e Z = ∅ p, p′ = e ! i = i′ " #

i e ! ei $ % ! t ∈ Z ∩ Z ′ j e

t & ! Z Z ′ j − 1 ' i = j = i′ ( ! (p, i) = (p′ , i′ )

⊔ ⊓

e n − w w = w(e) !) ' ' '

e * + * n−w i ! ei = 0

i

" i ! ' ( ' ! ' e , + n − w ⊔ ⊓

Ì

e n (p, i) (p′ , i′ ) |p| + |p′ | ≤ n + 1 (p, i) = (p′ , i′ )

w e w = 0 w = 1

w ≥ 2 - " p e e .' " 0, w = w(e) ! (p, i) (p′ , i′ ) / 0 ' '

(n − |p|) + (n − |p′ |) e |p| + |p′ | ≤ n + 1,

(n − |p|) + (n − |p′ |) ≥ 2n − n + 1 = n − 1,

w(e) ≤ 1

(p1 , i1 ) = (p2 , i2 )

⊔ ⊓

e (n + 1)/2

(n + 1)/2

[n, k, d] Fq ! " qn−k # d t = ⌊(d − 1)/2⌋ $ $

b

% b $

b & '

" (n + 1)/2 $

n

b 1 ≤ b ≤ (n + 1)/2 n2b−1 + 1 n F2

b (

)

$ *

* & ' + $ p " b 1, $ p 2b−1 b 1 -

n2b−1

0 & 1

⊓⊔ $

.

/

1 ≤ b ≤ (n + 1)/2 C n b 2n /(n2b−1 + 1)

n2b−1 + 1

b M C

C b M (n2b−1 + 1) V n (F2 ) M (n2b−1 + 1) ≤ 2n

⊓ ⊔

b

Ì 1 ≤ b ≤ (n + 1)/2 [n, k]

b

n ≤ 2r−b+1 − 1 , r = n − k r ≥ ⌈log2 (n + 1)⌉ + (b − 1) . ! [n, k]" M = 2k # $ % & 2k ≤

2b n2b−1

+1

.

' n ≤ 2r−b+1 + 2−b+1

(

n

n ≤ 2r−b+1 − 1,

) * + ) r ,

⊓ ⊔

!, $ #

-

b ≤ n/2

C n

b 2n−2b M C M > 2n−2b

n − 2b .

x, y ∈ C

n−2b

2b

x = VVVVVVVVVV AAAAAA y = VVVVVVVVVV BBBBBB,

V

A B 2b

n−2b

A A B B B z = VVVVVVVVVV A b

b

X Y b b ⊓ ⊔

b

Ì

C

0 ≤ b ≤ n/2

b

[n, k]

r ≥ 2b,

r =n−k

[n, k] 2k !

"# 2k ≤ 2n−2b . $"%&

2 "% ⊓ ⊔

' ( ) Fp ≃ Zp )

z

0 ≤ z ≤ p − 1

) p *

Fq q = ph h > 1 + , $

h Fp & , '

*

$- . / & ' 0 1 - # ) ( C ' 2 ) C Fp ( q m ≥ 1

e≤b Fq m

Fq

e′ ≤ (m − 1)b + 1 qm q

Ì [n, k] C Fq

b [mn, mk] C ′ Fq

(m − 1)b + 1

B Fq

Fq m

m

m [mn, mk] C ′ Fq

C

Fq

B π : C → C ′

r′ ∈ C ′

(m − 1)b + 1 π−1

r ∈ C

b !

r "

c

π

c′ = π(c) ∈ C ′ ⊓ ⊔ #

$ !!

C C ′

t

m Fq %

m

& e b (v, i)

e(x) ≡ xi−1 v(x)

(mod xn − 1),

' ()

v(x) =

b−1

vt xt

t=0

b − 1

e1 = (0101 0110 000) 6 V11 (Z2 ), e2 = (0000 0010 001) 5 V11 (Z2 ), e3 = (0100 0000 000) 5 V11 (Z2 ).

e1 (x) = x(1 + x2 + x4 + x5 ), e2 (x) = x6 (1 + x4 ), e3 (x) = x8 (1 + x4 ).

r s ! s " # $ e! % m = r − e

&

" $ ' Vn (F)/C ( ) &

" *$!

+ * ,

" $

-*

.

s

e ! +

Vn (F)/C q n−k + /%

Ì [n, k] C

Fq q n−k − 1

C 15 1 + x + x2 + x3 + x6 F2 [15, 9]

15 − 9 = 6

63 C 3 1

b ≤ 3 ! "

# b = 1 e(x) = xi ,

0 ≤ i ≤ 14.

$

%& '

( $

%& 2 # "

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14

b=1

100000 1 010000 2 001000 4 000100 8 000010 16 000001 32 111100 15 011110 30 001111 60 111011 55 100001 33 101100 13 010110 26 001011 52 111001 39

1+x x(1 + x) x2 (1 + x) x3 (1 + x) x4 (1 + x) x5 (1 + x) x6 (1 + x) x7 (1 + x) x8 (1 + x) x9 (1 + x) x10 (1 + x) x11 (1 + x) x12 (1 + x) x13 (1 + x) x14 (1 + x)

b=2

110000 3 011000 6 001100 12 000110 24 000011 48 111101 47 100010 17 010001 34 110100 11 011010 22 001101 44 111010 23 011101 46 110010 19 011001 38

Ì b = 1 b = 2

3 $

) e(x) = xi (1 + x2 ), 0 ≤ i ≤ 14, *%&+ i 2 e(x) = x (1 + x + x ), 0 ≤ i ≤ 14. *%,+ *%&+

$

%,

*%,+ -

1 + x + x2 111000 7 14 x(1 + x + x2 ) 011100 28 x2 (1 + x + x2 ) 001110 x3 (1 + x + x2 ) 000111 56 63 x4 (1 + x + x2 ) 111111 49 x5 (1 + x + x2 ) 100011 45 x6 (1 + x + x2 ) 101101 x7 (1 + x + x2 ) 101010 21 42 x8 (1 + x + x2 ) 010101 27 x9 (1 + x + x2 ) 110110 x10 (1 + x + x2 ) 011011 54 35 x11 (1 + x + x2 ) 110001 9 x12 (1 + x + x2 ) 100100 18 x13 (1 + x + x2 ) 010010 x14 (1 + x + x2 ) 001001 36

1 + x2 101000 5 10 x(1 + x2 ) 010100 20 x2 (1 + x2 ) 001010 x3 (1 + x2 ) 000101 40 31 x4 (1 + x2 ) 111110 62 x5 (1 + x2 ) 011111 51 x6 (1 + x2 ) 110011 x7 (1 + x2 ) 100101 41 29 x8 (1 + x2 ) 101110 58 x9 (1 + x2 ) 010111 x10 (1 + x2 ) 110111 59 57 x11 (1 + x2 ) 100111 61 x12 (1 + x2 ) 101111 53 x13 (1 + x2 ) 101011 x14 (1 + x2 ) 101001 37

Ì b = 3 60

63

b≤3

6

b ≤ 3

C

3

2 e(x) = x8 + x10 111111 ! x4 (1 + x + x2 )

3

" # $ !% &

&&' !( )*

r(x)

+

g(x)

! !% & ,

- )&.

/ !% &

r

3

e

/

s = rH T = eH T ,

e 3 r 3 e !"

# [n, k]$

b% # r

C

#

e r − e ∈ C e

i = 0

si (x)

xi r(x) si (x) b

e(x) = x(n−i) mod n si (x) i = i + 1 i = n & ' "

( C ) *+, b = 3 r = (1110 1110 1100 000) - r(x) = (x3 + x2 )g(x) + 1 + x + x4 + x5 , s0 (x) = 1 + x + x4 + x5 . - w(s0 ) > 3

i > 0% . s1 (x) = xs0 (x) (mod g(x)) = 1 + x3 + x5 / w(s1 ) = 3 s1 3% i > 1 0 ' *! s9 (x)

i 0 1 2 3 4 5 6 7 8 9

si (x) 110011 100101 101110 010111 110111 100111 101111 101011 101001 101000

Ì

3 e = (0000 0010 1000 00),

r → r − e = c = (1110 1100 0100 0000).

[n, k]

! "# $ ½ g(x) (x + 1)(x3 + x + 1) (x + 1)(x4 + x + 1) (x4 + x + 1)(x2 + x + 1) (x + 1)(x5 + x2 + 1)

[n, k, d] [7, 3, 4] [15, 10, 4] [15, 9, 3] [31, 25, 4]

b 2 2 3 2

t

t d − 1 = 2t ! d " #

$ % 1

&

h > t

! "

# $ %

& [n, k]'

b

e1 e2

(p1 , i1 ) (p2 , i2 ) |p1 | + |p2 | ≤ 2b e1 = e2

C

b1 , b 2

b b1 + b2 = 2b

b1 ≤ b2

b1

b2

( Ei i = 1, 2

bi e1 ∈ E1 e2 ∈ E2 e1 e2 2b )

e1 e2 * +

C ⊔ ⊓

b'

, f (x) ∈ Fq [x] f (x) - α f (x) f (x) n αn = 1

[n1 , k1 ]

g1 (x) b1

g2 (x)

[n2 , k2 ]

g2 (x) m

g1 (x) g2 (x) " g(x) = g1 (x)g2 (x) b " !

[n, k]

n = n1 n2 / gcd(n1 , n2 ) k = n − deg g1 (x) − deg g2 (x) b = min{b1 , m, (n1 + 1)/2}. n n - g1 (x) x 1 − 1 g2 (x) x 2 − 1. /

xn1 − 1 xn2 − 1 xn − 1 0 g1 (x) g2 (x) /

g(x) = g1 (x)g2 (x) xn − 1 + g(x) [n, k]'

k = n − deg g(x) = n − deg g1 (x) − deg g2 (x).

+ g1 (x) b1 '

g(x) b'

- g1 (x)

* ( ) e1 e2

n

(p1 , i) (p2 , j) (

|p1 | + |p2 | ≤ 2b.

e1 = e2 p1 (x) p2 (x) p1 p2 e1

e2

e1 (x) = xi p1 (x); e2 (x) = xj p2 (x).

S1 (x) = xi p1 (x)

(mod g(x))

S2 (x) = xj p2 (x)

(mod g(x)).

S1 (x) = S2 (x) xi p1 (x) = xj p2 (x)

(mod g(x)).

! g(x) g1 (x)g2 (x) " #

xi P1 (x) = xj P2 (x)

(mod g1 (x)).

l = |p1 | + |p2 | ≤ 2b ≤ 2b1

# g1 (x) e1 e2 $ xi p1 (x) = xj p2 (x)

(mod xn1 − 1).

% l ≤ 2b ≤ n1 + 1 & ' p1 = p2 i = j (mod n1 ) % xi p1 (x) = xj p2 (x) = 0

(xi − xj )p1 (x) = 0

(mod g2 (x)) (mod g2 (x)).

! 2|p1 | ≤ 2b p1 (x) # b−1 < m $ # (

g2 (x) ) # p1 (x) g2 (x)

xi − xj = 0 (mod g2 (x)).

* + " xt (mod g2 (x)) n2 i = j (mod n2 ) , n $ n1 n2 i = j (mod n) - i, j ∈ {0, 1, . . . , n − 1} i = j . , " ⊔ ⊓

b h(x) = (x2b−1 − 1) f (x) f (x) f (x) h(x) m = deg f (x) ≥ b f (x) n0

g(x) = f (x)h(x) [n, n − 2b + 1 − m]

! b n = lcm (2b − 1, n0 )

"# * # & g1 (x) = h(x) g2 (x) = f (x) ⊔ ⊓

g(x) = (x3 +1)(x3 +x+ 1) = x6 + x4 + x + 1 [21, 15, 4] 2 "

2! 1

3!

4

v = (1 1 0 0 0 0 0 1 0 1 1).

C [9, 3, 3] g(x) = x6 + x3 + 1.

2

r = (111 100 100).

C 3 [35, 27]

! "#$%&' ( ) "#$*$' ) )

) + Fq , . ./½ 0 ) u(x) n − 1 # 1 u(x) 2 n x

3 u(x) = u +u x+· · ·+u

4 n − 1 1 Fq 5 ω n! Fq u(x) 0

1

u ˆ(x) = u(ω 0 ) + u(ω)x + · · · + u(ω n−1 )xn−1 .

1

n−1 n−1 x

u(x) ∈ Rn (Fq ) ω n

Fq n uˆk k uˆ(x) u ˆk =

n−1

!"

ω ik ui .

i=0

# $ % &

' ( ' $ !"

u = (u0 u1 · · · un−1 ) u(x) = u0 + u1 x + · · · + un−1 xn−1 .

U (x) = uˆ(x) u u $ u(x) → uˆ(x) & $ $ Rn (Fqˆ) ) n − 1 Fqˆ Fq

ω ( & $ &

E = {1, x, x2 , . . . , xn−1 }

$ * ⎛

1 1 ⎜1 ω ⎜ ⎜ 1 ω2 ⎜ ⎜ ⎝

⎞ ... 1 . . . ω n−1 ⎟ ⎟ . . . ω 2(n−1) ⎟ ⎟. ⎟ ⎠

1 ω2 ω4

1 ω n−1 ω 2(n−1) . . . ω (n−1)

2

+ & ,

α ∈ Fn+1 n−1

αi =

i=0

α = 1 .

n 0

- α

n−1 i=0

i

α =

n−1 i=0

. αn = 1 = α0 α ∈ Fn+1

i+1

α

=

n i=1

αi .

n−1

(α − 1)

αi

i=0

!

= 0.

α = 1

Ì u1 x + · · · + un−1 xn−1

uk =

ˆ u ˆt =

u(x) = u0 + u ˆ(x) = uˆ0 + uˆ1 x + · · · + un−1 ˆ xn−1

⊓ ⊔

n−1 1 −ik ω u ˆi . n i=0

t

n−1

it

uˆi ω =

i=0

n−1 i=0

⎛ ⎝

n−1

uj ω

j=0

⎞

ij ⎠

it

ω =

n−1 j=0

uj

n−1

ω

i(j+t)

i=0

!

.

j + t = n ! " n

uˆ ˆt = nun−t , n # ! $ ˆ u ! u ⊓ ⊔ % & '

1 ˆ −t uˆ(ω ). ( n 1/n ) ut =

)

u % & j − 1 u ˆ(x)

j u

u = (u0 u1 . . . un−1 ) ui u

ˆ = (ˆ ˆ1 , . . . , u ˆn−1 ) u u u0 , u

!

* + ) , " $+

K C

Fq g(x) =

k∈K

(x − ω k ).

1, 2, . . . , d − 1 ∈ K c

C

c n − d

c(ωj ) = 0 1 ≤ j ≤ d − 1 c(x)

c g(x) n − 1, n − 2, . . . , n − d + 1 C(X) = cˆ(x) ⊓ ⊔

r n¾

C(X)

c

w(c) = n − r

r n C(X) r c ! ⊓ ⊔

Fq Fq d ≥ δ δ " # $

Fq % n = q m − 1 Fq & ' δ

g(x) Fq αl , αl+1 , . . . , αl+δ−2 ,

α n l ( ) l = 1 ! # $ &

' q n δ BCHq (n, δ) ) α Fq # $ n = qm − 1 * α n αj n m

x − αj

xn − 1+ g(x) ,( -. xn − 1 n n − deg g(x) 2

n δ Fq δ

Mi+ !"

l = 0 δ

Ì

δ

C

C

Fq

d ≥ δ

# $

l = 1 %

C & '

Fq Fq g(x) =(x − α)fe1 (x)(x − α2 )fe2 (x) · · · (x − αδ−1 )ff δ1 (x) =(x − α)(x − α2 ) · · · (x − αδ−1 )e g (x).

( ) * gˆ(x)

& g(x)

n+ $ %

)

αi

, - & .+# /

0 1 v = n−δ 2 / 3 " / C v − l = δ ⊔ ⊓

n! " # " $ Fq ⊇ Fq %

m n q m − 1 q m − 1 = kn ω Fq α = ω k n! " # " m

m

C = BCH2 (9, 2) &

" # F2 ' x9 − 1 F2 ξ F2

6

6

x6 + x4 + x3 + x + 1.

" ξ 26 − 1 = 63 ( α = ξ 63/9 = ξ 7

" # α, α2 .

x6 + x3 + 1.

9 − 6 = 3 C ⎛

⎞ 100100100 G = ⎝0 1 0 0 1 0 0 1 0 ⎠ . 001001001

8 C

0 3 G 3 3 G 2 2 6 3 G 9

C 3 !"#

Fq

$ % Fqm

& ' " & % (& $

) ( BCH2 (15, 7) $ 15 15* & ' α +

$ 15 = 24 − 1 α F16 )

x4 + x + 1.

, 2

15

C0 = {0}

C1 = {1, 2, 4, 8} C3 = {3, 6, 9, 12}

C5 = {5, 10} C7 = {7, 11, 13, 14}.

) ( m1 (x) m3 (x) m7 (x) 4 m5 (x) $ F2 -

g(x) = m1 (x)m3 (x)m5 (x),

g(x)

f (x) = x15 − 1 g(x)

[15, 5]

R = { αi : i ∈ {1, 2, 4, 8, 3, 6, 9, 12, 5, 10}.

C

R

α α2 α3 α4 α5 α6 δ = 7

m1 (x) = 1 + x + x4 m3 (x) = 1 + x + x2 + x3 + x4 m5 (x) = 1 + x + x2 ; d = δ

g(x)

g(x) = 1 + x + x2 + x4 + x5 + x8 + x10 .

7

g(x) !! g(x) = m3 (x)m5 (x)m7 (x)" α9 α10 α11 α12 α13 α14 δ = 7 # g(x) = 1 + x2 + x5 + x6 + x8 + x9 + x10 d = δ l = 9

Ì

C

d

C

δ

δ ≤ d ≤ 2δ − 1.

$ %

Fq

r − m = e,

r

m n

&

e = (e0 e1 · · · en−1 )

n Fq n (i, ei ) i 0 n − 1

= { (i, ei ) : ei = 0}. E

E e { i : (i, ei ) ∈ E}

e E

e i ei (i, ei ) ∈

ei ∈ Fq E Fq !

"

!

e

# " $ %

& α n %

'

σv (x) =

j∈Supp v

v n (1 − αj x).

' v v

& v i ∈ Supp v '

i v

σv(i) (x) =

j∈Supp v j =i

(1 − αj x).

i ∈ Supp v

σv(i) (x) = σv (x)/(1 − αi x).

v vi . ωv (x) = vi σv(i) (x) = σv (x) (1 − αi x)

i∈Supp v

i∈Supp v

σv ωv

v

v gcd(σv (x), ωv (x)) = 1 α(−i) i ∈ Supp v " J

! #

J = { i ∈ Supp v : ωv (α−i ) = 0}; $

gcd(σv (x), ωv (x)) =

i∈J

(1 − αi x).

# $ % ! " !

i ∈ Supp v$

ωv (α−i ) = vi σv(i) (α−i ).

$ % ' !

J =∅

gcd(σv (x), ωv (x)) = 1

& (i)

−i

σ (α ) = 0

vi = 0 ⊓ ⊔

# " ! (

v (x)$ σv (x) *

v

)

ωv (x)

v n σv (x) v (x) = ω(x)(1 − xn ).

, )

%

(i)

σv (x)$

v(x) =

i∈Supp v

vi

n−1 j=0

xj αij .

+

v-

i ∈ Supp v

σv (x) = σv(i) (x)(1 − αi x),

σv (x) v (x) =

i∈Supp v

=

i∈Supp v

vi σv(i) (x)(1 − αi x) vi σv(i) (x)(1 − xn )

n−1

xj αij

j=0

= ωv (x)(1 − xn ),

⊓ ⊔

v

i ∈ Supp v vi = −αi

ωv (α−i ) , σv′ (α−i )

σv′ (x) ¿ σv (x)

σv (x) v ′ (x) + σv′ (x) v (x) = ωv (x)(−nxn−1 ) + ωv′ (x)(1 − xn ).

x = α−i i ∈ Supp v ! σv (x) 1 − xn σv′ (α−i ) v (α−i ) = −nαi ωv (α−i ).

" # $ v(α−i ) = nvi %

⊓ ⊔

& !

σv v v σv vi

σv (x) = 1 + σ1 x + . . . + σd xd

v 0 ≤ j ≤ n − 1

vj = −

d i=1

σi vj−1 ,

n 3

P i

u(x) = u′ (x) = n−2 i=0 (i + 1)ui+1 x

Pn−1 i=0

ui xi ∈ Fq [x]

σv (x) v (x) = 0

(mod 1 − xn ).

0 ≤ j ≤ n − 1 xj σv (x)v(x) (mod 1 − xn ) d i=0 σi v(j−i) (mod n) 0 ≤ j ≤ n − 1 d i=0

σi vj−i .

σ0 = 1 n ⊓ ⊔

! " #

$ % C Fq n δ = 2t + 1 & g(x) C

c(x) c ∈ C e(x) e w(e) = l ≤ t = ⌊(δ − 1)/2⌋ ' r = c+e e(x) c(x)

s(x) r(x) r(x) g(x) ( r(x) = h(x)g(x) + s(x).

( )

l = 1 & * + $ l = 1 i 1 ≤ i ≤ δ − 1 ) Si = r(αi ).

( , αi 1 ≤ i ≤ δ − 1 r(αi ) = s(αi ) c(αi ) = 0 i r(x) = c(x) + e(x) r(αi ) = e(αi ) 1 ≤ i ≤ δ − 1 & ) Si e(x) % Si =

n−1 j=0

rj αij ,

i

n

r(x) =

r(αj )xj

j=0

r(x)

Si =

n−1

ej αij .

j=0

Si

S0 =

n−1

rj ,

j=0

E(x) δ−2

E(x) =

Si+1 xi .

i=0

! e = (e0 e1 , . . . en−1 ) ! E(x) =

n−1

(ei αi )xi ,

i=0

E = (e0 , αe1 , α2 e2 , . . . , αn−1 en−1 ).

"

E i = E

n−1 j=0

Ej αij =

n−1

ej αi(j+1) = Si+1 ,

j=0

E(x) # ei =

Ei . αi

e(x) = 0 E(x) = 0 $ σE (x) σe (x)

F2

σE (x)

r(x)

r

Si = r(αi ).

S(x) =

δ−2 j=0

Sj+1 xj = E(x)

r

δ

Fq

x

r(x)

1

!"# $

xδ

(

σE (x)E(x) = ωE (x)(1 − xn ).

n ≥ δ$

%!&'

ωE (x)(1 − xn ) = σE (x)S(x)

%!"#'

ωE (x) ≡ σE (x)S(x)

(mod xδ ),

%!""'

deg ωE (x) = p − 1 ≤ t − 1$ deg σE (x) = m ≤ t deg S(x) ≤ δ − 1

)

!

"

δ

m(x) δ−2

# $

% & ! ' %

(

* +,

-./℄

$ a(x) b(x) $ F$

(

deg a(x) ≥ deg b(x)$ $

1

23 3$

a(x)$ b(x)$

u(x)a(x) + v(x)b(x) = g(x).

g(x) = gcd(a(x), b(x)) u(x) v(x)

a(x) ∈ F[x] b(x) ∈ F[x]

r(x) ∈ F[x]

deg b(x) ≤ deg a(x) r(x) = gcd(a(x), b(x))

u(x), v(x) ∈ F[x]

u(x)a(x) + v(x)b(x) = r(x).

u−1 (x) = 1, v−1 (x) = 0, r−1 (x) = a(x) u0 (x) = 0, v0 (x) = 1, r0 (x) = b(x)

i≥1

qi

ri

ri−2 (x) = qi (x)ri−1 (x) + ri (x),

ui

vi

deg ri < deg ri−1

ui (x) = ui−2 (x) − qi (x)ui−1 (x); vi (x) = vi−2 (x) − qi (x)vi−1 (x);

i

deg ri

ri = 0 n = i − 1 r(x) = rn (x) u(x) = un (x)

v(x) = vn (x)

i

ri = 0

! " #

$ % &

vi (x)b(x) ≡ ri (x)

(mod a(x))

deg vi (x) = deg ri (x) < deg a(x).

vi ri−1 − vi−1 ri = (−1)i a ui ri−1 − ui−1 ri = (−1)i+1 b ui vi−1 − ui−1 vi = (−1)i+1 ui a + vi b = ri deg(ui ) + deg(ri−1 ) = deg(b) deg(vi ) + deg(ri−1 ) = deg(a)

0≤i≤n+1 0≤i≤n+1 0≤i≤n+1 −1 ≤ i ≤ n + 1 1≤i≤n+1 0≤i≤n+1

Ì a(x) b(x) deg a ≥ deg b

µ≥0

ν≥0

µ + ν = deg a − 1

j

0≤j≤n

deg vj (x) ≤ µ,

j

j < n

deg rj (x)

deg rj (x) ≤ ν.

rn (x) = gcd(a, b)

j

deg rj−1 (x) ≥ ν + 1

deg rj (x)

deg rj (x) ≤ ν. j

deg vj (x) ≤ µ

deg vj+1 (x) ≥ µ + 1, ⊓ ⊔

vj (x) rj (x)

a(x) b(x)

deg a(x) ≥ deg b(x)

!

"

(µ, ν)

µ + ν = deg a(x) − 1

(a(x), b(x), µ, ν) (vj (x), rj (x))

deg vj (x) ≤ µ,

deg rj (x) ≤ ν

℄ ! " m = µ n = ν

℄ ℄ ℄ !

! "#℄"# ℄ "℄ "#℄#" ℄%"# ℄ "#℄#" ℄% "# ℄ & '( "℄)! '("℄) * " "℄"℄℄ & + "℄%"℄ " "# ℄"# ℄"# ℄℄

Ì

a(x) b(x) v(x) r(x) v(x)b(x) ≡ r(x)

(mod a(x)),

deg v(x) + deg r(x) < deg a(x).

vj (x) rj (x) j = −1, 0, . . . , n + 1

(a(x), b(x)) j 0 ≤ j ≤ n λ(x) v(x) = λ(x)vj (x),

r(x) = λ(x)rj (x).

j

ν = deg r(x) µ = deg a(x) − deg r(x) − 1 deg v(x) ≤ µ

deg vj+1 (x) ≥ µ + 1 ≥ deg v(x) + 1

deg rj−1 (x) ≥ ν + 1 = deg r(x) + 1.

j uj (x)a(x) + vj (x)b(x) = rj (x),

u(x) u(x)a(x) + v(x)b(x) = r(x).

v(x) vj (x)

uj (x)v(x)a(x) + vj (x)v(x)b(x) = rj (x)v(x) u(x)vj (x)a(x) + v(x)vj (x)b(x) = r(x)vj (x).

rj (x)v(x) ≡ r(x)vj (x)

(mod a(x)).

deg rj (x)v(x) = deg rj (x) + deg v(x) ≤ ν + µ < deg a(x)

deg r(x)vj (x) = deg r(x) + deg vj (x) ≤ ν + µ < deg a(x).

rj (x)v(x) = r(x)vj (x) uj (x) vj (x) u(x) = λ(x)uj (x) v(x) = λ(x)vj (x).

r(x) λ(x)uj (x)a(x) + λ(x)vj (x)b(x) = r(x),

r(x) = λ(x)rj (x). ⊓ ⊔

v(x)

r(x)

v(x)b(x) ≡ r(x)

(mod a(x))

deg v(x) ≤ µ

deg r(x) ≤ ν,

µ

ν

deg r(x) − 1 = µ + ν.

vj (x) rj (x) λ(x)

(a(x), b(x), µ, ν)

v(x) = λ(x)vj (x) r(x) = λ(x)rj (x).

(x6 + x4 + x2 + x + 1)σ(x) ≡ ω(x)

(mod x8 ),

deg σ(x) ≤ 3 deg ω(x) ≤ 4 (x8 , x6 +x4 +x2 +x+1, 4, 3)

x2 + 1 x3 + x + 1.

σ(x) = λ(x)(x2 + 1) ω(x) = λ(x)(x3 + x + 1),

deg λ(x) ≤ 1 σ ω

! " # E(x) $%& % ' # (

t ≤ ⌊(δ − 1)/2⌋ ( (

Ì S(x) σ(x) ω(x)

) ω1 (x) σ1 (x) deg ω1 (x) ≤ p − 1 ≤ t − 1 deg σ1 (x) ≤ t

ω(x)1 ≡ σ1 (x)S(x)

(mod xδ ).

* +,

* , σ1 (x) * +, σ(x)

ω(x)σ1 (x) ≡ ω1 (x)σ(x)

(mod xδ ),

deg ω(x)σ1 (x) ≤ 2p − 1 2p − 1 ≤ δ − 2- deg ω1 (x)σ(x) ≤ 2t − 1 = δ − 2-

ω(x)σ1 (x) = ω1 (x)σ(x).

* .,

' / 0 gcd(ω(x), σ(x)) = 1 ' 1( * ., ω(x) ω1 (x) σ(x) σ1 (x) 2 ω1 (x) = 0 σ1 (x) = 0 deg ω1 (x) ≤ deg ω(x) deg σ1 (x) ≤ deg σ(x) (

λ1 , λ2 ∈ Fq

ω1 (x) = λ1 ω(x),

σ1 (x) = λ2 σ(x).

σ(x) ( σ1 (x)- (

⊓ ⊔

ω(x) σ(x)

t! deg σ(x) ≤ t deg ω(x) ≤ t − 1 " gcd(σ(x), ω(x)) = 1 # $

% & a(x) = x2t , b(x) = S(x), v(x) = σ(x), r(x) = ω(x),

"

(v(x), r(x))

µ=t ν = t − 1.

(x2t , S(x), t, t − 1)

v(x) = λσ(x) r(x) = λω(x),

' λ "

!

σ(0) = 1! λ = v(0)−1 ( ! ' σ(x) ω(x) !

σ(x)

( ' ! ' Fq )'' ! q = 2 " g(x) = 1 + x + x2 + x4 + x5 + x8 + x10

[15, 10]*

+,-

7 " ! α ∈ F16 1 + α + α4 = 0 α

' 15* . α2 , α3 , α4 , α5 , α6

g(x) g(x)

+,- l = 2 ) '

g(x) $ ' / m1 (x) = 1 + x + x4 m3 (x) = 1 + x + x2 + x3 + x4 m5 (x) = 1 + x + x2 .

# r = (10011 11110 00110).

r(x) = (1 + x + x6 + x8 + x9 )m1 (x) + (x2 + x3 ) 2

7

9

2

r(x) = (x + x + x )m3 (x) + (1 + x ) 2

5

8

9

11

r(x) = (1 + x + x + x + x + x )m5 (x) + x.

F16

S1 = r(α) = α2 + α3 = α6 S2 = r(α2 ) = (r(α))2 = α12 S3 = r(α3 ) = 1 + α6 = α13 S4 = r(α4 ) = (r(α))4 = α9 S5 = r(α5 ) = α5 S6 = r(α6 ) = (r(α3 ))2 = α11 .

! ei = Si+1

S(x) = α6 + α12 x + α13 x2 + α9 x3 + α5 x4 + α11 x5 .

" # S(x) F16 ! $ ! %& '

' xδ−1 = x6 S(x) ( )

# * si ti ri deg ri 1 0 x6 6 0 1 (α6 , α12 , α13 , α9 , α5 , α11 ) 5 1 (α13 , α4 ) (α4 , 0, α6 , α12 , α8 ) 4 (α2 , α3 ) (0, α11 , α7 ) (0, α2 , α3 , α14 ) 3 (1, α11 , α12 ) (α13 , α4 , α5 , α) (α4 , 0, α)

Ì

deg ri (x) < (δ −1)2 = 3+

! $ σ1 (x) = α13 + α4 x + α5 x2 + αx3 .

* σ1 (x) σ(x) $, α2 α14 α11 + $

E = (01001 00000 00010).

σE = σe e = E r → r − e = c = (11010 11110 00100).

F15 2 r = (101 000 111 111 01). 15

ω ! "

x4 + x + 1 = 0.

#$% C ! 31 11 & ' #$% F5 ! 24 5 ! ! 6 ( C

) *+ !

r = (111011 111101 0101001 0011011 11000).

℄

!" #

!

$ % & '!(#

%

) *#+ & #

p , t > 0 # ) n = p

Fn+1 [x]

ω

$

n Θf

Fn+1 #

"

t

−1 f (x) ∈

Θf = (f (ω 0 ), f (ω 1 ), · · · , f (ω n−1 )). Θ : Rn (Fn+1 ) → Fnn+1 f (x), g(x) ∈ Fn+1 [x] -&

$

Θf +g = ((f + g)(ω 0 ), · · · , (f + g)(ω n−1 )) = (f (ω 0 ), · · · , f (ω n−1 )) + (g(ω 0 ), · · · , g(ω n−1 )) = Θf + Θg ,

Θλf = (λf (ω 0 ), · · · , λf (ω n−1 ) = λΘf .

0 . k(x) Θ k(ω ) = k(ω 1 ) = . . . = k(ω n−1 )# ) / 0 ≤ i ≤ n − 1 ω i

k(x) n deg k(x) ≥ n#

p n t ω k ≤ n RS (n, k) n k

RS (n, k) = {Θf : f (x) ∈ Fn+1 [x], deg f < k}.

Θ

RS (n, k)

Fn

Θ n

Θ : Rk−1 (Fn+1 ) → Fnn+1 RS (n, k) k RS (n, k)

!

"

d − 1 ≤ n − k.

RS (n, k) n − k + 1

p(x) q(x) # k − 1 r(x) = p(x) − q(x)

p(x) = q(x) ⇔ r(x) = 0, deg r(x) < k r(x) I p(ω i ) = q(ω i ) $ r(x) = (x − ω i ). i∈I

r(x) % I & |I| ≤ k '

( ) " p q p(x) q(x)

" d(p, q) ≥ n − (k − 1) = n − k + 1.

(

⊓ ⊔

* + , - RS d (n)

+ d n

RS (n, n − d + 1) = RS d (n).

+ RS (15, 7) F16

. /0 15 − 7 + 1 = 9

RS (n, k)

c = (c0 c1 · · · cn−1 ) RS (n, k) c(x) ∈ Fn+1 [x] k − 1 ci = c(ω i ).

c(x) = c(ω −1 x) deg c(x) = deg c(x) < k ω −1 = ω n−1

c(ω i ) =

cn−1 ci−1

i = 0 0 < i ≤ n − 1.

Θec = (cn−1 c0 · · · cn−2 )

RS (n, k)

!

⊓ ⊔

g(x) = x8 + ω 14 x7 + ω 2 x6 + ω 4 x5 + ω 2 x4 + ω 13 x3 + ω 5 x2 + ω 11 x + ω 6 ,

ω

F2 [x] x4 + x + 1.

g(x) "## g(x) =

8

(x − ω i ),

i=1

$ %&

'

(

)

#

* +

Ì RS (n, k) Fn+1 n − k

⌊(n − k)/2⌋

" (

) , -

# # Fq q = pt . / "0 0 1 # # "0 2!3℄ 256℄ 7

#

[n, k]

C F2 [nm, km] C ′ b = m(⌊(n − k)/2⌋ − 1) + 1 # * 6 33 m

# C n − k + 1 8 C

⌊(n − k)/2⌋)

⊓ ⊔

m = (m0 m1 · · · mk )

RS (n, k) m m(x) =

n−d−1

mi xi .

i=0

m(x)

c = Θm = (m(ω 0 ) m(ω 1 ) · · · m(ω n−1 )).

Fkn+1 → Fnn+1 ci c ci =

k j=0

mj ω ij =

n j=0

mj ω ij = ci ,

mj = 0 j ≥ k c

! m "

#

m $

m(x)% c " & # ! r n m '# # ( #

) k − 1 * k

+

,

r

k 0- - m k

* mi

m

. "

½º q t ≤ n/2

n = q − 1 k = n − 2t

n x0 , x1 , x2 , . . . , xn−1 Fq n y0 , y1 , . . . , yn−1 ∈ Fq k > 0

P (x) ∈ Fq [x]

deg P (x) ≤ k − 1 P (xi ) = yi t i

½ ! " # $ % & ' ( ) * ) # * * +, # $ %& & C RS (n, k) xi = αi * r = (y0 y1 . . . yn−1 ),

c . * c Fq * c(x) ∈ Fq [x] deg c(x) ≤ k − 1 " t = ⌊(d − 1)/2⌋" * r c t +," " "

" c(x) " " c *" " " " c(x) 1

℄ ! " # $ P (x) # ! % P (x) ! #

& " ' & (

)

t(x) u(x)

v(x) = t(x) − u(x) 0 2t x Fq n + 1

v(x) n + 1 − 2t > k

v(x) = 0

deg S(x) > k

t(x) u(x) k − 1 ! " #

r

C

$ O(n3 ) %

&' ( ) *% +,

- $ O(npoly log n)

p(x)

y = (y0 y1 · · · yn−1 ) .

/ E(x) $ E(ω i ) = 0 p(ω i ) = yi 0 ω i

E(x)

E(x) 0

N (x) $

N (x) = p(x)E(x). 0

N (ω i ) = p(ω i )E(ω i ) = yi E(ω i ). N (x) 1

E(x) 0 $ % $ t , $ 2

#

⌊(d − 1)/2⌋

/ p(x)

ω i

n, k ,

n−k 2 1

t≤ 0

ω , ω , . . . , ω n−1 ∈ Fn+1

y0 , y1 , . . . , yn−1 ∈ F

p(x) deg p(x) < k i p(ω ) = yi i i

N (x) =

deg E(x) = t E(x)

0

n − 1

k+t−1 j=0

Nj xj

E(x) =

t

j=0

Ej xj

t

Et = 1;

N (ω i ) = yi E(ω i ).

deg N (x) ≤ k + t − 1 i = 0, . . . , n − 1

N (x)

%

E(x)

"

E(x)

$ !

&

#

n + 1 ! N0 , N1 , . . . , Nk+t−1 E0 , E1 , . . . Et−1 O(n3 )

k + t + t = k + 2t ≤ n

p(x)

p(x) = N (x)/E(x)

t

!

'

(N (x), E(x)) (' (N (x), E(x)) (N ′ (x), E ′ (x))

&

yi E(ω i ) = N (ω i )

N ′ (ω i ) = yi E ′ (ω i ),

yi E(ω i )N ′ (ω i ) = yi E ′ (ω i )N (ω i ).

)

E(ω i )N ′ (ω i ) = E ′ (ω i )N (ω i )

*

( )

i yi = 0

yi yi = 0

N (ω i ) = N ′ (ω i ) = 0

! deg E ′ (x)N (x) = deg E(x)N ′ (x) ≤ k + 2t − 1 < n,

E(x)N ′ (x) E ′ (x)N (x)

" p(x) = N (x)/E(x) # $ % deg p(x) < k&

ω i E(x)

p(ω i ) = N (ω i )/E(ω i ) = yi E(ω i )/E(ω i ) = yi ,

p(x) '

% () '

% * +

xi = αi Fq & ,

& # & " +-

'

()

.

() /

0.1 0.1 () & '

'

2% ,

n = q − 1 q

BRS (n, d)

d

0.1 Fq n = q − 1

BRS (n, d)

g(x) =

d−1 i=1

(x − ω i ),

α Fq n d Fq

! " # $ β $ % Fqm ⊇ Fq Fq Fq n

Ì C δ C k = n − δ + 1.

d C δ g(x) Fn+1 k k = n − deg g(x) = n − (d − 1) & d ≤ n − k + 1 δ ≤ d ⊓ ⊔ $ α = 2 Z5 &$ α 4 α

Z5 ' Z5 (

4 ) 4 4 *% +, BRS (4, 2) BRS (4, 3) BRS (4, 4)

g(x) (x − 2) (x − 2)(x − 4) (x − 2)(x − 4)(x − 3)

k 3 2 1

Ì 4 & -

% pt − 1 p .

-

n = 59 δ = 14 2 / $

59

1 BRS (59, 14)

BRS (63, δ) δ 1 62 F64 BRS (63, 14 + 4) F64 g(x) =

17 (x − αi ),

i=1

C

α

F64

!

"

G

#

$ G′

" %

C ′

4

G′

59

& 14 %

C = BRS (63, 18) 18 '

C ′

C

x = (x0 x1 · · · xn−4 xn−3 xn−2 xn−1 ) ∈ C

x4 = (x0 x1 , · · · xn−5 ), x4 = (xn−4 xn−3 xn−2 xn−1 ). x ∈ C

w(x) = w(x4 ) + w(x4 ) ≥ 18, w(x4 ) ≤ 4 y ∈ C ′ y = x4 y

x∈C

w(y) = w(x4 ) ≥ 18 − w(x4 ) = 14,

C′

14

'

C 47 C ′ k 47 47 − 4 = 43 !

k

43

& (

' ) *+, +-+

./ 01!

01!

BRS (n, d) RS d (n)

C = BRS (n, d) k

ω Fn+1 c = (c0 · · · cn−1 ) ∈ C

c(x) i 1 ≤ i ≤ d − 1 c(ω i ) = 0.

! " # $ 1, . . . , d − 1 % d = c(x)

n X

c(ω j )xj =

j=0

n X i=0

&

b ci xi ,

c1 = b b c2 = . . . = b cd−1 = 0;

1 ≤ j ≤ d − 1 & '" cj = b

n−1 X

ci ω ij = 0.

( )

i=0

* ( ) + d − 1 '" ci , 1 0 1 α α2 α3 . . . αn−1 B1 α2 α4 . . . α2n−2 C C B C B A @ 1 αd−1 α2(d−1) . . . α(d−1)(n−1))

* " . M (D) / & " D = {α, α2 , . . . , αd−1 } 0 " ( ) C 1 .2 d − 1 ' d − 1 0 % ' %

Ì

C BRS (n, d) Fq d c = (c0 , c1 , . . . , cn−1 ) ∈ BRS (n, d) P (x) = P0 + P1 x + . . . + Pk−1 xk−1

Fq [x] k − 1 = n − d ci = ω −i(d+1) P (ω −i ).

* Pc (x) c % 3 , " ' -

$ ( k) c ∈ C f = (f0 f1 . . . fn−1 )

fj = ω j(d+1−n) cj .

i ≥ d − 1 fbi =

n−1 X

fj ω ij =

j=0

n−1 X j=0

cj ω j(i+d+1−n) = b ci+d+1−n .

b c1 = b c2 = . . . = b cd = 0 8 fbn−1 = b cd = 0 > > > >fb < cd−1 = 0 n−2 = b > > > > :b c1 = 0. fn−d = b

f fb(x) n − d − 1 = k − 1 P (x) =

P (α−i ) =

1b f (x). n

k−1 1 X b −ij = fi , fi α n j=0

⊔ ⊓

ω j c ∈ C k − 1 F⋆q ! "# $ % & ' # " ## $ (

j(d+1−n)

C = RS 4 (10)

7 F11 C g(x) = (x − ζ)(x − ζ 2 )(x − ζ 3 ) = x3 + ζ 3 x2 + x + ζ,

ζ F11 ζ = 2 C ⎛

218 ⎜0 2 1 ⎜ ⎜0 0 2 ⎜ G=⎜ ⎜0 0 0 ⎜0 0 0 ⎜ ⎝0 0 0 000

10 81 18 21 02 00 00

00 00 10 81 18 21 02

⎞ 000 0 0 0⎟ ⎟ 0 0 0⎟ ⎟ 0 0 0⎟ ⎟. 1 0 0⎟ ⎟ 8 1 0⎠ 181

1, x, x2 , . . . , x6 6 F11 [x] C i! xi−1 F⋆11

" # $

ζ0 ζ1 ζ2 ζ3 ζ4 ζ5 ζ6 ζ7 ζ8 ζ9 1 x 2 G′ = x 3 x x4 x5 x6

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

ζ0 ζ0 ζ0 ζ0 ζ0 ζ0 ζ0

ζ0 ζ1 ζ2 ζ3 ζ4 ζ5 ζ6

ζ0 ζ2 ζ4 ζ6 ζ8 ζ0 ζ2

ζ0 ζ3 ζ6 ζ9 ζ2 ζ5 ζ8

ζ0 ζ4 ζ8 ζ2 ζ6 ζ0 ζ4

ζ0 ζ5 ζ 10 ζ5 ζ0 ζ5 ζ0

ζ0 ζ6 ζ2 ζ8 ζ4 ζ0 ζ6

ζ0 ζ7 ζ4 ζ1 ζ8 ζ5 ζ2

% C ′

ζ0 ζ8 ζ6 ζ4 ζ2 ζ0 ζ8

⎞ ζ0 ζ9 ⎟ ⎟ ζ8 ⎟ ⎟ . ζ7 ⎟ ⎟ ζ6 ⎟ ⎟ ζ5 ⎠ ζ4

&' (

% 43 ) # &

ϕ: *&

C C ′

C′

C ′ → F63 64 x = (x0 x1 · · · c59 ) → (x0 x1 · · · c59 0 0 0 0).

ϕ

C

+ ,

C ′

C = { p(x) ∈ C : p(α59 ) = p(α60 )p(α61 ) = p(α62 ) = 0},

C = {p(x) ∈ F64 [x] : deg p(x) ≤ 47, (x − α59 )(x − α60 )(x − α61 )(x − α62 ) | p(x)}. 47 − 4

*

-

C

[n, k, d]!

Fq

r

Fnq

# $ $ &

t = ⌊(d − 1)/2⌋

c∈C

. $ & r

d(r, c) ≤ t

c ∈ C

¿º

0 ≤ l ≤ n − 1 ∆l (r, C) c ∈ C d(c, r) ≤ l

i 0 n − 1 ∆i (r, C) ⊆ ∆i+1 (r, C);

r ∈ C

m < n ∆m (r, C) = ∅,

∆m+1 (r, C) = ∅.

! "" # !

m ∆m (r, C) = ∅ ∆m+1 (r, C) = ∅

$ % ∆m+1 (r, C) &

! ' ' '

" " c

r t ( ) " ! '

" # ! *

( " +

e

" e ≤ t = ⌊(d − 1)/2⌋ ( "" ' " " ! +

" '

' t

, $ " % +

r C " t r

-

"

C '

' "

.

&

!℄ #$

% &

& ' !(℄

q ! k" t n = q − 1

n x0 , x1 , x2 , . . . , xn−1 Fq " n # $ y0 , y1 , . . . , yn−1 ∈ Fq

P (x) ∈ Fq [x] % deg P (x) ≤ k − 1 ! P (xi ) = yi & n − t ' i

(

' !" '

) ' * '

+ " t = ⌊(n − k)/2⌋ *

' ,

y = (y0 y1 · · · yn−1 ) '

'

# $ ' E(x, y) ) E(xi , yi ) = 0

i ! E(x, y) - E(x, y)

p(x)

'

y" ) p(xi ) = yi E(xi , yi ) = 0 . ) Q(x, y) = E(x, y)(y − p(x))

(x, y) = (xi , yi ) / )

) ' ) ' % Q(x, y) ) Q(xi , yi ) = 0

(xi , yi ) Q(x, y) ≡ 0 ! (

Q(x, y) " '' pi (x) ) (y−pi (x)) ' Q(x, y) " ' " Q(x, pi (x)) ≡ 0 deg pi (x) < k - 0 pi (x)

. )" "

y − pi (x) Q(x, y) & ∆t (r, C) & + &

' Q(x, y) ) 12 )" p(x) ) '

*" ) ) y − p(x)

Q(x, y)

w1 , w2 " (w1 , w2 ) xi y j iw1 + jw2 + (w1 , w2 )

3 Q(x, y) (w1 , w2 )

) Q(x, y) (w1 , w2 )

Q(x, y)

deg(w1 ,w2 ) Q(x, y).

n k t

(xi , yi ) xi , yi ∈ F i = 0, . . . n − 1

p(x) deg p(x) < k p(xi ) = yi i t

r =1+

$

kn +

% k2 n2 + 4(t2 − kn) , 2(t2 − kn)

p

l = rt − 1.

Q(x, y) =

XX j1

qj1 j2 xj1 y j2

j2

deg(1,k) (Q) ≤ l qj1 j2 Q(x, y) !

i = 0, . . . , n − 1 Q(i) (x, y) = Q(x + xi , y + yi ),

" Q(i) (x, y) # r i = 1 . . . n j1 , j2 ≥ 0

j1 + j2 < r ! ! X X v w (i) qj1 j2 = qvw xv−j1 y w−j2 = 0. j j 1 2 v≥j w≥j 1

2

p(x) ∈ F[x] deg p(x) < k y − p(x) # Q(x, y) $ % p(xi ) = yi t i p(x) % ! % &'(℄

Q(x, y)

ËÙÔÔ

! r+1 l(l + 2) n . < 2k 2

Q(x, y)

qj1 j2 Q(x, y)

!

"

#

$ ` ´

%

& # n r+1 $ 2 # (1, k)

' l l ⌊X k ⌋ l−kj X2

j2 =0 j1 =0

1=

l ⌊X k⌋

(l + 1 − kj2 )

j2 =0

« — „— « „— k l l l +1 − +1 = (l + 1) k 2 k k « „— « „ l l l+1− ≥ k 2 ! r+1 l l+2 ≥ >n . k 2 2

# (

⊔ ⊓

$ ! (

)* + ,

Q(x, y) -

(xi , yi ) p(x) yi = p(xi ) (x − xi )r g(x) = Q(x, p(x))

- pe(x) = p(x + xi ) − yi . pe(0) = 0. * pe(x) = xu(x) ( u(x) ∈ F[x] / e g (x) = Q(i) (x, pe(x)) 0 e g (x − xi ) = g(x)

g(x) = Q(x, p(x)) = Q(i) (x − xi , p(x) − yi ) g (x − xi ). = Q(i) (x − xi , pe(x − xi )) = e

- Q(i) (x, y) r xu(x) y Q(x, y)

g (x) = Q(i) (x, xu(x)); e

xr

xr e g (x)

(x − xi )r e g (x − xi ) = g(x) #

⊔ ⊓

(xi , yi ) r Q(x, y) = 0

Ì p(x) k yi = p(xi ) t i rt > l y − p(x) Q(x, y) g(x) = Q(x, p(x)) ! " # $ (1, k) " Q(x, y) % & l

$ deg g(x) ≤ l.

(x−xi )r g(x) " i $ yi = p(xi ) S = { i : yi = p(xi )}.

$

h(x) =

Y

i∈S

(x − xi )r

g(x) $ |S| ≥ t deg h(x) ≥ rt

rt > l g(x) ≡ 0' " p(x) % Q(x, y) ( ) $ y − p(x) Q(x, y) % ⊔ ⊓

" $ r l ! ) " * #

+ , - $. t "

n k t t2 > kn l ` ´ r l(l+2) n r+1 ≤ 2 2k rt > l

$. l = rt − 1 , - % $ #

+ ! r+1 (rt − 1)(rt + 1) , n /0 < 2k 2

$ !

r 2 (t2 − kn) − knr − 1 > 0.

1 ! " $ # $ r & " ) " $ % p kn + k2 n2 + 4(t2 − kn) r ≥1+ /-0 . 2(t2 − kn) ( ) * ) " ⊔ ⊓ 2

+ 3 , - "

Ì

√

n t >

kn

n

!! "#

$ # %

# &2115 '()

X &8415

'()

#

X

S

#

S

*

* #

S 115.2 + #

10

2560

X

#

, -. ℄

0 " $

4

30 12

5.2

6 1.496 × 108

12, " 3

* &12)

5 6

* #

X

7

"

8 # & 9 :)

$

[24, 12, 8]

G24 & 9 0) 3

24 #

12 #

# $

;

% &

100%

1

#)

7

% &2 3) 6

G24 8 4

% < 1 6

RS (255, 223)

33

223 F28

16

255 1

16 32 14%

! " # $ % !

% & ! ' % % 5 × 10−3 ' 10−6

# ()(* + ,

$ 8.5 ( " % 120 15 * - . /" " ! / 0 - " ! !"

# " ! " "" + ,

1 2 3 % " " #

" - *

192×172 " B (

RS (182, 172) [182, 172, 11] # " B ′ 192 × 182 4

B ′ RS (208, 192) [208, 192, 17] % B ′′ 208×182 0 -

" 0 -

' % - 3 5 6 " # 37

5 6 -

B

B ′ - 11 × 17 = 187 3 [37856, 33024, 187] C 0 4832 6% % -

93 407

C = BRS (15, 11)

← −−−−−−−−−−−− − −−−−−−−−−−−→ B(0, 0) B(0, 1) ··· B(0, 171) B(1, 0) B(1, 1) ··· B(1, 171) ... ... B(191, 0) B(191, 1) · · · B(191, 171) B ′′ (192, 0) B ′′ (192, 1) · · · B ′′ (192, 171) B ′′ (193, 0) B ′′ (193, 1) · · · B ′′ (193, 171) ... ... B ′′ (207, 0) B ′′ (207, 1) · · · B ′′ (207, 171)

←−−−−−−−− −−−−−−−−→ B ′ (0, 172) · · · B ′ (0, 181) B ′ (1, 172) · · · B ′ (1, 181) ... ... B ′ (191, 172) · · · B ′ (191, 181) B ′′ (192, 172) · · · B ′′ (192, 181) B ′′ (193, 172) · · · B ′′ (193, 181) ... ... B ′′ (207, 172) · · · B ′′ (207, 181)

x ? ? ? ? ? ? y x ? ? ? ? ? ? y

ω F16 C = RS (15, 5) m = (1 ω ω 2 0 1). c = (ω 9 ω 12 ω 2 ω 5 ω 12 ω 2 ω 4 1ω 10 ω 13 ω 13 ω 10 ωω 6 0).

3/4 C = RS (10, 6) F11 r = (4 0 8 0 6 1 1 4 0 9 7).

½¼

! "# $

½¼º½

q

q

%

&

$

q ' ( ) * $

+

Fq , ?

!

+

$

( +

0 1

⎧ ⎪ ⎨0 φ(r) = ? ⎪ ⎩ 1

x ≤ 13 1 3 < x< x ≥ 23 .

2 3

φ : [0, 1] → {0, 1, ?}

¾¼¼

1 2

℄ !℄ "℄ #

$ % #

Fq

& F

q ' Fq # Fq ∪ {?} ( d(x, y) Fq Fq × Fq → Q

⎧ ⎨0 d(x, y) = 1 ⎩1

2

x=y x = y x = y

x =?

x, y =? y =?

( % )

n

Fq

*

d(x, y) =

n

d(xi , yi ).

i=1

&

?

+ ,

% &

Fq

Fq

* # *

c

r

d

e

-

& r = (r0 r1 · · · rn−1 ) * # &

r

0 ≤ i ≤ n − 1 ri =? $ .

Fq

x ∈ Fq

x+? = x.

½¼º¾

(#

* /0 1 / )

(#

&/ e 1 2 1

n

r ∈ Fq

c ∈ C %

d(c, r)

#2

q 3

½¼º¾

Ì

d

C e1

C

(n, M )

Fq

e0

e0 + 2e1 ≤ d − 1.

r e0

e1 e0 + 2e1 ≤ d − 1

c ∈ C r r c

d(c, r) =

1 1 e0 + e1 ≤ (d − 1). 2 2

!"#"$

c, c′ ∈ C r% d(c, c′ ) ≤ d(c, r) + d(r, c′ ) 1 1 ≤ (d − 1) + (d − 1) 2 2 = d − 1;

c = c′ & r ' ⊓ ⊔ ( '

) * "#+ (

C d

r , e0 = 0 * "#+ ' ( ( * - ) .

/ 0 ) [n, k]1 Fq

( % ( ' 2 C F16 α ∈ F16 1 + α + α4 = 0 β = α3 ) β ' ( 3 )

g(x) = (x − β)(x − β 2 )(x − β 3 )

BCH16 (5, 4) 2 4 ' 4 4 G 3 H C

¾¼¾

3 2 11 α α α 1 0 G= , 0 α3 α2 α11 1

⎞ 1 α11 α12 0 0 H = ⎝0 1 α11 α12 0 ⎠ . 0 0 1 α11 α12 ⎛

r = (? α6 ? ? 1),

?

! c ∈ C

r " #

c = (c0 c1 c2 c3 c4 ) = r + (e0 0 e2 e3 0). $% & cH T = 0 cH T = (e0 r1 e2 e3 r4 )H T

e0 e2 e3 ' ⎧ 2 ⎨ α = (α3 + α2 + α + 1)e2 + e0 α3 + α2 = (α3 + α2 + α + 1)e3 + (α3 + α2 + α)e2 ⎩ 3 α + α2 + α + 1 = (α3 + α2 + α)e3 + e2 = 0.

(

⎧ 2 ⎨ α = α12 e2 + e0 α6 = α12 e3 + α11 e2 ⎩ 12 α = α11 e3 + e2 .

) e0 = α3 e2 = α9 e3 = α12 * c = (α3 α6 α9 α12 1).

+

!) ,

" -. $ / 0% 1 2 3 4

! 1 ½¼º¿

* 56 57 / 0 $

-. % ' σe (x)

! ωe(x) 8 ! 9

./ : ;

"1 ;

½¼º¿

σe (x)º

σe

Ì

RS (n, k)

Fq

e0

e1

e0 + 2e1 ≤ r,

r = n − k

[n, k]

C

Fq

g(x) = (x − α)(x − α2 ) · · · (x − αr ),

α

!

Fq

r =n−k n

r = (r0 r1 · · · rn−1 ) ∈ Fq .

" "

c = (c0 c1 · · · cn−1 ) ∈ C ⊆ Fnq

d(r, c)

#

I0

I0 = { i : ri =?}

σ0 =

i∈I0

I0 = ∅"

" "

(1 − αi x);

σ0 = 1

Ë¾

? r 0

′ r′ = (r0′ r1′ · · · rn−1 ) ∈ Fnq ri′

=

? ri ri = 0 ri =?

r′ Fq

Fnq e1 + e2

r′ ! " # S(x) = S1 + S2 x + · · · + Sr xr−1

Sj =

n−1

ri′ αij ,

i=1

$

e′ = r′ − c

v = (e′0 , e′1 α, · · · , e′n−1 αn−1 ).

% & '

σ(x)S(x) = ω(x)

(mod xr ),

σ(x) ω(x)

( '

σ(x) =

(1 − αi x), i∈I

I & r I = I0 ∪I1

I1 = {i : ri =?

ri = ci }. )

σ(x) = σ0 (x)σ1 (x),

σ1 (x) σ1 (x) =

i∈I1

(1 − αi x),

½¼º¿

S0 (x) S0 (x) = σ0 (x)S(x)

(mod xr );

σ1 (x)S0 (x) = ω(x)

(mod xr ).

S0 (x) σ1 (x) ω(x)

deg σ1 (x) = e1 deg ω(x) ≤ e0 + e1 − 1 deg σ1 + deg ω ≤ e0 + 2e1 − 1 < r = deg xr .

!! gcd(σ(x), ω(x)) = 1 gcd(σ1 (x), ω(x)) = 1 σ1 (x) ω(x) "

# r − e0 deg σ1 (x) = e1 ≤ , 2

"#$% "

#

$

%

r − e0 r + e0 −1≤ − 1, "#&% 2 2 ⌊(r − e0 )/2⌋ + ⌈(r + e0 )/2⌉ = r. "#'% deg ω(x) ≤ e0 + e1 − 1 ≤ e0 + (

µ=

"

# r − e0 , 2

ν=

$

% r + e0 −1 2

(xr , S0 (x), µ, ν) v(x), r(x) ) µ ν σ1 (x) = λv(x),

ω(x) = λr(x).

( σ1 (0) = 1 ! λ

σ1 (x) = v(x)/v(0),

ω(x) = r(x)/v(0).

"#*%

ω(x) σ(x) + ,

ei e σ(α−i ) = 0 0 −i ei = ) αi σω(α σ(α−i ) = 0. ′ (α−i )

C = RS (10, 6) C

5 F11

1 2 C g(x) = 1 + α3 x + α4 x2 + α8 x3 + x4 ,

α F11

r = (α5 α6 ? α4 α3 α3 α6 ? α4 α).

Ë½ !

I0 = {2, 7};

σ0 (x) = (1 − α2 x)(1 − α7 x) = α9 x2 + 1.

r′ = (α5 α6 0 α4 α3 α3 α6 0 α4 α).

S(x) = 1 − x + x2 + α2 x3 + α7 x4 + α3 x5 + α7 x6 + α6 x7 + α3 x8 + x9 . "

S0 (x) = 1 − x + α7 x2 + α6 x3 + αx4 . µ = 1 ν = 2 # $

σ1 (x) = x + 1,

ω(x) = α9 x3 + α4 x2 − 1.

σ1 (α−5 ) = 0%

r " σ(x) = σ0 (x)σ1 (x) = α9 x3 + α9 x2 + x + 1

σ ′ (x) = α7 x2 + x + 1.

I = {2, 5, 7},

−i

) i αi σω(α ′ (α−i ) 2 α5 5 α4 7 α6

e = (0 0 α5 0 0 α4 0 α6 0 0).

c = r′ − e = (α5 α6 1 α4 α3 α8 α6 α α4 α).

½¼º½º [15, 5, 7]

r = (0?1 111 0?1 110 00?)

½¼º¾º α ∈ F16

! "

# α4 +α+1$

RS (15, 7)

r = (α13 1 ? α10 α12 α6 ? α5 α13 ? α α8 α7 α2 α9 ).

½¼º¿º C

[7, 3, 4]

g(x) = x4 + x3 + x2 + 1. %

& 3$

4 C $

½½

℄

½½º½

S = (P, B, I)

! #

P B I ⊆ P × B

$

P

"

&

p

%

P

B

B

(p, B) ∈ I

B

B ⊆ 2P

p (p, B)

%

I

&&

'

I = {(x, Y ) : Y ∈ B, x ∈ Y }. (

(P, B, I)

S = (P, B, I) S ′ = (P ′ , B′ , I ′ ) φ P ∪ B P ′ ∪ B′ φ(P) = P ′ φ(B) = B′ (p, B) ∈ I (φ(p), φ(B)) ∈ B′

¾½¼

φ

S S ′ S = S ′ φ

φ(P) = B′ φ(B) = P ′ (p, B) ∈ I (φ(B), φ(p)) ∈ I ′ φ S S ′ S = S ′ φ

S

S !

Aut (S) " #

$ %

% &

'

Ì

S = (P, B, I) B, B ′ ∈ B B = B ′ p ∈ P (p, B ′ ) ∈ B.

(p, B) ∈ B,

˜ ∈) S˜ = (P, B,

S

( B ∈ B ˜ = {p ∈ P : (p, B) ∈ I}, B

˜ : B ∈ B}. B˜ = {B ˜ ∈) ' #

˜ = B˜′ ) (P, B, (

B = B ′ B

φ : P ∪B → P ∪B˜ ( x x ∈ P φ(x) = x ˜ x ∈ B.

( % ' φ & ˜ φ(p) = p ∈ φ(B) * φ (p, B) ∈ I B

⊔ ⊓ + % '

p &

B ' (p, B) ∈ I &

˜ ' + % ' B B

, -

'

S = (P, B, I)

)

S T = (B, P, I T ) ' I T = { (x, y) : (y, x) ∈ I}

S

½½º¾

½½º¾

v t !

k " t

λ

v = k

v < k # $

D = (P, B, I) t − (v, k, λ) t t, v, k, λ

% |P| = v D !

B ∈ B

k " t

λ

& v > 0 k > 0 t = 0

' t = 0 λ = 0 ( 4

λ ≤ k ≤ v k = v

)

φ S φ · φ

p

B p ∈ φ(B) φ(B) ∈ B φ

φ

( $ t− (v, k, 1) t ≥ 2 S(t, v, k) 1*

2*

+ , t − (v, 2, λ)

+

,

¾½¾

K5 K3′

K3 c

3

b

2

a

1

K3,3 t ≤ 2

t ≥ 1

λ = r 2

5

2 − (5, 2, 1)

P = {1, 2, 3, 4, 5}

B

2 P

K3 = {a, b, c} K3′ = {1, 2, 3} !

" K3,3 = (P, B)

P = K1 ∪ K2 ,

B = {{x, y} : x ∈ K3 , y ∈ K3′ }.

# 1 − (6, 2, 1) $ (P, B)

% P = K1 ∪ K2 K1 ∩ K2 = ∅

½½º¾

5

4

7 6

1

3 2

B ∈ B

B ∩ K1 = ∅,

B ∩ K2 = ∅

(P, B)

P = {1, 2, 3, 4, 5, 6, 7}, B = {{1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 4, 7}, {2, 5, 6}, {3, 5, 7}, {3, 4, 6}}.

|P| = 22 + 2 + 1

B∈B

|B| = 2 + 1

2 − (7, 3, 1)

!

0≤s≤t

D = (P, B)

λs

λs = λ

D

t − (v, k, λ)

s

(v − s)(s − s − 1) · · · (v − t + 1) . (k − s)(k − s − 1) · · · (k − t + 1)

s − (v, k, λs )

1 ≤ s ≤ t

s

S s ≤ t m

S T = {(V, B) : S ⊂ V ⊆ B, |V | = t, B ∈ B}.

|T | v−s |T | = λ t−s

!

! k−s =m ; t−s

m S

⊔ ⊓ λs λs =

v−s λs+1 . k−s

! t − (v, k, λ) t ≥ 2

n = λ1 − λ2

" λ1

r

b

t = 2 D (v, k, λ) λ(v − 1) = r(k − 1) vr = bk # $ $ %

λs s = 1

# $

& !% A′ = { (x, B) ∈ P × B : x ∈ B}. r

|A′ | = vr;

!

b k |A′ | = bk,

⊔ ⊓

2' v = b(

r = k

½½º¿

K3,3 ½½º¿

S = (P, B, I) |P| = v |B| = b

{p1 , p2 , . . . , pv }

S {B1 , B2 , . . . , Bb } b × v A = (aij ) 0 1 1 (pj , Bi ) ∈ I aij = 0 (pj , Bi ) ∈ I.

S

S

A S T

AT

! ""# ⎛

1 ⎜0 ⎜ ⎜0 G=⎜ ⎜1 ⎜ ⎝0 0

110 001 000 001 100 010

00 11 00 00 10 01

00 00 11 10 01 00

⎞T 0 0⎟ ⎟ 1⎟ ⎟ . 0⎟ ⎟ 0⎠ 1

$ ""# t − (v, k, λ) k r

$ ! ""%

P

B

7

g

6

f

5

e

4

d

3

c

2

b

1

a

⎛

11 ⎜1 0 ⎜ ⎜1 0 ⎜ A=⎜ ⎜0 1 ⎜0 1 ⎜ ⎝0 0 00

100 011 000 010 001 101 110

⎞ 00 0 0⎟ ⎟ 1 1⎟ ⎟ 0 1⎟ ⎟. 1 0⎟ ⎟ 0 1⎠ 10

0

S = (P, B, I)

Γ (S)

S

! " # Γ (S) P ∪ B $ % x, y " # (x, y) ∈ I Kv,b

!! &

½½º¿

P

B ½ j

Bj χBj (Pi ) i 1 v

r n

0

Ì 2−(v, k, λ) D

A Q r = λ1

D n = r − λ Iv Jv

v × v v × v

1 AT A = (r − λ)Iv + λJv ,

det(AT A) = rknv−1 .

D A Q v (i, j) AT A ! i A j i = j r " i = j λ

⎛

⎞ r λ ... λ ⎜λ r ... λ⎟ ⎜ ⎟ AT A = ⎜ ⎟ = (r − λ)Iv + λJv . ⎝ ⎠ λ λ ... r # ! $ %

AT A

j = (1, 1, . . . , 1) AT A r + (v − 1)λ = rk v − 1 (1, −1, 0, . . . , 0) (0, 1, −1, 0, . . . , 0)

r−λ = n &

D rank Q (A) ≥ rank Q (AT A)

" A v

'

(

1

℄

det(AT A) = 0º È det(AT A) = rknv−1 r, k = 0 n = 0 r = (v − 1)λ/(k − 1) = λ v = k

D ⊓ ⊔

AT

AAT = (k − µ)Ib + µJb

µ

A

t

det(AAT ) = rk(k − µ),

2 − (v, k, λ) D

k > λ

r

λ 2! r = k " k ≥ λ # A D ! det(A2 ) = det(AT A) = rk(r − λ)v−1 = 0,

k = λ

⊔ ⊓

# t! t ≥ 2 2! "

t D

r ≥ 2

b ≥ v $ t! D 2!

% A & rank Q (A) = v ' v v " b ≥ v ⊔ ⊓

v

n = k−λ

ËÙÔÔ 2 − (v, k, λ)

v n = k − λ

2 − (v, k, λ)

A

det(A)2 = det(AT ) det(A) = rknv−1 = k 2 (k − λ)v−1 .

v−1

k−λ

⊓ ⊔

π = (P, B, I) n ≥ 2 a, b ∈ P a = b

B ∈ B

a b

a b

B, C ∈ B B = C

p ∈ P

B C

B ∈ B p ∈ P p

B

n + 1 ! 2

"# $ % ! # n ≥ 9 & n' (%℄

!

! " # $ $ ! % ! # &

! *

n 2 − (n2 + n + 1, n + 1, 1)

+ ,

- $ n

B ∈ B p ∈ P \ B

p B ' #

, ' n + 1

p . |P| = (n + 1)n + 1 = n2 + n + 1 /

# D = (P, B) $

B, C ∈ B B ∩C = ∅ " b ∈ B # n + 1

B C '

n2 + n + 1 0$ # B n #

# v = n2 + 2n + 1# $ . B ∩ C = ∅ D ⊓ ⊔ !

#

¾¾¼

n

! "#

Ì

n n ≡ 1, 2 (mod 4) n

n n

q

V

3 Fq

PG (2, q) = (P, B, I) P 1 V B 2 V P ∈ P

B ∈ B P ⊆ B

PG (2, q)

!"

1 V (q 3 − 1)/(q − 1) = q 2 + q + 1 2 V (q 2 − 1)/(q − 1) = (q + 1) 1 1 P, Q # P ⊕ Q 2 !

PG (2, q)

q !

$ "

%

PG (2, 2)

&

n = pt p

! ' "

S = (P, B, I)

Fq (

FP q " P → Fq &

' "

P !

S

! Fq S CFq (S) FP q "

S

CFq (S) = χB : B ∈ B.

CF (S) S

FPq P

B q

χB =

χ{x} .

(x,B)∈I

B vB χB FPq

! "

B CF (S) q

Z #

24 $ F 2 3 CF (Π) = FP C2 (Π) % H3 (2) $ Π & Π H3 (2) 3 H3 (2) 7 ' ((( ) {a, . . . , g} Π Π 7

Π

a b c d e f g

1 [1 [1 [1 [0 [0 [0 [0

2 1 0 0 1 1 0 0

3 1 0 0 0 0 1 1

4 0 1 0 1 0 1 0

5 0 1 0 0 1 0 1

6 0 0 1 1 0 0 1

7 0] 0] 1] 0] 1] 1] 0]

Ì 3 H3 (2) 4

' ((* + 2

H3 (2)

2 − (7, 4, 2)

¾¾¾

a b c d e f g

1 [1 [1 [1 [1 [0 [0 [0

2 1 1 0 0 1 1 0

3 0 0 1 1 1 1 0

4 1 0 1 0 1 0 1

5 0 1 0 1 1 0 1

6 0 1 1 0 0 1 1

7 1] 0] 0] 1] 0] 1] 1]

Ì 4 H3(2)

C3 (Π) 7 1 −1 0 [7, 6, 2] S = (P, B, I) p p S CF (S)

F3

p

rank Fp (S) = rank p (S) = dim(CFp (S)). 2 Π 4 3

6 5 Π 7 F

n ≥ 1 j = (1 1 1 · · · 1) ∈ Fn (Fjn )

D = (P, B, I) 2 − (v, k, λ)

n p n F p rank p (D) ≥ (v − 1),

p k CF (D) = (Fj)⊥ CF (D) = FP

p n C = CF (D) w=

v B = rj.

B∈B

x ∈ P wx =

(x,B)∈I

v

B

r wx (t) = λ

t = x

t = x.

w − wx = n(j− vx ) ∈ C

j− vx ∈ C x ∈ P vx − vy ∈ C ! (F j)⊥ ⊆ C rank p (D) ≥ v − 1 "#

C = (Fj)⊥ #$

vB ∈ (Fj)⊥ B ∈ B p k vB , j = |B| = k ⊓⊔

S = (P, B, I) F

S F

HF (S) = CF (S) ∩ CF (S)⊥ .

BF (S) = HF (S)⊥ = CF (S) + CF (S)⊥ .

F

F

Π = (P, B, I)

Π Σ = σ ¾ P

Σ Π

Σ = σ

Π = (P, B, I)

Σ

n

B

|P| = |B| Σ B n2 + n + 1 Σ !

B ∈ B θ ∈ Σ B ∈ B θ(B) = B "

θ n2 + n + 1# θ B $ # n + 1 % θ gcd(n2 + n + 1, n + 1) = 1 θ ⊓ ⊔ ! % &'() *'+℄ # PG (2, q) %

2

1 ≤ i ≤ n2 + n + 1

P = σ i (Q)

P, Q ∈ Π

P = {1, 2, 3, 4, 5, 6, 7} B0 = {1, 2, 4} σ = (1 2 3 4 5 6 7).

B B0 Σ = σ B P

1 2 3 4 5 6 7

2 3 4 5 6 7 1

4 5 6 7. 1 2 3

Ì

B0 Σ

Π = (P, B) ⎞ ⎛ 1 ⎜0 ⎜ ⎜0 ⎜ G=⎜ ⎜0 ⎜1 ⎜ ⎝0 1

10 11 01 00 00 10 01

100 010 101 110 011 001 000

0 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 1⎠ 1

F CF (Π) g(x) = 1 + x + x3 .

! F p" G F = F2 gcd(1 + x + x3 , x7 − 1) = 1 + x + x3 ,

CF2 (Π) 7 − 3 = 4 F = F3 gcd(1 + x + x3 , x7 − 1) = x − 1,

CF3 (Π) 6 #

$ % & ' Π $ ( ψ:P→ P

⎧ 1 → ⎪ ⎪ ⎪ ⎪ 2 → ⎪ ⎪ ⎪ ⎪ ⎨ 3 → ψ : 4 → ⎪ ⎪ 5 → ⎪ ⎪ ⎪ ⎪ 6 → ⎪ ⎪ ⎩ 7 →

1 2 4 3 . 7 6 5

ψ

Π

PG (2, q)

!!

" (v, k, λ)# m

D = {d1 , d2 , . . . , dk } ⊆ N

{1, 2, . . . , v},

$

% di − dj

(mod m)

λ

& D = {1, 2, 4, 7} %

7 (7, 4, 2)

" (v, k, 1) %

v

(v, k)

' D %

i1 , i2 , i3 , i4 ∈ I

i1 = i2 i1 − i2

(mod v) = i3 − i4

(mod v)

ii = i3 i2 = i4 & D = {1, 2, 4} % (7, 3)

& % D (v, k, λ)

(

2 − (v, k, λ)

D = (P, B) P = {1, 2, . . . , v}

¾º B = { Bi : i = 1, 2, . . . v} Bi = { d + i (mod v) : d ∈ D}

v

Bi

k Bi 2

λ

a, b ∈ P a = b

Ωa,b = { Bi : a, b ∈ Bi }. a, b ∈ Bi a = a + i b = b + i a, b D a − b = a − b a = b + (a − b) l = (a − b)

λ D!

λ (a, b, i) a = a + i, b = b + i. |Ωa,b | = λ (n2 + n + 1, n + 1)

"

n # $ (n2 + n + 1, n + 1) D! %

D xi . j(x) = i∈D

2

&

g(x) ' j(x) xn +n+1 − 1 ( )

j *

D xj g(x)

2

(mod xn

+n+1

− 1).

½½º½º

(13, 4)*

½½º¾º PG (3, 4)

3 F4 +

' ,$-

.

½¾

!

"

! #

! $%&

½¾º½

!

&

'

( [24, 12, 8]%

) !

G24 12

#

8

24

*

[24, 12, 8]%

[n, k, d] C n = 24 k ≥ 12 d ≥ 8 [24, 12, 8]

½º C 12 V = F24 2 C i Si

v ∈ V w(v) ≤ 4 w(v) = 4 vi = 1 24 t < 4

Si t t !

Si

4 " #$

i% & 1

23 3 " 24 24 24 24 23 |Si | = + + + + = 0 1 2 3 3 1 + 24 + 276 + 2024 + 1171 = 4096 = 212 .

$ a, b ∈ Si

8 a + b 8

" "" 4

"" ai = bi = 1

1 + 1 = 0 F2 '

f = a + b ( 7 $ f )

C $ v ∈ Si v + C * 212 ≤ |V /C| = |V |/|C| ≤ 224 /212 ,

|C| ≤ 212 + $

C

$ dim C = 12 s212 + C s2 + C s1 + C C

C

b ∈ V 4 bi = 0 , b ∈ Si

$ a ∈ Si b ∈ a + C $ a + b ∈ C

a + b ( 8

w(a + b) ≥ 8. C

8

½¾º½

⊓ ⊔

8 C

c, d f Supp (f ) = Supp (c)∩Supp (d) f = c ∩ d Ì C [24, 12, 8] C 8 5 − (24, 8, 1)

v ∈ V (P, B) P = {1, 2, . . . , 24} ! " v P # i ∈ a a = 1 i ∈ Supp (a) # v = 24 k = 12 5

$ 1 ≤ i, j, k, l, m ≤ 24 S # b 4 {j, k, l, m} a ∈ S b ∈ a + C % i ∈ (a + b)& a + b ∈ C # c = a + b 8& j, k, l, m ∈ a + b # 5 ! 8

c ∈ C

# c, c ∈ C

w(c) = w(c ) = 8 5 ≥ w(c ∩ c ) ≤ 7 w(c + c ) ≤ 2(8 − 5) = 6 ' C 8 (

i

i

i

′

′

z z

≤3

}|

≤3

}|

′

′

≥5

{z

}|

{

≥5

}|

z

{

{z z

c + c′

=8 ≤3

}|

≤3

}|

{ {

=8 ≤6

8 C 5 − (24, 8, 1) $

24 8 5

/

5

= 759.

⊓ ⊔

*

)

¾¿¼

4 − (11, 5, 1), 5 − (12, 6, 1),

4 − (23, 7, 1), 5 − (24, 8, 1).

3 − (22, 6, 1),

M11 M23 M22 M12 M24 [24, 12, 8] M24 3 PG (2, 4) M24 M24 5 1

! " #

$ % PG (2, 4) & '()℄

8

Ì

C

[24, 12, 8]

[24, 12, 8]

8

C

C ′ C C C ′ [24, k, 8] |C ′ | = |C| Si C F24 2

C ′

!

" a ∈ V s ∈ Si # i $ a ∈ s + C ′ % 8 C ′ " C & '(( 5−(24, 8, 1) ) a ∈ F24 2 w(a) ≥ 5

c ∈ C ′ w(c) = 8 c " a 5

8

w−5

z z

}| z

w−8+t

}|

{

3−t

}|

{z {z

5

}| 5

}|

{

{z

z

t≤3

}| t

}|

! a + c

{ {

=w≥5 =8 = w − 8 + 2t ≤ w − 2

a′ = a + c w(a′ ) ≤ w(a) − 2 * " a ∈ V

½¾º¾ (24, 212 , 8)

a) ≤ 4

w( a a

C ′ i Si ∈ Si a a) ≤ 3 a w( C V w( a) = 4 ai = 1 a ∈ Si C V a ∈ Si ! w( a) = 4 ai = 0 ! + x ∈ C w( x ∈ Si a a + x) = 8 " # a

C V

! # # # V # C ′ Si

|V /C ′ | ≤ |Si | = |V /C|. |C| = |C ′ | #

⊓ ⊔

$ % && & '(

[24, ≥ 12, ≥ 8]2

[24, 12, 8]2

C

2 − (24, 8, 1) C 2 − (24, 8, 1) M24

[24, 12, 8]2

M24 [24, 12, 8]2

Ì

G24

½¾º¾

(24, 2

12

, 8)

[24, 12, 8]

! " # $ %

G24

& $

#

(24, 212 , 8) $' G24 #

' (

2 − (11, 6, 3)

$ "

2 − (11, 6, 3)

¾¿¾

0

1 B1 B B1 B B1 B B1 B N =B B1 B0 B B0 B B0 B @0 0

1 1 1 0 0 0 1 1 1 0 0

1 1 0 1 0 0 1 0 0 1 1

1 0 1 0 1 0 0 1 0 1 1

1 0 0 1 0 1 0 1 1 1 0

1 0 0 0 1 1 1 0 1 0 1

0 1 1 0 0 1 0 0 1 1 1

0 1 0 1 1 0 0 1 1 0 1

0 1 0 0 1 1 1 1 0 1 0

0 0 1 1 1 0 1 0 1 1 0

1 0 0C C 1C C 1C C 0C C 1C C 1C C 1C C 0C C 0A 1

2 − (11, 6, 3)

D = (P, D) 2 − (11, 6, 3) N

10λ = 5r D r = 6

! b = 11 D "

N T N = 3I + 3J. # $ % &' # $ F2 N T N = I + J (mod 2) ( 2) N " 10 &

ker N " j * +

k = 6 " v = 11 +

(p, B) p ∈ P B ∈ B P ∈ B ' x ∈ B

3 +

x p ( '

3 × 6 = 18 +

,&

+

p " 6 = r +

"

3 +

p B 3

& + - p B

+

D µ = 3

N N k = 6 # 2$ i j "

bij & (Bi ∪ Bj ) \ (Bi ∩ Bj ).

- |(Bi ∪ Bj ) \ (B1 ∩ Bj )| =

(|Bi | + |Bj | − |Bi ∩ Bj |) − |Bi ∩ Bj |) =

2k − 2µ = 12 − 6 = 6,

bij 6 . '

/

,

'

N

0 1 2 "

# !$

' ⊔ ⊓

½¾º¾ (24, 212 , 8)

[24, 12, 8]

G24 N

G 8

12 × 24

G = (I12 P ) I12 12 × 12

0 1 0 1 ··· 1 B1 C B C P =B C. @ N A 1

G ! "

24 G

4 ! G 0 #

G

0 4 $

! N

! G 8 %

G

12 ! [24, 12, 8]

&

" ' "

' ( )℄ + G24 ,

+ 0

Ì C

24 |C| = 212

8 0 ∈ C C G24

- C

%

C

C ′ 23

7

|C ′ | = 212

C ′ !

"

.

/ ! , 0 , !

1 C 0, 8, 12, 16 24

2 3

4 "

½ C 0, 8, 12, 16, 24

41

5 C

C = C ⊥ 6 !

7 12 !

7 212 5

C ! C

8 G C %

! 12 9 !

„ « 1 ... 1 0 ... 0 G= . A B

1

!

4

B 0 B 11 B [12, 11, 2] B I11 1 !"

G # G′

# (I12 |P ′ ) $ P ′ = P P % & ' 1 P ′

B (

N ′ P ′

N ′ 6 6 ) N

* 2 − (11, 6, 3) ( ⊔ ⊓

½¾º¿

G24

G24

G23

7 ! " [23, 12, 7] [24, 12, 8] 2 c = (c0 . . . c22 ) c23 = i=0 2ci #

G24 [23, 12, 7] $ " 7 [23, 12] C % B3 (x) 3 C

[23, 12]

G24

|B3 (x)| =

3 23 i=0

i

=

23 23 23 23 + + + = 211 , 0 1 2 3

&

2

23

=2

12

& 3 ' n i=0

i

' (

[23, 12, 7]

)

G23

G23 * "

%

½¾º¿ i 0 7 8 11

Ai 1 253 506 1288

Ai 1288 506 253 1

i 12 15 16 23

Ì G23

[23, 12] 211 − 1 = 2047 = 23 · 89 F211 23 β

β (x − β i ), g(x) =

F211

i∈R

R = {1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12}.

g(x) = x11 + x9 + x7 + x6 + x5 + x + 1.

11 F2 [x] g(x) x23 − 1 [23, 12]

C

! C " #$ G23

%&

7

C

5

C 5

c(x)

'

4

g(x)

(

c ∈ C

c(β) = c(β 2 ) = c(β 3 ) = c(β 4 ) = 0. ) *+, - ./0 +

C

Ai i C Ai = A23−i .

β 22 = β −1 g(x) = (x − β j ),

e j∈R

⊓ ⊔

= {22, 21, 19, 15, 7, 14, 5, 10, 20, 17, 11}. R

g(x) g (x) = (x23 − 1)/(x − 1) = 1 + x + x2 + · · · + x22 g(x) g (x) ∈ C

j = (1 1 . . . 1) 1

23 − i ⊓ ⊔ j i

c ∈ C w(c) = 0

(mod 4)

c(x)

c

c(x) = xe1 + xe2 + · · · + xew ,

0 ≤ e1 < e2 < · · · < ew ≤ 22

c(x) c(x) = x−e1 + x−e2 + · · · + x−ew . !!"

# 23 $ % c ∈ C &&

c(x) = 0

(mod g(x));

c(β) = 0 ! $ % c c(1) = 0

(mod x − 1).

c(x) = 0 ' $ c(x) &&

c(β −1 ) = c(β) = 0,

c(x) = 0

(mod g(x)).

( &

c(x) c(x) = 0

(mod x23 − 1).

) w w = 0 2*

c(x) c(x) =

w

xei −ej

i,j=1

=w+

w

xei −ej

i,j=1 i=j

=

n

i,j=1 i=j

(mod x23 − 1)

xei −ej

(mod x23 − 1)

(mod x23 − 1).

c(x) c(x) =

22

µb xb ,

b=1

µb (i, j) ei − ej = b (mod 23) 0 2 µb b ei − ej = b (mod 23) ej − eb = 23 − b (mod 23) µb = µ23−b .

22

µb = w(w − 1).

22

µb = 2

b=1

w(w − 1) =

b=1

11

µb = 0

w(w − 1)

(mod 4).

b=1

w−1 w 4 ! ⊓ ⊔

Ì

i

c∈C

W = {0, 7, 8, 11, 12, 15, 16, 23}.

" # C 2, 3, 4 $ % & C 22, 21, 20 ' ( & C 6, 10, 14, 18 ) $ ! 17, 13, 9, 5 * ⊓ ⊔ ' + ! ! C 7 [23, 12, 7], C -. / G23

[24, 12, 8]

[6, 3, 4] ω

F4

x2 + x + 1.

ω = ω + 1! " F4 = {0, 1, ω, ω}

H6 [6, 3] F4

H6 = { (a, b, c, a + b + c, ωa + ωb + c, ωa + ωb + c) : a, b, c ∈ F4 }.

0 H6 4 6 H6 4 ! "

H6 A(Z) = 1 + 45Z 4 + 18Z 6 . # $% & H6 0 1 1001ωω @0 1 0 1 ω ω A . 0011 1 1

' ( F4 0 1 ω ω

1 2 3 4

5 6 7 8

9 10 11 12

c 13 14 15 16

17 18 19 20

21 22 23 24

) H6 G24

c 24 c i (v0 , v1 , v2 , v3 ) wi = 0v1 + 1v2 + ωv3 + ωv4 c ∈ G24 w = (w1 w2 · · · w6 ) F4

c = (1100 1010 0110 0000 0110 0000).

! * *

F4 c 0 1 1 1 1 1 1 . ω 1 1 1 ω w 1ω ω0ω0

' w = (1ω ω0 ω0) ∈ H6 c ∈ G24

C 1 ≤ v ≤ r y(v) ∈ C ⊥

x, y(v) = 0

i C

yi(v) = 1 j = i yjv = 0 v

x t

t ≤ r/2

x, yv = 0

≤ t v

≥ r − (t − 1) v

xi

xi .

r−(t−1) > t x, y(v)

xi G24

y(v) ∈ G24

1 ≤ i ≤ 253 yji 253 8 G24 1 j ! x "

# x, yji $ %

1

t & ' (

4

j )

4

$

* t 0 1 2 3 4

xj xj

0 0 77 253 112 176 125 141 128 128

Ì G24

+ %

% G11 [11, 6, 5]3 , G23 +

- G12 [12, 6, 6]3 .

G11 %

M11 7920 M11 AG (2, 3) 4 − (11, 5, 1) [11, 6, 5]3 M12 95040 M12 M11 5 − (12, 6, 1) [12, 6, 6]3 M22 443520 M22 PG (2, 4) 3 − (22, 6, 1) ‡[22, 12, 6]2 M23 10200960 M23 M22 4 − (23, 7, 1) [23, 12, 7]2 M24 244823040 M24 M23 5 − (24, 8, 1) [24, 12, 8]2 ‡ G23 [22, 12, 6]2 M22 2

Ì

Fq

Fq χ : F⋆q → {1, −1} χ(x) =

−1 1

x F⋆q x F⋆q .

χ F⋆q C⋆ χ(x) = −1 x

χ

Fq χ(0) = 0

q

ai aj Fq

S q

Sij = χ(ai − aj ),

S I J

1

SJ = JS = 0 SS T = qI − J S T = (−1)(q−1)/2 S

S5 5 ⎛

0 ⎜+ ⎜ S5 = ⎜ ⎜− ⎝− +

+− 0 + + 0 −+ −−

⎞ −+ − −⎟ ⎟ + −⎟ ⎟, 0 +⎠ + 0

+ +1 − −1

Á Ú

χ

11

G11

⎞ 1 ... 1 G = ⎝ I6 S5 ⎠ . ⎛

F3

G11

Z3 F3 ≃ Z3

F3

g(x) = x5 + x4 − x3 + x2 − 1.

Ì

G11

G11

G11 5 G12 = G11 G11 ! G12 " # 3 G G12

G

(0 − 1 − 1 − 1 − 1 − 1)T $ G 6%

# 2 + 2% " # 6 S5

# G

2 6 % " # 3 3 + 1 # 3 6

G12 6% G11

5 " (11, 36 , 5)

& (11, 36 , 5) '

$ ' ( 2 ! 2 X 11 i |B2 (x)| = 2 = 35 = 311−6 i i=0 % " )

' ⊔ ⊓

(11, 36 , 5) G11

Ì

q

G11

q G23

i 0 5 6 8 9 12

G11

G23

Ai 1 132 132 330 110 24

Ì G11

G24 (1010 1101 0100 1101 1011 1101)

G24

½¿

RS (n, k) k−1 F⋆n+1

!

"

# ! ½¿º½

q

$

p(x1 , x2 , . . . , xm ) =

i1 ,i2 ,...,im

ai1 ,i2 ,...,im xi1i xi22 · · · ximm

Fq [x1 , x2 , . . . , xm ] x1 , x2 , . . . , xm p r i1 , i2 , . . . , im % i1 + i2 + · · · + im = d & ai1 ,i2 ,...,im = 0 p(x) = x21 x2 + x1 x2 x3 − x32 x3 + x23

$ F3 [x1 , x2 , x3 ] 4

x = (x1 , x2 , . . . , xm ) Fq [x] = Fq [x1 , x2 , . . . , xm ].

x = (x1 , x2 , . . . , xm )

Fq [x]r

Fq x1 , x2 , . . . , xm r Fq [x]r Fq r+m , r

B

r

B xti11 xti22 · · · xtikk ,

i1 < i2 < · · · < ik t1 + t2 + · · · + tk ≤ r

p(x1 , x2 , . . . , xm ) ∈ Fq [x]r p(x)

v ∈ Frq

Fq p(v) =

i1 ,i2 ,...im

im ai1 ,i2 ,...,im v1i1 v2i2 · · · vm .

Fm q Fm q = {v1 , v2 , . . . , vqm }.

! "

Θ : Fq [x]r → Fq

p(x) Θp = (p(v1 ), p(v2 ), . . . , p(vqm )) .

#

! "$%

RM

q (r, m)

r

RM q (r, m) = { Θp : p ∈ Fq [x], deg p ≤ r}.

½¿º¾

RM q (r, m) r m

r, s

r ≤ s

RM q (r, m) ⊆ RM q (s, m). m

q r+m Fq [x]r r

RM q (r, m)

f, g ∈ Fq [x] f = g deg f, deg g ≤ r f (x) = g(x)

rq m−1

x ∈ Fm q

m = 1

m

m≥2

! "#

m − 1 m−2 $ %

xm rq m Fq f g

xm &

q ⊓ ⊔

RM q (r, m) Fq [x]r q > r r > q r v ∈ Fm q

'()* q (

RM q (r, m)

q m − rq m−1

rq m−1 Fm q

Fq [x]r RM q (r, m)

+ q m − rq m−1 ⊓ ⊔

$ %

q > r

r ≥ q

!,, -

½¿º¾

!,,

'()* #

r = 1

.

/

0

f : Fm 2 → F2 f

m " v ∈ V f (v) = 1

RM 2 (r, m)

f

Ωm

m

2m

w w ∈ Fm 2 v w

Fm 2 = {v1 , v2 , . . . , v2m }.

f : Fm 2 → F2

Θf = (f (v1 ), f (v2 ), . . . , f (v2m )).

V

(w1 w2 . . . wm ) ∈ V

K = {1, 2, . . . , m}

Fq

m

w =

Iw = {i ∈ K : wi = 1}

w

xw =

m Y

(xk + 1 + wk ) =

Y

xj .

Iw ⊆J ⊆K j∈J

k=1

X

m Y

(xk + 1 + wk )

k=1

0 k xk = wk ! " ⊔ ⊓ # " $% & m &

F2 [x]

r

r

m

!

r

m

RM 2 (r, m). "#$%

r m

r = 0, 1, . . . , m "#$% RM 2 (r, m) r

½¿º¿

RM 2 (r, m) V = Fm 2 RM 2 (r, m) r F2 [x] RM 2 (r, m) =

(

i∈I

)

xi : I ⊆ {1, 2, . . . , m}, 0 ≤ |I| ≤ r .

! RM 2 (r, m) 2m

m

r k=

"

m = 4 r = 2

m . i i=0

#

RM 2 (2, 4)

B = {1, x1 , x2 , x1 x2 , x3 , x1 x3 , x2 x3 , x4 , x1 x4 , x2 x4 , x3 x4 }.

" [16, 11]$

Ì RM 2(r, m)

r m 2m−r

! %$ &' (

f=

j∈J

xj ,

J ⊆ {1, 2, . . . , m}, |J| = r.

# f (x) = 0 j ∈ J xj = 1 ) 2m 2m−r ⊓ ⊔ ½¿º¿

* %$&' RM 2 (1, m) xi 1+ 0 j V = Fm 2 1 , m 2m−1 RM 2 (1, m) 0 j 2m−1 " B = {x1 , x2 , . . . , xm , 1}+ RM 2 (1, m) 2m vi

vi = (w1 w2 . . . wm , 1),

w = (w1 w2 . . . wm ) ∈ Fm 2

(0 0 . . . 0 1)

RM 2 (1, m) Hm 0 T Bm = , j Hm

Hm (2)

H

2m

Bm

4 !"# “ ”⊥ RM 2 (1, m) = Hm (2) . m (2)

RM 2 (1, 3) ! ⎛ ⎞ 1001011 H3 = ⎝ 0 1 0 1 1 0 1 ⎠ . 0010111

"

⎛ ⎞ 10010110 B ′ = (H3 0T ) = ⎝0 1 0 1 1 0 1 0⎠ ; 00101110

B3 =

B j

′

=

Hr 0 j

T

⎛

10 ⎜0 1 ⎜ =⎝ 00 11

01 01 10 11

011 101 111 111

⎞ 0 0⎟ ⎟ 0⎠ 1

RM 2 (1, 3) #

0 j 4$ RM 2 (1, 3) [8, 4, 4]

%

& r = 1 &&

Ì RM 2 (r, m)

r < m

ÍÒ 1 m 1 w ∈ V = Fm 2 vw m RM 2 (r, m) r < m ! " m # x1 x2 · · · xm

V $%

1 ! m & RM 2 (r, m) ⊓ ⊔

0 ≤ r < m r RM 2 (r, m)⋆ RM 2 (r, m) !

0 V ⋆ → F2 V ⋆ = Fm 2 \ {0}

r < m RM 2 (r, m)⋆

m

[2

m − 1, 0

!

+

m 1

!

+··· +

! m ]. r

" π π : RM 2 (r, m) → RM 2 (r, m)⋆ # dim ker π ≤ 1 $ π π m F22 (1 0 0 . . . 0) % (1 0 0 . . . 0) ∈ RM 2 (r, m) & # ker π = ker π ∩ RM 2 (r, m) = {0}' RM 2 (r, m)⋆ ( )* RM 2 (r, m) ⊔ ⊓

f, g Θf Θg 2m m

Θf , T hetag =

)

2

(Θf g )i .

i=1

m

Θf , Θg =

2 i=1

Θf i Θg i .

'(

m F22

f g v

f (v) = g(v) = 1.

f g

⊓ ⊔

f, g =

f g

f (x)g(x).

m x∈F22

m ≥ 1 r 0 ≤ r < m

RM 2 (r, m)⊥ = RM 2 (m − r − 1, m). f ∈ RM 2 (m − r − 1, m) g ∈ RM 2 (r, m)

f xi m − r − 1 g r f g m − 1 f g ∈ RM 2 (m − 1, m) ! " f g ! #

f, g

f g RM 2 (r, m)⊥ ⊇ RM 2 (m − r − 1, m). $

m

m + m + · · · + m−r−1 *1 m m+ m = 2m − m−r + · · · + m−1 + m m m m m = 2 − r + ···+ 1 + 0 = 2m − dim(RM 2 (r, m)) =

dim(RM 2 (m − r − 1, m)) =

0

= dim(RM 2 (r, m)⊥ ).

⊓ ⊔ %

Hm (2)

RM 2 (1, m)⊥ = Hm (2) = RM 2 (m − 2, m).

(m − r) AG (m, 2)

RM 2 (r, m)

T AG (m, 2) m − Fm 2 # $ %

"

m

aij Xj = bi ,

i = 1, 2, . . . , r.

r !

r $&'%

j=0

( # r

i=1

⎛

⎝bi + 1 +

T) m j=1

⎞

aij xj ⎠ ,

# * r+ )

"

A = A(m, r)

AG (m, 2)

RM 2 (r, m) ⊓ ⊔

r

,

&-)

CF2 (A(m, r)) ⊆ RM 2 (m − r, m). . " *

CF2 (A(m, r))

RM 2 (m − r, m)

#/ "

H

#

F2 ) (t + 1) AG (n, 2) !

t )

r ≤ s ≤ m)

" 0

CF2 (A(m, r)) ⊆ CF2 (A(m, s)). ) CF2 (A(m, r)) " AG (2, m) s ≥ r CF2 (A(m, r))

"

"

*

r

A(m, r)

(

RM 2 (m − r, m) ⊆ CF2 (A(m, r)),

# !

⊓ ⊔

½¿º½º RM 5 (2, 2) ½¿º¾º RM 2 (1, 4) RM 2 (1, 4)⋆ ½¿º¿º RM 2 (1, 4)

RM 2 (2, 4)

pn − 1 p

C !" #$% &

' "

( C (n, M, d) " A a ∈ A " ! 1 ≤ i ≤ n c = (c1 c2 . . . cn ) ∈ C ci = a

i )

a* (n − 1, M ′ , d′ ) Ci,a = { (c1 c2 . . . ci−1 ci+1 . . . cn ) : c = (c1 c2 . . . cn ) ∈ C, ci = a}.

C a = 0 ∈ Fq

+

i Ci = Ci,0

C i

C q M

q Cs

C s M/q s

Cs

C

M/q

q

M/q C

M/q Cs ⊓ ⊔

Ì C [n, k, d] 1 ≤ i ≤ n

c ∈ C ci = 0

Ci [n − 1, k − 1, d′ ] Ci C ′ = { c : c ∈ C, ci = 0} = C

C Ci k − 1 ⊓⊔ ci = 0 c ∈ C

i [n − 1, k − 1, d] !

k [n, k]

"

"

! C

n Fq ! C G = (Ik | P ) .

#

Cs

s

0 ≤ s ≤ k C Gs Cs $ G s

i1 , i2 . . . , is Ik s i1 , i2 . . . , is

G C k × n Cs % (k − s) × (n − s) &

C ds

Cs ' ds ≥ d,

ds = d

( H3 (2) ) % ⎛ ⎞ 1 ⎜0 ⎜ G=⎝ 0 0

0 1 0 0

00 00 10 01

1 1 1 0

01 1 1⎟ ⎟. 1 0⎠ 11

[5, 2, 3]

Gs =

G

10110 01011

.

C=

n Fq ½ C

(c1 c2 . . . cn cn+1 ) : (c1 c2 . . . cn ) ∈ C,

n+1 i=1

ci = 0 .

G n × k H C G (n + 1) × k G G

G 0 H ⎛ ⎞ 1 1 1 ... 1 ⎜ 0⎟ ⎜ ⎟ ⎜ H 0⎟ H=⎜ ⎟. ⎜ ⎟ ⎠ ⎝ 0

C

C

C

C

d+1

d

C

H3 (2) [7, 4, 3]# H3 (2) [8, 4, 4]# ⎛ ⎞ 11111111 ⎜1 1 1 0 1 0 0 0⎟ ⎜ ⎟ ⎝0 1 1 1 0 1 0 0⎠. 11010010

! "

H3 (2)

1$

% &

1 + 14x4 + x8 . 1

C C

C n k d

l l l

¾ 1 1 [7, 3, 4] C ⎛ ⎞ 11001100 G = ⎝0 1 1 0 0 1 1 0⎠ . 00101101

w(x) = 1 + 7x4 .

C ′ C ! j = (1 1 1 1 1 1 1) ∈ C " # [7, 4, 3]$ % & C ′ ⎛ 1 ⎜0 ′ G =⎜ ⎝0 1

10 11 01 11

01 00 01 11

100 110 101 111

⎞ 0 0⎟ ⎟. 0⎠ 1

[8, 4, 4] w(x) = 1 + 14x4 + x8 .

& ' 2

1

C H

H H = (H0 H1 · · · Hk−1 | In−k ) ,

In−k (n − k) × (n − k)

Cp ¿ p C Cp H

p ! " p

H Cp (n − k − p) × (n − p)

′ Hp = H0′ H1′ · · · Hk−1 | Ij′0 Ij′1 · · · Ij′k −p−1 ,

′ H0′ , H1′ , . . . Hk−1 ′

Ij 1

H0 , H1 , . . .

p

# [5, 2, 3]$ C ⎛

⎞ 10100 H = ⎝1 1 0 1 0⎠. 01001

H [4, 2, 2] Hs =

1110 0101

.

[n, k, d] n − 1 k k − 1

d ! d − 1

3

Ì

!

"

[n − l, k − l, ds ≥ d] [n + l, k + l, ds ≤ d] [n + l, k, de ≥ d] [n − l, k, dp ≤ d] [n, k + l, da ≤ d] [n, k − l, de ≥ d]

Ì

# " !

! $

"

"

"

%

C1 C2 |C1 |C2 | C1 C2 |C1 |C2 | = { (c1 c2 ) : c1 ∈ C1 , c2 ∈ C2 }.

& u = u1 u2 u′ = u1 u′2

v = v1 v2 v′ = v1 v2′ ' 4 5 6

! "

C1 v1

u1 u1 u2 = u

u2

v = v1 v2

v2

C2

d(v, v′ ) = d(v1 v2 , v1 v2′ ) = d(v1 , v1 ) + d(v2 , v2′ ) = d(v2 , v2′ ),

|C1 |C2 |

m Ci

[ni , ki , di ] 1 ≤ i ≤ m [n, k, d] n=

m

ni ,

k=

m

d = min {di }.

ki ,

1≤i≤m

i=1

i=1

Gi Ci G

Ci

⎛ ⎞ ⎜ ⎜ G=⎜ ⎝

G1

G2

Gm

⎟ ⎟ ⎟. ⎠

C1 [4, 1, 4] C2

[7, 4, 3] ! C1 C2 "

[11, 5, 3] # C

C " $ "

!

C

n = 8

32

4

|C|C|C|C|

n k

Ci 1 ≤ i ≤ m [n, ki , di ]

Ci

m , i=1

Ci = {v : v = v1 + v2 + · · · + vm , vi ∈ Ci }.

k ≤ k1 + k2 + . . . + km d ≤ mini {di }

Gi Ci

C ⎛ ⎞ ⎜ ⎜ G=⎜ ⎝

G1 G2 ⎟ ⎟ ⎟ . ⎠ Gm

Ci

C1 C2 [n1, k1, d1] [n2, n1, d2]

C1 C2 ! C1 C2 " #$% k1 C1 n1 & C2

C1 C2

|u|u + v| v0 = u1

C1

u0

C2

v1

n1 − k1

k1

n2 − n1

C2 C1

C1 C2

Ì

d ≥ max{d1 , d2 }

C1 C2 C1 C2

[n2 , k1 , d]

n2 k C1 C2 k1 n2 C1 C2 C2 p ∈ C1 C2 n1 p ! C1

w(p) ≥ d1 "# p C2 w(p) ≥ d2 $ ⊓ ⊔

|u|u + v|

% &

! 7

C1 C2 n1 n2 G1 G2 n1 = n2

|u|u + v| C = |C |C 1

G1 G1 G= . 0 G2

1 + C2 |

C (2n, k1 + k2 , d) n = max{n1 , n2 } d = min{2d1 , d2 } ! RM 2 (r + 1, m + 1) = |RM 2 (r + 1, m)|RM 2 (r + 1, m) + RM 2 (r, m)|.

" # $ % &

C1 C2 [n1 , k1 ] [n2 , k2 ] C1 C2

% k2 C1 n1 % C2 '

()) * # ' ()+

A, B m1 × n1 m2 × n2

A ⊗ B A B # n1 n2 × t1 t2 aij A aij B

10 A = 1 2 B = , 12 1020 . A⊗B = 1224

V, W V ⊗ W v ⊗ w v ∈ V1 w ∈ V2 (v1 + v2 ) ⊗ w = v1 ⊗ w + v2 ⊗ w, v ⊗ (w1 + w2 ) = v ⊗ w1 + v ⊗ w2 , α(v ⊗ w) = (αv) ⊗ w = v ⊗ (αw).

C1

2

u

uk2

1

C1

v⊗w=0

C1

v1

v2

...

2

v

n1

vn1

k2

v=0

w = 0.

G1 C2 C1 ⊗ C2

G2 G1 ⊗ G2

u2

C2

1

C2

C1

u1

C2

C1 C2 G1 G2 C1 ⊗C2 G1 ⊗G2

C1 C2 [n1 , k1 , d1 ] [n2 , k2 , d2 ]

C1 ⊗ C2 [n1 n2 , k1 k2 , d1 d2 ]

C = C1 ⊗ C2 n1 n2 C1 ⊗ C2 " v1 ⊗ v2 v1 ∈ C1 v2 ∈ C2 v1 ⊗ v2 ∈ C1 ⊗ C2 v1 C1 v2 C2 #

!

w(v1 ⊗ v2 ) = w(v1 )w(v2 ) = d1 d2 . $

G1 ⊗ G2

8

C1 ⊗ C2

d1 d2

k1 k2

⊔ ⊓

0

k1 − 1

1

n1 − 1

0 1

k2 − 1

n2 − 1

Ì

C1 , C2 C = C1 ⊗ C2

ti = ⌊(di − 1)/2⌋

b = max{ n1 t2 , n2 t1 }. ≤ j ≤ n2 −1 n2 (j+kn1 ) C2 C2 t2 n1 t2 n1 C1 t1 n2 ⊓ ⊔

b

k 0

C

C1 C2 [7, 4, 3] [49, 16, 9] 4

C 1 ⊗ C2

!

7

"

# $%& ' ( Ai 0

H3 (2)

Ai

An−i

i 0 9 12 16 17 20 21 24

Ì

Ai 1 49 98 931 1764 5292 7826 16087

i 25 28 29 32 33 37 40 49

Ai 16087 7826 5292 1764 931 98 49 1

H3 (2) ⊗ H3 (2)

v1,1 v1,2 · · · v1,n1 v2,1 v2,2 · · · v2,n1

vn2 ,0 vn2 ,1 · · · vn2 ,n1

n Fq [n, n, 1] I(n)

½¼ n (n) C C C (n) = C ⊗ I(n) .

n2 C [n1 , k1 ]! n2 × n1 !

" #! vi = (vi,1 vi,2 · · · vi,n1 )

C 1 ≤ i ≤ n2

9 10

b

n2

n1

C (n2 )

C C (n2 )

v = (v1,1 v2,1 · · · vn2 ,0 v1,2 · · · v1,n1 · · · vn2 ,n1 ).

C (n2 )

! r

C

b

n1

e b/n2 #

b < en2 n1

C e$

"

C (n2 )

%

%

%

C

255

159

7

[63, 12, 9]

3/2

300

%

r

R !

"

#

$ % %

1 &'

n %

$

( ) * n

+,- .

% %

n / n %% $ % %

- C = {Ci } # qi 0 [ni , ki , di ]

{ni }

-

R(C) = lim inf i→∞

C $

C = {Ci }

ki . ni

δ(C) = lim inf i→∞

di . ni

C

R(C), δ(C) > 0.

Fq

Fq Hr (q) Fq q n−r = q n−log(n(q−1)+1) n − log(n(q − 1) + 1) = 1. n 3

δ(Fq ) = 0

R(Fq ) = lim inf n→∞

! "

# q

δ ≤ 1

αq (δ) = lim sup n→∞

! ( αq (δ)

logq A(n, δn) . n

$%&%'

) A(n, δn) q * n δn # A(n, δn) ≤ q n−δn+1 1 logq A(n, δn) n

1+ αq (δ) " δ 0 1

αq (δ) ≤

n − δn + 1 ≤ (1 − δ). n

$%&,'

! %&,

- . /%0

1

δ α2 (δ) ≤ 1 − H2 . 2

αq (δ) ≤ max

0, 1 − Hq (δ/2) −

δ logq (q − 1) . 2

Ì 0 ≤ δ ≤ 1

θ=

q q−1

αq (δ) ≤ max {0, 1 − δθ} .

θ ≤ δ ≤ 1 Aq (n, δn) ≤ δ (δ − θ)−1 .

!

n lim

n→∞

1 1 −1 logq Aq (n, δn) ≤ lim logq (δ − θ) = 0. n→∞ n n

"

q#

θ n $ 0 ≤ δ ≤ θ C = {Ci } q # (ni , Mi , di ) δ {ni } %

lim

n→∞

di = δ, ni

lim

n→∞

logq Mi = αq (δ). ni

i n′i = ⌊(di − 1)/θ⌋ $

n′i ≥ lim i→∞ i→∞ ni lim

1 di − ni θ ni θ

=

δ . θ

! % Ci ni Ci′ n′i

ni −n′i

& C ′ M ′ ≥ M/qn −n ' d′i

d′i ≥ di ( n′i )) i

θn′i ≤ di − 1 ≤ d′i − 1,

′ i

%

) * Mi ′ n q i −ni

≤ Mi′ ≤ d′i (d′i − nθ)−1 ≤ d′i .

Mi ≤ q ni −ni d′i

′

αq (δ) = limi→∞ ≤ limi→∞ ≤ limi→∞ ≤ 1 − δθ.

1 ni 1 *ni

logq Mi ′ logq (q ni −ni di ) + ′ log d′ n 1 − nii + nqi i ⊓ ⊔

Fq q

θ−1 ! Rδ = (RS δn (n)) RS δn (n) "

# F(n+1) δn !$ Rδ n n − δn + 1

R(Rδ ) = lim inf n→∞

(1 − δ)n + 1 = 1 − δ. n

% q "

# Fq q − 1

t

C

n (t, ℓ) r ∈ An ! t r " ℓ # (e, 1) $ e

(n, M, d) q (t, ℓ) M Vq (n, t) ≤ ℓq n .

# t

% An " ℓ

lim

n→∞

1 logq Vq (n, nτ ) = Hq (τ ) + τ logq (q − 1), n

C = {Ci } q

i Ci (iτ, i) (i, Mi ) lim

i→∞

1 log Mi ≤ 1 − Hq (τ ) − τ logq (q − 1). i

(t, ℓ)

Ì (n, M, δn) q C

(τ n − 1, (q − 1)n)

τ=

√ 1 (1 − 1 − θδ). θ

α(δ) !

"#℄% (n, M, d) q A V & % % ' β : A → {1, 2, . . . , q} Rq = {e1 , . . . , eq } Rq

( A → Rq ε := a → eβ(a) . ! ε ( ! Eq : An → Rqn % )

Eq (c) = Eq (c1 c2 . . . cn ) = (ε(c1 ) ε(c2 ) . . . ε(cn )).

*+% E , % (Rq )n ≃ Rqn + ||x|| x ∈ Rqn a, b a, b 1 ≤ i ≤ n% + Hi ! - Rqn q X Hi : . / xq(i−1)+j = 1. j=1

' + Hi ci c0 % ! C Eq , H=

n \

Hi .

i=1

' + + Q = ( q1 , q1 , . . . , 1q ) ∈ H0 % H−Q = {x−Q : x ∈ H} ,

+1 Hi ½

+ q Rqn

{E(x) − Q : x ∈ An }

(q − 1)n

2 3 An Rnq

a, b ∈ An ||E (a)||2 = n, 1

E(a), E (b) = n − d(a, b).

4 + + +

a ∈ An E (a) n α = E (a) β = E (b) α β

α, β =

qn X i=1

αi βi = |{ i : αi = βi = 1}| = |{ i : ai = bi }|.

! " ⊔ ⊓

a b α, β = n − d(a, b) #

" E

v1 , . . . , vm RN vi , vj ≤ 0

1 ≤ i < j ≤ m u ∈ RN u, vi > 0 i = 1, 2, . . . , m

m ≤ N;

u ∈ RN

u, vi ≥ 0

i = 1, 2, . . . , m

m ≤ 2N − 1;

m ≤ 2N.

m ≥ N + 1 $ v1 , . . . , vm % S ⊆ {1, . . . , m} ! &

X ai vi = 0, i∈S

& ai = 0 ' ai

& (

T + = { i ∈ S : ai > 0};

w=

X

ai vi =

i∈T +

T − = { i ∈ S : ai < 0}, X

(−aj )vj ,

j∈T −

T + ⊂ S ! S & w = 0 w, w > 0 " * + X X X w, w = (−aj )vj = ai vi , −ai aj vi , vj ≤ 0, i∈T +

j∈T −

i,j

−ai aj > 0 i ∈ T + j ∈ T − vi , vj ≤ 0 # ai > 0 & i ∈ S ) * ai > 0 & i ∈ S vi , u > 0

0 = 0, u =

X

ai vi , u =

i∈S

X i∈S

ai vi , u > 0.

v1 , . . . , vm

m ≤ N

N N = 1 ! m ≤ N " m > N

# v1 , . . . , vm S ⊆ {1, . . . , m}

"

$

X

ai vi = 0,

i∈S

ai = 0 % vi & S = {1, 2, . . . , s} ' # V = {v1 , . . . , vs }

$ &

W

RN

(s − 1)

s X i=1

j = s + 1, . . . , m

Ps

i=1

ai vi = 0

vi , vj = 0,

1 ≤ i ≤ s ai > 0( vi , vj ≤ 0( vi vj

i = 1, 2, . . . , s j = s + 1, . . . , m ) vs+1 , . . . , vm ⊥ W

(N − s + 1) ' s > 1

m − s ≤ 2(N − s + 1) − 1,

m ≤ 2N − s + 1 ≤ 2N − 1

%

u = −vm

m − 1 ≤ 2N − 1

v1 , . . . , vm−1 ⊔ ⊓

ǫ > 0 w1 , . . . , wm m wi , wj ≤ −ǫ 1 ≤ i < j ≤ m m≤1+

0≤

m X i=1

wi ,

1 . ǫ

"

m X i=1

wi =

m X i=1

wi , wi + 2

X

1≤i<j≤m

'

θ=

wi , wj ≤ m − m(m − 1)ǫ.

m ≤ 1 + 1/ǫ

⊔ ⊓

q ) ! q−1

! *

Ì

C (n, M, d) q A d=

1 (1 − σ)n. θ √

c ∈ C e = θ1 (1 − γ)n 0 < γ < 1 γ > σ j

1−σ |Be (c) ∩ C| ≤ min n(q − 1), 2 γ −σ

√

ﬀ

.

γ = σ |Be (c) ∩ C| ≤ 2n(q − 1) − 1 c1 , . . . , cm t b ∈ An 1 ≤ i ≤ m vi = E (ci ),

r = E (b).

vi

(q − 1)n r v = v(α) = αr −

1−α j. q

! "

α #

vi − v(α), vj − v(α) < 0,

i = j 1 ≤ i, j ≤ m

vi − v(α), v(α) ≥ 0.

e = ⌊ d−1 ⌋ ei = d(b, ci ) $

% 2 vi , vj = n − d(ci , cj ) ≤ n − d v, v = α2 n + 2(1 − α)α vi , v = αvi , r +

n n n α2 + (1 − α)2 = + n q q q θ

1−α n n vi , j = α(n − ei ) + (1 − α) ≥ α(n − e) + (1 − α) . q q q

& i = j vi − v(α), vj − v(α) ≤

´ 1 ` n σ + α2 − 2αγ , θ

e = (1 − 1/q)(1 − γ)n d = (1 − 1/q)(1 − σ)n ” 1 “σ +α , γ> 2 α

'

(

α √ )

√ * + , #

( α = σ γ > σ # ( " , vi √ u v( √ σ) H - m ≤ (q − 1)n ! γ > σ

,

m≤

1−σ . γ2 − σ

α = σ

vi − v, vj − v ≤

1 n(σ − γ 2 ) < 0. θ

1 ≤ i ≤ m wi =

vi − v . ||vi − v||

wi 1 ! " wi , wj ≤ −

γ2 − σ . 1 − γ2

# $ % „ « 1 − γ2 1−σ m≤ 1+ 2 , = 2 γ −σ γ −σ √ & γ = σ ! ' m ≤ 2n(q − 1) − 1 ⊔ ⊓ ( )

|Bτ − 1 (w) ∩ C| ≤ (q − 1)n, n

w ∈ A * (

$ ! + $

n

τ−

1 1 ≤ (1 − γ), n θ

γ>

√

σ,

δ=

1 (1 − σ). θ

, σ δ √ γ > 1 − θδ, τ≤

√ 1 (1 − 1 − θδ). θ √

⊔ ⊓

- τ ≤ 12 (1 − 1 − 2δ) * . ! α(δ)

1

Elias-Bassalygo McEliece, Rodemich, Rumsey, Welch

0.8

0.6

0.4

0.2

0 0

0.1

0.2

0.3

0.4

0.5

τ

0 ≤ δ ≤ θ1

αq (δ) ≤ 1 − Hq (τ ) = 1 − Hq

1 θ

„

« √ 1 (1 − 1 − θδ) . θ

≤ δ ≤ 1 αq (δ) = 0

! " (n, M, δn) δ ≤

1 θ

# " (nτ − 1, (q − 1)n) $ ! % ! lim

n→∞

1 Mn ≤ 1 − Hq (τ ) − τ logq (q − 1) ≤ 1 − Hq (τ ), n

# % % " " % " "

" " & " " ' ⊔ ⊓ ( ) α2 (δ) ≤ 1 − H2

“

1 (1 2

−

√

” 1 − 2δ) ,

* " # % * " " '

α2 (q) ≤

min

0≤u≤1−2δ

{1 + g(u2 ) − g(u2 + 2δu + 2δ)},

!"#"$

√

g(u) = H2 [(1 − 1 − u)/2] % &'℄ &)℄ * α(δ) + , - ./

Ì 0 ≤ δ ≤ θ

αq (δ) ≥ 1 − Hq (δ).

k [n, k, δn]

qn qn n−1 . = i Vq (n − 1, δn − 2) (q − 1) i=0 i

qk <

δn

k

k < n − logq Vq (n − 1, δn − 2).

Vq (n, δn) > Vq (n − 1, δn − 2),

[n, k, δ] k = n− logq Vq (n, δn) logq A(n, δn) 1 ≥ lim sup 1 − logq Vq (n, δn) = 1 − Hq (δ). αq (δ) = lim sup n n δ→∞ δ→∞ !

⊓ ⊔ !"#

αq (δ) ≤ 1 − δ αq (δ) ≤ 1 − δθ αq (δ) ≤ 1 − Hq (δ/2) αq (δ) ≥ 1 − Hq (δ) √ αq (δ) ≤ 1 − Hq ((1 − 1 − δθ)/θ)

Ì $ %

& α(δ) '( ) ( *) ' % + , % )

) )

"

# $ %

& "

1

R(δ)

(q = 2)

1 2

0

C1 , C2 , . . . , Ct

i = j n i=1 Ci = F2

.t

Fn2

2

k

1

δ

!

Ci ∩ Cj = {0}

t=

2k − 1

2n − 1

2n − 1 ; 2k − 1

C = {Ci }

ǫt ≥ V2 (n, d − 1) d

C = {C1 , C2 , . . . , Ct } t(1 − ǫ) C

!

Fn2

Ci ∈ C

d vi ∈ Ci

0 Bd−1 (0)

vi = vj i = j # |Bd−1 (0)| − 1 = V2 (n, d − 1) − 1

d $ tǫ ≥ V2 (n, d − 1)

tǫ %

& C t ⊓ ⊔ "

d

(1 − ǫ)

C

C Fn2

Ci ∈ C k F2 Ci ≃ Fk2 c n F2k F2 = F2k F2

w ∈ Ci c

F2k n

k

w = (w1 w2 . . . wc ) ∈ Fc2k

!" #

w = 0 i

$

%

wi = 0

wi = 1

&' ( PG (c − 1, 2k ) ((

2n − 1 2ck − 1 = = t. 2k − 1 2k − 1

Cw

'

Cw = { (w1 x, w2 x, . . . , wc x) : x ∈ F2k }. Cw 1 F2k w k F2 ) C

w = v

Cw ∩ Cv = {0}

Cw ∩ Cv = {0}

'

= { Cw : w ∈ PG (c − 1, 2k )}. C

Fn C 2

&

C = { Cw : w ∈ PG (c − 1, 2k )}

Fc2k

y = (y1 , y2 , . . . , yc ) ∈ Fc2k

y

w = y1j y (( Cw y ∈ Cw * ' ⊓ ⊔

1 ≤ j ≤ c

*

'

+

* , ( (

Ì

0 < ǫ ≤ 1 k 2(c−1)k (1 − ǫ)

c

δ = H −1 (1 − 1c ) − ǫ

(ck, k, δ(ck)) .

1 c = 1 − H(δ)

k

δ

R =

Fck k

C 2 (c−1)k

! t = 2 " ##

lim

k→∞

1 1 V2 (ck, δ(ck)) ≤ (1 − ), ck c

k ǫt = ǫ2(c−1)k ≥ V2 (ck, δck).

"

$

⊓ ⊔

!

" # $ % & " ' ( ) *

+ ,

- $

$ .'/℄ .!12℄ .2#℄

3 .'4℄ .!1!℄ ./1℄ .45℄

+ 4! 6 &

[n, n − k]q

,

, %

k − 1 , , Fq 7 6

8 !'! %

Fq X Ω ω : X → Fq ,

n D = (d0 , d1 , . . . , dn−1 ) X (Ω, D)

E Fnq

(Ω, D) n Ω

! " #

Ω

E(Ω, D) = {(ω(d0 ) ω(d1 ) . . . ω(dn−1 )) : ω ∈ Ω}.

(λω)(x) = λ(ω(x))

(ω + θ)(x) = ω(x) + θ(x),

E(Ω, D) D $

Ω X → Fq D

E(Ω, D)

% l ∈ Fq &

ω ∈ Ω

$$

lω ∈ Ω

'

(c0 c1 . . . cn−1 ) = (ω(d0 ) ω(d1 ) . . . ω(dn−1 )) ∈ E, (lω(d0 ) lω(d1 ) . . . lω(dn−1 )) = (lc0 lc1 . . . lcn−1 ) ∈ E.

% (c0 c1 . . . cn−1 ) ! ω, ω′ ∈ Ω

(c′0 c′1 . . . c′n−1 )

(c0 . . . cn−1 ) = (ω(d0 ) . . . ω(dn−1 )), (c′0 . . . c′n−1 ) = (ω ′ (d0 ) . . . ω ′ (dn−1 )).

%

Ω

ψ = ω + ω′ ∈ Ω

(ψ(d0 ) . . . ψ(dn−1 )) = (c0 + c′0 . . . cn−1 + c′n−1 )

E

⊓ ⊔ [n, n − k] ( % (Ω, D) Ω # Fq ) k − 1 D

q − 1 F⋆q

Ω

E(Ω, D)

(Ω, D)

!

"

! #

Ω

$

D

n

%

!

k

d

% # & '() * +,- .)℄0 & $ 1 2 $ Fq 3 ,

C

$

Fq

Fq

!

C1

P = {P1 , . . . , Pn };

Q = {Q1 , . . . , Qs }

P ∩ Q = ∅

. 1

D=

n

Pi ;

E=

mi Q i ,

i=1

i=1

s

deg D > 0

E f

L(E)

$

C

L(E)

C

div f + E > 0. 4

E

L(E)

5

'

L(E)

+222℄

Qi

6

mi

Q

#

4

5 ' 7 8

CL (D, E)

[n, k, d]

E(L(E), P)

CL (D, E)

Θ : L(E) → Fq

Θ: $

L(E)

L(E) −→ (Fq )n λ −→ (λ(P1 ) . . . λ(Pn )).

ℓ(E) = dim L(E)

⎜ G := ⎝

! "#

λ1 , λ2 , . . . , λℓ(E)

&

%

$

⎛

G = (gij )

λ1 (P1 ) · · · λ1 (Pn )

λℓ(E) (P1 ) · · · λℓ(E) (Pn )

⎞

⎟ ⎠.

! '#

G CL (D, E) Fnq (

PG (1, q)

Pj = (j, 1);

j

Fq

α

Q∞ = (1, 0),

D=

q−1

F⋆q

Pαi .

i=0

k < q CL (D, kQ∞ ) *

)

L(kQ∞ )

φ = Y t f (X/Y ), f Fq [x]

t = deg f ≤ k

φ(Pj ) = 1f (αj /1) = f (αj ),

! +#

Θφ = (f (1), f (α), . . . , f (αq−1 )).

f (x) CL (D, kQ∞ )

φ

Θ

Fq , - k F⋆ q $ CL (D, kQ∞ )

RS (q − 1, k)

C : XZ 2 + X 2 Z + Y 3 = 0,

F4 = {0, 1, ω, ω 2} ω x2 + x + 1 F2

9 F4 Q1 (1, 0, 0), Q2 (1, 0, 1), Q3 (0, 0, 1), P1 (ω, 1, 1), P2 (ω 2 , 1, 1), P3 (ω, ω, 1), P4 (ω 2 , ω, 1), P5 (ω, ω 2 , 1),P6 (ω 2 , ω 2 , 1).

D := P1 + P2 + P3 + P4 + P5 + P6 ;

E := Q1 + Q2 + Q3 .

L(E) f1 := 1,

f2 := X/Y,

f3 := Z/Y.

Θ ⎛

⎞ 1 1 1 1 1 1 G = ⎝ ω ω2 1 ω ω2 1 ⎠ . 1 1 ω2 ω2 ω ω

G G CL (D, E) [6, 3]!

" # $ 3 % &'(

% )

*++℄-

" F C D E " q

CΩ (D, E)

n

c = (c1 , . . . , cn )

cj φ(Pj ) = 0

j=1

φ ∈ L(E)

D ! " Ω(D) # ! $$% % % &'℄

CΩ (D, E) CL (D, E)

ℓ(E) L(E) E deg E < 0

ℓ(E) = 0;

!

ℓ(E) ≤ 1 + deg E;

!

ℓ(E) = deg(E) − g + 1.

%!

" # g C deg E > 2g − 2$

& g C

m = deg E =

s

mi .

i=1

' d CL (D, E)

k n ( )

Ì ËÙÔÔ 2g − 2 < deg E CL (D, E)

< n

k = deg(E) − g + 1 d ≥ n − deg E ( *& %! L(E) + deg(E) − g + 1 , - *

Θ : L(E) → Fn q + & f ∈ ker Θ , f (Pi ) = 0 Pi ∈ P .

f ∈ L(E − D). /*

deg(E − D) = deg E − deg D ≤ (n − 1) − n ≤ 0. 0 L(E − D) = {0}$ Θ +

¾º

f ∈ L(E)

Θf d n − d P Pi1 , . . . , Pin−d

f (Pij ) = 0

G = Pi1 + Pi2 + · · · + Pin−d .

f ∈ L(E − G)

E−G

deg(E − G) = deg E − (n − d).

deg E − n + d ≥ 0

n − deg E

dL (E)

⊓ ⊔

CL (D, E)

C : X 3 Y + Y 3 Z + Z 3 X = 0,

F8

x3 + x + 1

g = 3 F2 (ξ) ξ ! F8 24

"#"

F8

C

Q1 (0, 0, 1) P1 (1, 1, ξ) P4 (1, ξ, 1) P7 (1, ξ 2 , 1) P10 (1, ξ 3 , ξ 2 ) P13 (1, ξ 4 , 1) P16 (1, ξ 5 , ξ) P19 (1, ξ 6 , ξ 3 )

Q2 (0, 1, 0) 2

P2 (1, 1, ξ ) P5 (1, ξ, ξ 2 ) P8 (1, ξ 2 , ξ 4 ) P11 (1, ξ 3 , ξ 3 ) P14 (1, ξ 4 , ξ) P17 (1, ξ 5 , ξ 5 ) P20 (1, ξ 6 , ξ 4 )

Q3 (1, 0, 0) P3 (1, 1, ξ 4 ) P6 (1, ξ, ξ 6 ) P9 (1, ξ 2 , xi5 ) P12 (1, ξ 3 , ξ 5 ) P15 (1, ξ 4 , ξ 3 ) P18 (1, ξ 5 , ξ 6 ) P21 (1, ξ 6 , ξ 6 )

Ì F8

D=

21

Pi ;

E = 2(Q1 + Q2 + Q3 ).

i=1

$

deg E = 6

2g − 2 = 4 < 6 = deg E < 21 = n. CL (D, E) k = 6 − 3 + 1 = 4 d > n − 6 = 15 ′ %

E = 3(Q1 + Q2 + Q3 ) & "##

k = 9 − 3 + 1 = 7 d > n − 9 = 12 "##

n − deg E ≤ d ≤ n − deg E + g.

g

dL g = 0

!" [n, k, d] # CΩ (D, E) $ C

2g − 2 < deg E < n CΩ (D, E)

k = n − deg E + g − 1 d ≥ deg E − (2g − 2)

% # CΩ (D, E) & dΩ (E) = deg E −(2g −2) dL (E)

' ( ) $ Fq ( H ' 2

X q+1 = ZY q + Z q Y.

*+,

- . H q3 + 1 Fq / 0 H & g = q(q − 1)/2/ H & Fq ' (1 2 0/ P Fq H & 3

P H H

P / 4 & $ - - * , g = q(q − 1)/2 N = q3 + 1 & ' ( % H &

$ l∞ : [Z = 0] Q∞ = (0, 1, 0) 2

2

2

r Hr := CH (D, rQ∞ ),

D =

P,

P ∈H P =Q∞

r

n = q3 r ≤ s Hr ⊆ Hs r ≤ 0 Hr = {0} r > q3

dim Hr = n 1 r !" " # $ Hr % ! q2 − q − 2 < r < q3 k = dim Hr = r + 1 − g = r + 1 − q(q − 1)/2.

& ' d Hr

d ≥ q 3 − r.

( # ) q = 3 F9 ' H

X 4 = ZY 3 + Z 3 Y

g = 3 '* 28 F9 H ) + , !" & ξ F9

* x2 − x − 1 = 0 - r = 7 (0, 1, 0) (1, 2, 1) (ξ, , ξ 7 , 1) (ξ 2 , ξ 3 , 1) (2, ξ 3 , 1) (ξ 6 , 2, 1)

(2, 2, 1) (ξ 5 , ξ 5 , 1) (1, ξ, 1) (ξ, 1, 1) (0, ξ 2 , 1)

(0, 0, 1) (ξ 7 , ξ 7 , 1) (1, ξ 3 , 1) (ξ 3 , 1, 1) (0, ξ 6 , 1)

(ξ, ξ 5 , 1) (ξ 7 , ξ 5 , 1) (ξ 2 , ξ, 1) (ξ 5 , 1, 1)

(ξ 3 , ξ 7 , 1) (ξ 5 , ξ 7 , 1) (ξ 6 , ξ 3 , 1) (ξ 7 , 1, 1)

(ξ 3 , ξ 5 , 1) (ξ 6 , ξ, 1) (2, ξ, 1) (ξ 2 , 2, 1)

Ì F9 H

!" " ' L(7Q∞ ) k = 7 + 1 − 3 = 5.

n = 27 k = 5 dL = 20 ½ 5 1,

X , Z

Y , Z

Y 3 + Y Z2 , X 2Z

Y 4 + Y 2Z 2 . X 3Z

C [27, 5, 20]

! " # $% " & ' ( !) *" + &

,- " " . / . 0 & 1 " C = {Ci : i ∈ N} Ci [ni , ki , di ] ni δ(Ci ) = di /ni ' R(Ci ) = ki /ni

0 2 " ! 0 $

Ì

0≤R≤1

q

1 . R(C) + δ(C) ≥ 1 − √ q−1

C

345

6 78!℄ 0 : 6 1

℄"

111 1 1 1 1 1 1

2 2 ξ7 ξ5 ξ ξ3 ξ3 ξ ξ5

ξ6 ξ2 2 2 1 1 2 2 1

ξ5 ξ7 ξ ξ3 ξ ξ3 ξ ξ3 ξ

ξ7 ξ5 ξ6 ξ2 1 1 ξ2 ξ6 2

1 1 1 1 1 1 1 1 1

⎞

⎟ 2 ξ ξ3 ξ5 ξ7 ξ2 ξ6 0 0 ⎟ ⎟ 3 2 6⎟ ξ 1 1 1 1 2 2 ξ ξ ⎟ ⎟ 2 6 2 6 1 ξ ξ ξ ξ 2 2 0 0⎟ ⎠ 7 3 5 7 6 2 ξ ξ ξ ξ ξ ξ ξ 0 0

1 1 1 1 1 1 1 1 1

H7

⎜ ⎜ 1 2 0 ξ ξ3 ξ3 ξ1 ξ5 ξ7 ⎜ ⎜ C = ⎜ 2 2 0 ξ5 ξ7 ξ5 ξ7 ξ5 ξ7 ⎜ ⎜ 1 1 0 ξ2 ξ6 ξ6 ξ2 ξ2 ξ6 ⎝ 2 1 0 ξ6 ξ2 1 1 ξ2 ξ6

⎛

1 . αq (δ) ≥ (1 − δ) − √ q−1

δ

q ≥ 49

#

$

"

αq (δ) ≤ (1 − δ).

!" #

Nq (C) Fq $ C g(C) " " CL (Dm , Em ) %

" $ Xm Fq limm N (Xm ) = ∞& " $ Dm Xm limm deg Dm = ∞& '" $ Em Xm deg Em > 2g − 2" ( ) " $ E $ Xm & " Dm = Xm \ E

" # Fq $ "

Ì

C Fq g

√ |Nq (C) − (q + 1)| ≤ 2g q. √ * $ C (q + 1) + 2g q % " + q % , - . % / "

C Fq g

√ |Nq (C) − (q + 1)| ≤ g⌊2 q⌋.

$ $ Fq $ %

$" # , 0 "

* $ Fq {Xm} %

"

lim g(Xm ) = ∞;

m→∞

"

lim

m→∞

Nq (Xm ) = ∞. g(Xm )

( ! $ " 1 $ "

F4

m F4

AG (m, 4) AG (m, 4)

(X1 , X2 , . . . , Xm ).

F (X, Y )

2

2

F (X, Y ) = XY + Y + X 2 .

F (X, Y )

m−1

m

AG (m, 4)!

"

#

% #

Xm

$

AG (m, 4)

F (X1 , X2 ) = F (X2 , X3 ) = . . . = F (Xm−1 , Xm ). &$

Xm

gm =

(

2m + 2m−1 − 2 m

2

m−1

+2

' ( %

3 · 2m−1

−2

{Xm }

m+3 2 m 2

+1

−2

m+2 2

+1

m

m

.

)

!

"#$ %&℄ (

) %&*℄

%*℄ + , -. / (0! 1( -. 023 4 56

-

r

e

7

)) -

Q8

& ))

Q

C = CΩ (D, E)

dΩ

t = ⌊(dΩ − 1 − g)/2⌋ Q

F

D

F ! !

! " # φ ∈ L(F ) φ

e $

" %& #

$

φ ' #

L(F ) $ #

t t L(F ) L(F − Q)( ' Q ! L(F − Q) = {0} ! Q "!! dim L(F ) ≥ t + 1 ) # deg F ≥ t + g * r = (r1 r2 · · · rn ) e = (e1 e2 · · · en )

r − e ∈ CΩ (D, E).

r

f |r =

n

f ∈ L(E)

ri f (Pi ).

i=1

+ r ∈ CΩ (D, E) f |r = 0,

f ∈ L(E) ψ ∈ L(F ) ϕ ∈ L(E−F )( ψϕ ∈ L(E) Θψϕ ∈ CL (D, E)( c ∈ CΩ (D, E) ψϕ|c = 0.

" ψ ψϕ|r =

n i=1

ri ψ(Pi )ϕ(Pi ) =

n

ei ψ(Pi )ϕ(Pi ) = 0.

,-,,

i=1

L(F − Q)

K(r, F ) = {ψ ∈ L(F ) : ψϕ|r = 0 ϕ ∈ L(E − F )}.

r ∈ C

K(r, F ) = L(F )

L(F − Q) ⊆ K(r, F ).

deg(E − F ) > t + 2g − 2

CΩ (Q, E − F ) = {0}

ψ ∈ K(r, F ) 0=

ri ψ(Pi )ϕ(Pi ) =

ei ψ(Pi )ϕ(Pi ),

ϕ ∈ L(E − F ) w wi = ei ψ(Pi ) CL (Q, E − F )

ei ψ(Pi ) = 0 i ψ

!

"

L(F − Q) = K(r, F ).

# $ Q ! ! K(r, F ) % !

& $ ' (

CΩ (D, E)

[n, k]' (

CΩ (D, E) Fq ' ! C g F C deg F ≥ t + g deg(E − F ) > t + 2g − 2

) H CΩ (D, E) r

* c ∈ CΩ (D, E) d(r, c) ≤ (dΩ − 1)/2

K(r, F ) K(r, F ) = {0} r ψ ∈ K(r, F ) ψ = 0

J = {j : ψ(Pj ) = 0}

L(r)

!

xi = 0 HxT = HrT ;

i ∈ J

L(r) e

c = r − e;

L(r)

ÍÒ C

G H

H

! "##$ % H

& &

H (dv , dc) ' ( ) dc 0* + ) dv 0 H , (dv , dc ) , dc , dv % , d

, (d, d)

F = {Hn,m} ,

d H ∈ F , d - % . ((% H

m × n , S / & & ΓH H , S = (P, L, I) H

¿¼¼

[7, 4] S ΓH ΓH = (V, E) V = R ∪ L R L H {ri , lj } ∈ E Hij = 1

ΓH H C C

ΓH = (R ∪ L, E) ! "

n L n

# $ n − k R H # # % li rj j & H i&

Γ [7, 4] H H H 7 3 Γ Γ7,3 10

H F32 1 H 4 × 3 = 12 Γ ! "# H $ % i 1 7 & ' i i ( ) Ë½ * 7 + , 3 2 + * 0 + *

1 - 0 1 1 0. + * 1 % & Γ = (P, E) v ∈ P v * Γ / L R %

' Γ 0 Γ = (L ∪ R, E) (l, r) v ∈ L l c ∈ R r 0 Γ = (P, E) |E| Γ ' |P | / Γ ' |E| |E| 2

Γ d &)% $ ) )

n * G = {Γn = (Pn , En)}∞n=0 ) |Pn| = n * ) G |En | = O(n);

¿¼¾

G

|En | = O(n2 ).

½

C

(l, r)"

!

ln

C

!

C

(l, r)

C

(l, r)" C n

C !

#

H

⎛

10 × 20

⎞ 00001000111000010001 ⎜ 00000011001101010000 ⎟ ⎜ ⎟ ⎜ 01100010000000010101 ⎟ ⎜ ⎟ ⎜ 00000101010000001110 ⎟ ⎜ ⎟ ⎜ 11001000000010001010 ⎟ ⎟ H =⎜ ⎜ 00000010001101100001 ⎟ . ⎜ ⎟ ⎜ 00011101000010100000 ⎟ ⎜ ⎟ ⎜ 10000000100011100000 ⎟ ⎜ ⎟ ⎝ 11110000010000001000 ⎠ 00010100100100000110

$ H (3, 6)" 3 0 6% H (3, 6)" & '() C H [20, 10]" * G C & '(+ 4

n

*

Cn (dv , dc )

,

(dv , dc )"

!

- *

!

1

,

1 10011000100000000000 B 01001110010100000000 C C B B 11001010001010000000 C C B B 11000010000001000000 C C B B 11100001011000100000 C C B G=B C B 01011101000000010000 C B 01110100000000001000 C C B B 01101110001000000100 C C B @ 00011000010000000010 A 01001001011000000001 0

c c c c c c c c c c

v20 v19 v18 v17 v16 v15 v14 v13 v12 v11 v10 v9 v8 v7 v6 v5 v4 v3 v2 v1

10

9 8 7 6 5 4 3 2 1

(l, r)

Γ = (L ∪ R, E)

v ∈ L l w ∈ R r

[7, 3] !

"

(3, 4) #

$ % % &' " ( Ë½

# )

Ë¾

2

Ë¿

0

Γ C

0

c

0

n ! "

#

$ Γ = (L ∪ R, E) # S ⊆ L Γ (S) ⊆ R % &

&

S Γ1 (S) % &

&

S # % Γ (S) ! S ' Γ1 (S) ! S #

Γ = (L ∪ R, E) ! (l, r)(

n & L ! (l, r, γ, δ) ¾ ' S ⊆ L |S| ≤ δn |Γ (S)| ≥ lγ|S|. )#* + ' (l, r)( , ! # + " ' & (α, β) S ⊆ L |S| ≤ βn |Γ (S)| ≥ α|S|.

- (l, r, γ, δ)( !

(α, β)( & β = δ α = lγ # + " ! (γ, δ)(

R & L# . ! , & , %

#

Γ = (L ∪ R, E) (l, r, γ, δ) S ⊆ L |S| ≤ δn |Γ1 (S)| > l(2γ − 1)|S|. 2

/0

u = |Γ1 (S)| d = |Γ (S)| − u

(lγ, δ) u + d ≥ lγ|S|

S ⊆ L

S l|S| S

! " # 2 u + 2d ≤ l|S| 2d ≤ l|S| − u.

$ %&

' $ &

( u+d 2u + 2d 2u 2u u

≥ ≥ ≥ ≥ ≥

lγ|S| 2lγ|S| 2lγ|S| − 2d 2lγ|S| − (l|S| − u) l(2γ|S| − 1)|S|.

)'

⊔ ⊓

* Γ1 (S) #

' S + ,

- r ∈ Γ1 (S) s ∈ S * ' S " % s = 1 r ,

! s * S . Γ1 (S) C ,

Ì G = (L ∪ R, E) (l, r, γ, δ) γ > 21 C G nδ # c ∈ C δ′ n <

δn / S ⊆ L 0 δ ′ n # |Γ1 (S)| = 0 ' ) 1 |Γ1 (S)| ≥ (2γ − 1)l|S| ≥ |S| > 0,

c = 0 #

⊔ ⊓

! " " #

C

Γ r

v ∈ C r

! " v1 , v2 , . . . , vn ri r ! 2 " !

# 0$ v = (v1 v2 · · · vn ) v ! % i " vi

i

" ! vi % v = (v1 v2 . . . vn ) ! &

! k

' k ( " 3 ) y ∈ C r & y′ r * +

, -

%

Ì

Γ

(l, r, γ, δ) γ ≥ 34 Γ

δ n

2

C C

vi

0 ! " |R|

y # bj j $ % sj uj & & % ' {uj }j=1,2,...,k &

& l ' b1 ≤

δ n. 2

( 0 < bj < δn ! l bj+1 < δn ) uj = 0* + , - . % ! # 0 < bj < δn ' ! & sj + uj >

3 lbj . 4

, -

&

%

2 *

lbj bl 2sj + uj ≤ lbj .

, /-

, - , /- uj >

1 bj . 2

, -

. &

" l/2 , -

" &

& 0 j + ' bj+1 < δn 1

bj+1 ≤ δn # bj+1 = δn uj+1 >

lδ 1 bj+1 = n. 2 2

, 2-

3 , 2- {uj }j % n

u1 ≤ lbi ≤ lδ 2

! " " #" 0 $$ " bl δn % " %

& δn " '" 0 " %" " ( ⊔ ⊓

n

℄ !℄ "

# $

$

% & %

"

' (

' " ! ( ) * + ,

$ "

- ./ 0

1

$ % & % 2 3 %

n×k

C

( " k 2 O(nk) ≃ O(n2 )

% ( 4 2 5 %

4 6 37 ( C 1 " O(2n−k )

6 $ $ $ 7 ( , " 8 9 : ; <

¿½¼

℄

! "# $ ! % &'℄ ( ) ! % * +, - % $

O(n). # * / ! ! % 0

% !! n % ' !

!! n

#$ ! ½

,#

# $ !

% $! !! +#

!! ! 1 2 3 4 !!

# !! $ !

0

" !!

!! n %

$ ! % . !

2

!

!3 %

+# ! % !!

() / w # /

!!

2 ! 3 ! ! /! $ ! $

! 1

Fq m, n, k k ≤ n φ, ξ, ψ, ϑ φ ξ ψ ϑ

: : : :

Fm Fk Fm Fk

→ → → →

Fm Fm Fn Fn .

ϑ (n, k) m (φ, ξ, ψ, ϑ) ui ∈ Fkq i = 0, 1, . . . ci ∈ Fnq s0 = 0 ci = ψ(si ) + ϑ(ui ) si+1 = φ(si ) + ξ(ui ).

!

s m A, B, C, D φ ξ ψ ϑ " A: B: C: D:

m k m k

×m ×m ×n × n.

(A, B, C, D)

# # $ ! $ s0 = 0 ci = si C + ui D si+1 = si A + ui B.

% & '

( D [n, k]

A, B, C #

(A, B, C, D) ) D * m = 0 +

0

(

, -

. ,

¿½¾

(A, B, C, D) ri i = 0, 1, . . . , j

uj

rj

i < j ci cj

C D

j = 0 s0 = 0 r0 = e0 + u0 D,

e0

C D c0 = r0 − e0 ,

! ! D j > 0 0 ≤ i < j

ui " sj # sj = sj−1 A + uj−1 B = sj−2 A + uj−2 BA + uj−1 B = · · · =

(

vj = rj − sj C.

j−1 i=0

ui BAj−1−i ;

$%&' $%%'

vj = ej + uj D,

ej

C

D

uj

uj D = vj − ej = rj − sj C − ej .

(n, k)

[n, k] C

C ! "

#

ui

uj j > i $ %&' %' (

$

) *" "

# ! + , -

. $ )

{xi}i xi z i ,

X(z) =

i

0z = 0 ) i Z Fnq ((z)) ) z / Fnq 0 C G

{ui}i "

)

j

U (z) =

ui z i .

i≥0

1 ui "

ci = ui G (

"

C(z) =

i≥0

⎞ ⎛ (ui G)z i = ⎝ ui z i ⎠ G. i≥0

Fn [[z]]

G

F (z) → F (zG) = F (z)G,

C(z) = U (z)G.

ui

A, B, C, D

+ * + ui z i D si z i C + + * + * ui z i B. si+1 z i = si z i A +

S(z) =

ci z i =

si z i ,

*

C(z) =

ci z i ,

U (z) =

i

i

! "!

#! $!

ui z i ,

!

i

%%

S(z)z −1 = S(z)A + U (z)B C(z) = S(z)C + U (z)D.

&! '!

( &! '!

E(z) = B(z −1 I − A)−1 G(z) = D + E(z)C = D + B(z −1 Im − A)−1 C,

)! !

S(z) = U (z)E(z)

"!

C(z) = U (z)G(z).

*!

(

G(z)

+ G(z) (

G(z) = G.

Fq ((z)) z

Fq

1 Fq

Ftq

Fq ((z))t gi z i ,

F (z) =

i

fi t zi ) = (F1 (z), F2 (z), . . . , Ft (z)). fi 2 zi , . . . fi 1 zi , F (z) = ( i

i

i

Ì

G(z)

(n, k)

(A, B, C, D) Fq (z)k → Fq ((z))n Γ : F (z) → F (z)G(z)

Γ ! U (z) = i vi z i

vi ∈ Fk "

Fq ((z))k → Fq ((z))n U (z)G(z) =

i

vi z

i

!

D + B(z −1 Im − A)−1 C = i

vi D + B(z −1 Im − A)C z i ,

# Fnq $ Γ

$ % ⊓ ⊔

Γ & G n × k

Fq ((z))

' (n, k) C Fq k ( Fq ((z)) ! C

Fq ((z)) % C

2 [2, 1] G = [1, 1] G′ = [1, 0] 2 1

[2, 1] ! " # (2, 1) C m = 2 $%%℄ ' 4 (A, B, C, D)

2 × 2 1 × 2 2 × 2 1 × 2 ( A=

01 , 00

B= 10 ,

C=

10 , 11

D= 11 .

( C ) *

E(z) E(z) = B(z

'

−1

−1

I − A)

z −1 1 −1 2 = 10 = zz . 0 z −1

10 G(z) = D + E(z)C = 1 1 + z z 2 = 1 + z + z2 1 + z2 . 11

+,+, +,+-

) C

G(z) = 1 + z + z 2 1 + z 2 .

. /

0

W (F (z)) F (z) =

+∞

fi z i

i=−∞

Fq ((z)) 1 fi 0 ( C ⊆ Fq ((z))n ( C(z) = (C1 (z), . . . , Cn(z)) ∈ C Wl (C(z)) =

n i=1

W (Ci (z)).

Fq ((z))n

dl (C) C ⊆

dl (C) = min Wl (C(z)). C(z)∈C

C

dl = 5

!

"

# Γ = (V, E) V E V × V $ (v1 , v2 ) ∈ E (v2 , v1 ) ∈ E

Γ #

#

¾ T = (V, E) n "

(V, E) V E φ : V → {0, 1, 2, . . . , n} e = (vi , vj ) ∈ E φ(vj ) = φ(vi ) + 1.

φ(v) v % & "

n & 0 & n

p (i, vi ) i ∈ 0, . . . n p = v0 v1 · · · vn . '

$ C [n, k]( q (

& H = (h1 T , h2 T , . . . , hn T ) (n − k) × n

ΘH (C) = (V, E) H

(V, E) n & φ ) 2

Ì

Σ q n − k V = {0, 1, . . . , n} × Σ.

si = (i, s) ∈ V i si s i (vi , wj ) E j = i + 1 c ∈ C c = (c1 c2 . . . cn ) i = 0 v0 = (0, 0) i > 0 ! i (i, v) = 1, ct ht , t=1

hj j H ! i+1 (j, w) = j, ct ht . t=1

(vi , wj ) ∈ E

Ì

w = v + cj hj .

! "#$

ΘH (C) = (V, E) C H λ : E → Fq (vi , wj ) ∈ E

c = (c1 c2 . . . cn ),

λ(vi , wj ) = cj ;

λ

e = (vi , wj ) ! $ c d cj = dj % &

'( )) i i v= ct ht = dt ht . t=1

t=1

* +

, ! "#$

cj hj = dj hj .

- ) +

hj = 0 ( & )) j )) C )) 1 . cj = dj

λ ( E ⊓ ⊔

000 001 010 011 100 101 110 111 0 1

2 3 4 5 6 7

H3 (2) e λ(e) = 1 (1 1 0 0 1 1 0)

00 07

[7, 4] H3 (2) ⎛

⎞ 0001111 H = ⎝0 1 1 0 0 1 1⎠ . 1010101

7

H 8 H3 (2) 7 − 4 = 3

23 = 8 e

i i+1 ci+1

!

" #$#

%

Ì ξ C

λ

ΘH (C)

&

ΘH (C) c ∈ C ξ(c) = v0 v1 v2 · · · vn−1 vn

¿¾¼

p = v0 v1 · · · vn−1 vn

v0 = 00 , c = (c1 c2 · · · cn ) p = ξ(c)

n

ci hi =

ci = λ(vi−1 , vi ) λ p n

(ci h1i , ci h2i , . . . , ci h(n−k)i ) = cH T .

i=1

i=1

n

vn = 0n .

cH T = 0;

c ∈ C

⊓ ⊔

!"#

H

C

[n, k]

w

n

sn

w

s w

H

$ !"%

&

'

n

ΘH (C)

&

( ) * + C [n, k], ΘH (C) = (V, E)

(

H

r = (r1 r2 . . . rn )

c ∈ C r &

n

ψ : E → N

-

.

ψr (e) = d(λ(vi , vi+1 ), ri+1 ),

(vi , vi+1 ) ∈ E

λ d(x, y)

d(x, y) ¿ d(x, y)

r v0 v1 · · · vn−1 vn

ΘH (C) lr (v0 v1 · · · vn−1 vn ) =

n−1

ψr (vi , vi+1 ).

i=0

r p = v0 v1 · · · vn−1 vn

ΘH (C) lr (p) = 0 r ∈ C

p

r = (r1 r2 · · · rn )

i < n λ(vi vi+1 ) = ri+1 .

ψr (λ(vi vi+1 ), ri+1 ) = d(ri+1 , ri+1 ) = 0.

0 ! p = v0 v1 · · · vn−1 vn ΘH (C) lr (v0 v1 · · · vn−1 vn ) = 0.

ψr (vi vi+1 ) = 0 i = 0, 1, . . . , n − 1 λ(vi vi+1 ) = ri+1 " p r ⊓ ⊔

# r

#

r

n lr) (p) p = v0 v1 · · · vn−1 vn

ΘH (C) c ∈ C p r d(r, c) = lr (p). 3

¿¾¾

000 001 010 011 100 101 110 111 0 1

2 3 4 5 6 7

r = (1001111) H3 (2)

ψr 1

(0001111) 1 r

r ∈ C r ∈ C = γ0 γ1 · · · γn g

ΘH (C) g lr ( g) =

n i=1

d(λ(γi−1 γi ), ri ) =

n

d(gi , ri ) = d(g, r).

i=1

r C ⊓ ⊔

! " d # $ r

% " & "

'

ΘH (C) = (V, E) λ(

ψ : E → N n

s ∈ Σ µj (s)

00 sn s

n

Bn (s)

00 sn sn s n 00 µn (s) = ∞

s t B(s, t) B(s, t) = λ(s, t) s t

B(s, t) µ0 (0) = 0 µ0 (s) = +∞ s = 0 B0 (0) = ∅ j = 1 s ∈ Σ t ∈ Σ µj−1 (t) + λ(tj−1 , sj )

µj (s) = µj−1 (t) + λ(tj−1 , sj ) Bj (s) = Bj−1 (t) ∗ B(t, s),

∗ j = n Bn (s) µn (s) j ← j + 1

!

"

C (n, m)#

$ F2 (A, B, C, D)

C Σ $

Σ

2m

m % & ' Θ(C) = (V, E) V = N × Σ;

$ (i, v) vi ( e ∈ E

vi wj

½º j = i + 1 u wj = vi A + uB;

e

c = uD

Θ(C) t + 1

! t t i i ui z G(z), ci z = i=0

i=0

G(z)

C

! t ri z i , R(z) = i=0

" ψ Θ(C) # $ %

&

'

0

R(z) ri

D D ( '

) *

Θ

+ & #

C(z) =

t

ci z i ,

i=0

Fn2 , & "

D

ui

(R, +, ·) R

R + · a, b, c ∈ R a · (b + c) = a · b + a · c;

(b + c) · a = b · a + c · a.

e ∈ R

a ∈ R a · e = e · a = a;

· e ab = 0

a = 0 b = 0 ! R⋆ = R \ {0} · " # R⋆

$ R + ab = 0 a, b ∈ R % (R, +, ·)

Zpq pq p, q > 1 pq 1

Z

Zp p Q R C

R a ∈ R

b ∈ R c ∈ R ac = b a ! " # $ b ∈ R ǫ ∈ R a = ǫb a R a % 0 Z +1 −1 4 ∈ Z 2 2 4

R

& R = {a1 , a2 , . . . , an } ' a ∈ R a = 0 aa1 , aa2 , . . . , aan .

( ! ) aai = aaj a(ai − aj ) = 0,

a = 0 ai − aj = 0 * R +

aai ! i % e = aai

R "" e = ai a * ai a ! R \ {0}

⊓ ⊔ , S R R

R

, J R

J R a ∈ J r ∈ R ar ∈ J ra ∈ J

R a ∈ R

R

a (a) (a) = {ra + na : r ∈ R, n ∈ Z}.

R (a) = {ra : r ∈ R}

J = R R

a ∈ R R = (a) a, b ∈ R ab ∈ R a ∈ R b ∈ R

M J ⊆ M M = R M = J J R

R

R/J

(a + J) + (b + J) = (a + b) + J;

(a + J)(b + J) = ab + J.

R S φ : R → S ker φ = {r ∈ R : φ(r) = 0S } ℑR ⊆ S R/ ker φ

J R ψ : R → R/J

ψ(a) = a + J R R/J J

R

R P R R/M ! M R

R/M " #!

R

P R ! R/P 1 + P = 0 + P (a + P )(b + P ) = (0 + P ) "" ab ∈ P

P

a ∈ P b ∈ P a + P = 0 + P b + P = 0 + P # R/P

¿¿¼

¾º

M

R

a ∈ M

a ∈ R

J = {ar + m : r ∈ R, m ∈ M } R

M J = R

r ∈ R m ∈ M ar +m = 1 (a+M )(r +M ) = (ar + M ) = (1 − m) + M = 1 + M R/M R/M J M ⊆ J J = M a ∈ J \ M (a + M )

(r + M )

(a + M )(r + M ) = (1 + M ) r ∈ R ar + m = 1 m ∈ M

ar, m ∈ J

1 ∈ J J = R M

⊓ ⊔

K

K

(0)

nr = 0

R

r∈R

R

R

R

!

n

n > 0

0

0

R = {0} 0

"

r∈R

e

n R ab

a, b > 1 R⋆

0 = ne = (ae)(be), ⊓ ⊔

"

p Zp = Z/(p) p

p

# $

'

Z3

[a]

%( )

(p)

⊓ ⊔

% &

3

a

· [0] [1] [2]

+ [0] [1] [2] [0] [0] [1] [2]

[0] [0] [0] [0]

[1] [1] [2] [0]

[1] [0] [1] [2]

[2] [2] [0] [1]

[2] [0] [2] [1]

Ì

Z3

p Fp {0, 1, . . . , p − 1} φ : Z/(p) → Fp φ([a]p ) = a a [a]p Fp φ p

F5 5

+ 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 0

2 2 3 4 0 1

3 3 4 0 1 2

· 0 1 2 3 4

4 4 0 1 2 3

0 0 0 0 0 0

1 0 1 2 3 4

2 0 2 4 1 3

3 0 3 1 4 2

4 0 4 3 2 1

F5

R p

a, b ∈ R n ∈ N n

n

n

n

(a + b)p = ap + bp ;

n

n

(a − b)p = ap − bp .

p p(p − 1) · · · (p − i + 1) ≡0 = i 1 · 2···i

(mod p).

p

p p−1 p p 2 (a + b) = a + a b + ··· + abp−1 + bp = ap + bp . 1 p−1

n ! n

n

n

n

ap = ((a − b) + b)p = (a − b)p + bp . ⊓ ⊔

¿¿¾

R R

a(x) =

n i=0

ai xi = a0 + a1 x + · · · + an xn ,

n i ai ∈ R x R

ai = 0

a(x) =

n

ai xi ,

b(x) =

bi xi

i=0

i=0

n

a(x) + b(x) =

n

(ai + bi )xi ;

i=0

a(x)b(x) =

n+m k=0

⎛

⎞

⎟ k ⎜ ⎜ ai b j ⎟ ⎠x . ⎝ i+j=k; 0≤i,j≤n

R x R

R[x]

R R[x] R R[x] R R[x] R f (x) = ni=0 ai xi R

an = 0

! an f a0 " # n f f (x) an = 1

$ deg(0) = −∞

Ì

f, g ∈ R[x] deg(f + g) ≤ max(deg f, deg g); deg(f g) ≤ deg f + deg g.

R

deg(f g) = deg f + deg g.

F F [x]

g = 0 F [x] f ∈ F [x] q, r ∈ F [x]

f = qr + g,

deg r < deg g.

F [x] J = (0) F [x] g ∈ F [x] J = (g) F [x]

J = (0) h(x) J b

g(x) = b−1 h(x) g ∈ J g f ∈ J q, r ∈ F [x] f = qg + r,

deg r < deg g.

!

J "" f − qg = r ∈ J # $% g

r = 0 J = (g) J J = (g) = (g1 ) g1 g = c1 g1 ,

g1 = c2 g,

c1 , c2 ∈ F [x] ! g = c1 c2 g c1 c2 = 1 c1 c2 " g g1 g = g1 ⊔ ⊓

f1, . . . , fn ∈ F [x] d ∈ F [x]

d fi c ∈ F [x] fi d

d

d = b1 f1 + · · · + bn fn ,

b1 , . . . , bn ∈ F [x]

p

p ∈ F [x]

F p = bc b, c ∈ F [x] b c

F [x]

f ∈ F [x]

f = ape11 · · · pekk ,

a ∈ F

p1 , . . . , pk

F [x]

e 1 , . . . , ek

F F [x]

F

f ∈ F [x]

F [x]/(f )

f

f (x) = x2 + x + 1 ∈ F2 [x] F2 [x]/(f )

4 [0] [1] [x] [x + 1]

+

[0]

[1]

[0]

[0]

[1]

[1]

[x]

[x]

[x + 1] [x + 1]

[0]

[1]

[x + 1]

·

[0]

[0]

[0]

[0]

[0]

[x]

[1]

[0]

[1]

[x]

[x + 1]

[x]

[0]

[x]

[x + 1]

[1]

[1]

[x]

[x]

[x + 1]

[1]

[x]

[0]

[x + 1]

[x + 1]

[0]

[1]

[x]

[1]

[0]

[x + 1] [0] [x + 1]

[x]

[x + 1]

Ì

F2/(f )

b ∈ F

f (b) = 0

x−b

f (x)

b∈F

f

f ∈ F [x]

∈ F [x]

f (x) = q(x)(x − b) + c

c ∈ F b x

c = 0 ⊓ ⊔ b f ∈ F [x] f (x) (x − b)k (x − b)k+1 k ≥ 1 k b k = 1 b ! k ≥ 2 b

Ì

f ∈ F [x] deg f = n ≥ 0 b1 , . . . , bm ∈ F

f k1 , . . . , kn (x − b1 )k1 (x − b2 )k2 · · · (x − bm )km f (x) k1 + · · · + km ≤ n f n F

f (x) = ni=0 ai xi F [x]

f f ′ (x) =

n

iai xi−1 .

i=1

f ∈ F [x] b ∈ F f f (x) f ′ (x) p p

k f ∈ F [x] f [k] (x) =

1 (k) f (x). k!

b ∈ F f ∈ F [x] k b f [i] (x) 0 ≤ i < k f [k] (x)

k ! xn ! n n−k . x k

F [x]

n ≥ 0 a0 , . . . , an n + 1 F b0 , . . . , bn ∈ F

f ∈ F [x] d ≤ n f (ai ) = bi i ! n n f (x) =

i=0

bi

k = 0 k = i

(ai − ak )−1 (x − ak ).

F K F F F K F

F

(0) F

Q

Fp

!"

# F

F

F Fp Q F p 0

K F M

F

K(M )

F K M K M M $ θ L = K(θ) K θ L K

K F θ ∈ F θ

$ % & K an , . . . a0 ∈ K an θn + an−1 θn−1 + · · · + a0 = 0,

θ K L K

K

'

K

K

K

C R ( i

$

x2 + 1 = 0.

Q

θ ∈ F K g ∈ K[x] J = {f ∈ K[x] : f (θ) = 0} θ K θ

θ ∈ F K g θ

g K f ∈ K[x] f (θ) = 0 g f g K[x] θ

θ K θ L K L

K K L

L K L K L K L K L K

[L : K] L K M L M K

[M : K] = [M : L][L : K].

! "

θ ∈ L

K K L K m = [L : K] m + 1 {1, θ, θ 2 , . . . , θ m }

a0 + a1 θ + · · · + am θm = 0,

ai ∈ K

θ

K

⊔ ⊓

! Q Q ! R Q # g K θ K K/(g) $ " K(θ)

θ ∈ F n K g K !

½º K(θ) K[x]/(g)

n−1 ) K(θ) K

[K(θ) : K] = n (1, θ, . . . , θ α ∈ K(θ) K K n

τ : K[x] → K(θ) τ (f ) = f (θ)

τ ker τ = (g) S τ K(θ) S ≃ K[x]/(g) S

θ K ! " K(θ) S = K(θ) #

S = K(θ) α ∈ K(θ) $

α = f (θ) f ∈ K[x] f = qg + r

q, r ∈ K[x] deg r < deg g = n ! α = f (θ) = q(θ)g(θ) + r(θ) = r(θ),

α

% 1, θ, . . . , θ n−1 & a0 + a1 θ + · · · + an−1 θn−1 = 0,

ai ∈ K h(x) = a0 + a1 x + · · · + an−1 xn−1

θ g

deg h < deg g %% h = 0 1, . . . , θ n−1 K(θ) K α %

K ' K(α)

K(θ) n = [K(θ) : K] = [K(θ) : K(α)][K(α) : K],

⊔ ⊓

K θ

F

Ì

f ∈ K[x]

K

α

K f

( L = K/(f ) ) [h] = h + (f )

h ∈ K[x] a ∈ K

[a]

ax0 ( K → L a → [a]

K

K ′ L '

L

K ' h(x) ∈ K[x] [h] = [a0 + a1 x + · · · + am xm ] = [a0 ] + [a1 ][x] + · · · + [am ][x]m .

' [a]

a

[h] = a0 + a1 [x] + · · · + am [x]m ,

L [x] K L K [x] f = b0 + b1 x + · · · + bn xn f ([x]) = b0 + b1 [x] + · · · + bn [x]n = [b0 + b1 x + · · · + bn xn ] = [f ] = [0],

[x] f

Ì

⊔ ⊓

α, β

F

K

ψ : K(α) → K(β)

f ∈ K[x]

F [x]

f

f ∈ K[x] α β

K

F

f

α1 , . . . , αn ∈ F

f (x) = a(x − α1 )(x − α2 ) · · · (x − αn ). F F = K(α1 , α2 , . . . , αn )

F

f

f

K

f

F

f K

K

K

f

K

f ∈ K[x]

f

K

f

K

K ! f ∈ K[x] n ≥ 2 f (x) = a0 (x − α1 )(x − α2 ) · · · (x − αn ) K D(f ) f D(f ) = a02n−2

Y

1≤i<j≤n

(αi − αj )2 .

" D(f ) = 0 f (x) #

$ D(f ) ∈ K

! f (x) = a0xn + a1 xn−1 + · · · + an g(x) = b0 xm + m−1 + · · · + bm K[x] % n m b1 x R(f, g) f g m + n

˛ ˛ a0 ˛ ˛ 0 ˛ ˛ ˛ ˛ ˛ ˛ 0 R(f, g) = ˛ ˛ b0 ˛ ˛ 0 ˛ ˛ ˛ ˛ ˛ 0

˛ a1 · · · an 0 · · · 0 ˛˛ a0 a1 · · · an 0 · · · 0 ˛˛ ˛˛ ˛ ˛ · · · 0 a0 a1 · · · an ˛ ˛, b1 · · · bm 0 · · · 0 ˛ ˛ bm · · · 0 ˛ b0 b1 · · · ˛ ˛ ˛˛ · · · 0 b0 b1 · · · bm ˛

m ai n bi

R(f, g) ∈ K f (x) = a0 (x − α1 )(x − α2 ) · · · (x − αn )

K R(f, g)

R(f, g) = am 0

n Y

g(αi ),

i=1

R(f, g) = 0 f g

K[x]

f

′ D(f ) = (−1)n(n−1)/2 a−1 0 R(f, f ),

f ′

n − 1 ! "!℄

Ì

F

F pn p

F n F

F p

K Fp F [F : K] = n K F pn ⊓ ⊔

aq = a

F q

a ∈ F

a = 0 F q − 1 ⊓ ⊔ q q

Ì

F q K F

xq − x ∈ K[x] F [x] (x − a), xq − x = a∈F

F xq − x K

p n

pn q = pn xq − x Fp

xq − x Fp [x] F Fp q F qxq−1 − 1 Fp [x] −1

S = { a ∈ F : aq − a = 0} S

F S xq − x xq − x S

S = F • F q = pn

F p F Fp

F xq − x Fp ! F

⊓ ⊔ •

q = pn

q = pn

q

"! #

$

% Fq % &

%

Fq

Fq

Fq q = pn

Fq pm m n m n

Fq pm !" Fp30

p

(

)

F⋆q

Fpi

Fp30

30

p30 ' 1, 2, 3, 5, 6, 10, 15, 30 "

* +

Fq

# F⋆q Fq

q − 1

Fp30

F p6

Fp10

Fp15

F p2

F p3

F p5

Fp

Fp 30

Fq q − 1 0 ∈ Frq r q = 2 q ≥ 3 h = p1 p2 · · · prm ! h = q − 1 " 1 ≤ i ≤ m fi (x) = xh/p − 1 # h/pi Fq

h/pi < h Fq 0

fi ai bi = ah/p $ i r p bi = 1 bi pi ! psi 0 ≤ si ≤ ri % 1

2

m

i

ri i

ri

i

i

r −1 p i

bi i

h/pi

= ai

= 1,

bi pri b = b1 b2 · · · bm h ! ! b h m h/pi h/p1 i

h/p1 h/p1 b2

1 = bh/p1 = b1

m · · · bh/p . m

% 2 ≤ i ≤ m pri h/p1 bh/p = 1 & i h/p = 1 b1 h/p1 b1 ⊔ ⊓ b1 pr1 & b F⋆q 1

i

1

1

F

Fq

⋆ q

Fq ϕ(q − 1) ϕ

Ì

Fq

Fq

Fr

Fr

Fq [x]

Fq

n

n

Fr Fq q n [Fr : Fq ] = n ζ ∈ Fr Fr = Fq (ζ) ⊓ ⊔ ζ n Fq

F9 z z 4 ∈ F3 F⋆3

! 2" z 8 = 1 # F9

F9 z

$ %

+ 0 1 z z2 z3 2 z5 z6 z7

0 0 1 z z2 z3 2 z5 z6 z7

1 1 2 z2 z7 z6 0 z3 z5 z

z z z2 z5 z3 1 z7 0 2 z6

z2 z2 z7 z3 z6 2 z 1 0 z5

z3 z3 z6 1 2 z7 z5 z2 z 0

2 2 0 z7 z z5 1 z6 z3 z2

z5 z5 z3 0 1 z2 z6 z z7 2

z6 z6 z5 2 0 z z3 z7 z2 1

z7 z7 z z6 z5 0 z2 2 1 z3

· 0 1 z z2 z3 2 z5 z6 z7

0 0 0 0 0 0 0 0 0 0

1 0 1 z z2 z3 2 z5 z6 z7

z 0 z z2 z3 2 z5 z6 z7 1

z2 0 z2 z3 2 z5 z6 z7 1 z

z3 0 z3 2 z5 z6 z7 1 z z2

2 0 2 z5 z6 z7 1 z z2 z3

z5 0 z5 z6 z7 1 z z2 z3 z4

z6 0 z6 z7 1 z z2 z3 2 z5

z7 0 z7 1 z z2 z3 2 z5 z6

Ì F9

! Fq f ∈ Fq [x] & F = Fq [x]/(f ) ! ' ( ' deg f = m F ≃ Fqm

Ì

f ∈ Fq [x] m f

α Fqm f Fqm 2

α, αq , αq , . . . , αq

m−1

.

f ∈ Fq [x] Fq α

Fq f

h ∈ Fq [x] h(α) = 0 a

f g(x) = a−1 f (x) α Fq

⊔ ⊓

n f ∈ Fq [x] Fq m f (x)

xq − x m

n

f (x) xq −nx α n

Fq αq = α α ∈ Fqn ! Fq (α)

Fqn "

[Fq (α) : Fq ] = m n # m n Fqn Fqm

α f Fq [Fq (α) : Fq ] = m n α ∈ Fqm $ %% α ∈ Fqn αq = α n α xq − x & f (x) n xq − x ⊔ ⊓

α f ' Fq %% [Fq (α) : Fq ] = m ( ) Fq (α) = Fqm α ∈ Fqm * f (β) = 0 f (β q ) = 0 f (x) = am xm + am−1 xm−1 + · · · + a1 x + a0 ai ∈ Fq 0 ≤ i ≤ m f (β q ) = am β m + am−1 β m−1 + · · · + a1 β + a0 = aqm β qm + · · · + aq1 β q + aq0 m−1

= (am β m + · · · + a1 β + a0 )q = f (β)q = 0.

! α, αq , . . . , αq

f $

+ j k αq = αq j, k 0 ≤ j < k ≤ m − 1

q m−k

αq

m−k+j

= αq

m

= α. m−k+j

$ , - f (x) %% xq − 1 m m − k + j .( m − k + j < m ⊔ ⊓

f Fq [x] m

f Fqm

Fq [x]

Fqm Fq σ Fqm Fq

Fqm Fq Fqm Fq

Gal(Fqm : Fq ) Fqm Fq

Fqm Fq α ∈ Fqm 2 m−1

α Fq α, αq , αq , . . . , αq

σj :

Fq m → Fq m j α → αq

Fqn ! Fqn " #$ β ∈ Fq i

σi (β) = β q = β,

σi Gal(Fqm : Fq ) [Fqm : Fq ] α # Fqm Fq 0 ≤ i < j ≤ m − 1 σi (α) = σj (α) σi

0 ≤ i ≤ m σi σj = σi+j ;

σ1i = σi ,

σi

Fqm : Fq

m

σ0 , σ1 , . . . , σm−1

σ Fqm Fq % β # Fqm f (x) = xm + am−1 xm−1 + · · · + a0 ∈ Fq [x] Fq

0 = σ(β m + am−1 β m−1 + · · · + a0 ) = σ(β)m + am−1 σ(β)m−1 + · · · + a0 ,

σ(β) f Fqm &' σ(β) = β q j 0 ≤ j ≤ m − 1

σ j σ(α) = αq α ∈ Fqm σ = σj # σ1 Gal(Fqm : Fq ) j

⊓ ⊔

K = Fq F = Fq

F

m K

F K

F α ∈ F F K m

{1, α, α2 , . . . , αm−1 }

K

F α ∈ F F K ! F K {1, α, α2 , . . . , αm−1 } F K α ∈ F

α K " K TrF /K (α) = α + αq + · · · + αq

m−1

.

K

F TrF /K (α)

TrF /K : F → K TrF /K (α + β) = TrF /K (α) + TrF /K (β) α, β ∈ F TrF /K (cα) = cTrF /K (α) c ∈ K α ∈ F TrF /K F K F K K

TrF /K (a) = ma a ∈ K ! TrF /K (αq ) = TrF /K (α) α ∈ F

!

# F K

" F # $ $ K % K

F K %

Lβ β ∈ F Lβ (α) = TrF /K (βα) α ∈ F & β = γ Lβ = Lγ

" K $ F #% $ K E $ F '

TrE/K (α) = TrF /K (TrE/F (α)).

α ∈ F N

F /K (α)

NF /K = α · αq · · · · · αq

m−1

= α(q

m

α K

−1)/(q−1)

.

g(x) = xm + am−1 xm−1 + · · · + a0

α K TrF /K (α) = −am−1 ;

NF /K (α) = (−1)m a0 .

Ì NF /K : F → K

NF /K (αβ) = NF /K (α)NF /K (β) α, β ∈ F NF /K F K F ⋆ K ⋆ NF /K (a) = am a ∈ K NF /K (aq ) = NF /K (a) a ∈ F

K F K E F

α ∈ E NE/K (α) = NF /K (NE/F (α)).

K

A

A = {α1 , . . . , αm } B = {β1 , . . . , βm } F 1 ≤ i, j ≤ m j 0 i = j TrF /K (αi βj ) = 1 i = j

B

ψ1 , . . . , ψm G F ⋆ ! F

a1 , . . . , am ∈ F

g ∈ G " a1 ψ1 (g) + a2 ψ2 (g) + · · · + am ψm (g) = 0.

T

f (x) = an xn + · · · + a1 x + a0

T

n

an T + · · · + a1 T + a0 I = 0,

I

0

!

" #

T! g(x)

T

T

g(x) = det(xI − T );

T

!

α V!

$ #

%

T

T

V

T kα

k = 0, 1, . . .

&

' '

T

q

q m−1

{α, α , . . . , α #

K = Fq

}' F K

F = Fqm ( F K ) α ∈ F #

*

Ì

K F F K

K = Fq F = Fq

0 2 m−1 m

F K

1 = σ , σ, σ , . . . , σ

F

K ! σ m = 1

xm − 1 ∈ K[x]

σ σ i

F ⋆ " #$%&

m '

σ

xm − 1 (

σ )*

σ ( m α ∈ F (

σ i α, σ(α), σ 2 (α), . . .

F K + σ i (α) = αq

α, αq , . . . , αq

m−1

F ,

-

⊔ ⊓

. $ ' /

F

Fqm Fq (

α ∈ Fq A = {α, αq , αq , . . . , αq } Fq Fq

xm −1 αxm−1 +αq xm−2 + · · · + αq x + αq Fq [x] 2

m−1

m

m−2

m m−1

m

n F xn −1 n F F(n) xn − 1 F(n) n (n) E

E (n)

n F

p p n E (n) n p n n = mpe m, e p m F(n) = F(m) E (n) = E (m) xn − 1 F(n)

E (m) pe

p

F

p

n

F p n p E (n) n n φ(n) φ

ζ n F Qn (x) =

n

(x − ζ s )

s=1,gcd(s,n)=1

n

F

Qn (x)

ζ

F p n p

xn − 1 =

/

d|n

Qd (x)

Qn (x) F F Qn (x) ∈ Z[x]

! " n F d d n ζ n F ζ s E (n) d = n/ gcd(s, n) d ζ s E (n)

xn − 1 =

n

s=1

(1 − ζ s ),

(x − ζ s ) ζ s d # $

n "

Qn (x) % n = 1 Q1 = x − 1 n > 1

Qd (x) 1 ≤ d < n & /

Qn (x) = (xn − 1)/f (x),

f (x) = d|n,d
( F Z ) p = 0 xn − 1 f (x) ( Qn (x) * Z+ ⊓ ⊔

q n i q

n

i

Ci = {i, iq, iq 2 , . . . , iq t−1 },

n

t

iq t ≡ i (mod n)

α

n αi

d =

n/ gcd(n, i)

2

αiq , αiq , . . .

d

iq t ≡ i (mod n)

!

t≤t

αiq

t

mαi (x) =

t−1

j=0

Ci

t

" #

n

f (x) = x − 1

j

(x − αiq )

! !

q

f (x) = xn − 1

n

" $

Fq

Fq f (x) =

xn − 1 q n

" f (x) = x15 −1 Z2 %

2

15

C0 = {0}

C1 = {1, 2, 4, 8} C3 = {3, 6, 12, 9}

C5 = {5, 10} C7 = {7, 14, 13, 11}.

&'

f (x)

(

) * $ + * $

! "#$℄ "&'℄ "()℄ "*+#℄ "'#℄ "',℄ ")'℄ "-$℄ "#)℄ "#&℄ .

/ *,

0 "$,℄ "##℄ "*++℄ "$(℄ /

*,

0 /

F

AG (n, F) n F S = (P, L) W = Fn

L = { x + L : x ∈ W, L ≤ W, dim L = 1},

x + L x + L = { x + y ∈ W : y ∈ L}.

AG (n, q)

Fq 2 − (q n , q, 1)

AG (n, F)

F⋆q = Fq \ {0} V = Fn+1

n + 1 F V ⋆ = V \ {0}

V ⋆ x, y ∈ V x ∼ y ⇔ ∃λ ∈ F⋆ : x = λy.

x F⋆ x V

∼ VF

W V ⋆

⋆

π:

W ⋆ → V ⋆ /F⋆q v → F⋆ v.

PG (n, F) n F !

S = (P, L) V ⋆ /F⋆

L ! π

2" V

L = { π(W ⋆ ) : W ≤ V, dim W = 2}.

# PG (n, q) PG (n, Fq )

2

! "

2 PG (2, q)

q ≥ 9

θq =

q n+1 − 1 = q n + q n−1 + · · · + q + 1. q−1

$ p ∈ PG (n, F) p = Fp

p ∈ V ! 1 % AG (n, F) PG (n, F)

ξ p = (p1 , p2 , . . . , pn ) ∈ AG (n, F)

p = F⋆ (1, p1 , p2 , . . . , pn ) ∈ PG (n, F).

AG (n, F)

PG (n, F)

x

ξ

L

AG (n, q)

x∈L

ξ(x) ∈ ξ(L).

PG (n, F)

= (T0 , T1 , . . . , Tn ) T F[T] n n+1 T1 , T2 , . . . , Tn F[T] T 0 , T 1 , . . . , Tn

T = (T1 , T2 , . . . , Tn )

V ⊆ AG (n, F)

F = {f1 (T), f2 (T), . . . , fs (T)} ⊆ F[T]

V = {x ∈ AG (n, q) : f1 (x) = f2 (x) = . . . = fs (x) = 0}.

V = V (F). V ⊆ PG (n, q)

g2 (T), . . . gl (T)} ⊆ F[T] G = {g1 (T),

V = { F⋆ x ∈ PG (n, F) : g1 (x) = g2 (x) = . . . = gs (x) = 0}.

V

V

ξ

j

(fi )j (T)

f1 , f2 , . . . , fs

!

deg fi

Ve

e = fei (T)

X

T0deg fi −j (fi )j (T),

i=0

fi (T)

Ve

V

S ⊆ PG (n, q) S X ∩ S X

ξ AG (n, q)

AG (n, q)

V

X

V

V1 , V2 ⊆ V V = V1 ∪ V2

PG (n, q)

V V

X X n = dim X ! n

X = X0 ⊂ X1 ⊂ . . . ⊂ Xn ⊂ ∅,

Xi Xi−1 i = 1, 2, . . . , n

" dim AG (n, F) = dim PG (n, F) = n # p ∈ AG (n, F)

dim p = 1,

$ $

X

& % F, G ∈ Fq [T] G(x) = 0 x ∈ X F/G

X

$ F/G F ′ /G′ (F ′ G − F G′ ) = 0 '(")

x ∈ X X G '(")

(F ′ G − F G′ ) ∈ I(G), G

G

F[T]

X

F(X )

f ∈ F(X ) f = F/G G(x) = 0

X

x∈X

f

f (x) = F (x)/G(x) ∈ F

x

F(X ) = F(Y)

X

Y

X

F(X )

p

p∈X

p

Op

Op

mp

mp = { f ∈ Op : f (p) = 0}. !

Op /mp

PG (n, q) F (x, y, z) = 0

PG (2, q)

"

#

Fq

F

"

$

F

" #

X

#

p∈X

&

p∈X

mp

p

X

p X tp mp % p

V " X

0

X

1

#

D X div(X ) X

np p, D= p∈X

np ∈ Z np = 0 p

D p np = 0 Supp (D) = { p : np = 0};

D deg D =

np .

p∈Supp (D)

np ≥ 0 p ∈ Supp (D)

div(X ) X D D′ X D ≥ D′ D − D′ 0 div(X ) div0 (X ) ! " p ∈ X # tp φ(t) ∈ Op ∞ a i ti . φ(t) =

i=r

i ai = 0 φ p ordp (φ)

f = φ/ϑ ∈ F(X )

p

ordp (f ) = ordp (φ) − ordp (θ).

ordp (φ) f p ordp (θ)

f ∈ K(X )

(f ) =

ordp (f )p.

p∈X

deg(f ) = 0.

D L(D) = { φ ∈ K(V) : div(φ) + D ≥ 0}.

L(D) F ℓ(D) = dim L(D)

Ì

X

g ≥ 0 D ℓ(D) ≥ deg D + 1 − g;

g

X deg D > 2g − 2

ℓ(D) = deg D + 1 − g.

X ! !

X g

Fq Ni X Fqi ! |Ni − (1 + q i )| ≤ 2g q i .

¾º½

E, A, I, O, N, L, R, T, S, C, D, U, M, P, V, G, H, F, Q, B, Z, X, W, J, K, Y. 0.502

A0 = {E, A, I, O, N},

A1 = {L, R, T, S, C, D, U, M, P, V, G, H, F, Q, B, Z, X, W, J, K, Y}.

!

E A I O N L R T S C D U M

000 001 0.2388 0100 0101 0110 1000 1001 1010 1011 1100 0.5466 11010 11011 11100 0.1013

P V G H F Q B Z X W J K Y

111000 111001 111100 111101 1111100 1111101 11111100 11111110 111111110 11111111100 11111111101 11111111110 11111111111

0.0811 0.0204 0.0117 0.0002

0.000

Ì

¾º¾

n = 4.134.

p H(p) =

pi log

i

1 . pi

H(p) = 2.77.

! !

"

3# ! $ % & '( $ )*+℄ - .

/

26 0 ! t ≥ log(26)/ log(2) = 4.7;

5 - ! 5

/ 26 ! 1 ¾º¿ -

2 $

3 4 0 !

# 000 - !

! 1 0 ! ∗! & 5 4 ! 1 - 6

! 1

+! 9.68 8.98 0 1 !

3.54 3.42 - ! 26 /

3 !

2 7

! 1 .

/ !

A B C D E F G H I J K L M

8 12 14 10 4 12 12 10 11 16 12 12 12

3 5 5 4 2 5 4 5 3 5 4 5 3

N O P Q R S T U V W X Y Z

8 14 14 16 10 8 6 10 12 12 14 16 14

3 4 5 6 4 4 2 4 5 5 5 5 5

Ì

¿º½ σ0 = (acb),

σ1 = (acb),

σ2 = (bca).

C ′ = { (σ0 (x)σ1 (y)σ2 (z)) : (xyz) ∈ C}

C ′ = {(ccc), (caa), (bac)}.

¿º¾ !! M < 26−4+1 = 8.

" # $ M

1 6 i=0

i

≤ 26 .

ËÚ M ≤ 9

6 4 4

¿º¿ C (t) t C n A tn C (t) M k M ci C (t)

C c = c1 c2 · · · ct

c′ = c′1 c′2 · · · c′t

C (t)

d(c, c′ ) =

t

d(ci , c′i )

i=1

! x, y ∈ C a = x x · · · x ; k−1

Ú

b = y x · · · x k−1

C " a b d(x, y) # C (t) (tn, M t , d) (t)

r

$ % &' % (

) % & 8 " 1 r' B1 (r) 28

B2 (r) ) ) % ( 4 "

d(r, (1001 0100)) = 4 d(r, (0111 0011)) = 2

d(r, (1000 1011)) = 3 d(r, (0110 1100)) = 7

* c = (0111 0011).

% ( + C

28

H 8 n C 8 H 4 k = 8 − 4 C [8, 4] H j = (1111 1111)

C C i 8 − i C 4 3 H 3 !! d ≥ 4 d = 4 "

# AC (Z) = 1 + 14Z 4 + Z 8 .

$

% C & ⎛

01 ⎜1 0 ⎜ ⎜1 1 ⎜ ⎜1 1 ⎜ ⎜1 0 H=⎜ ⎜0 1 ⎜ ⎜1 1 ⎜ ⎜0 0 ⎜ ⎝1 1 11

10 10 10 01 11 11 11 11 10 10

10 01 00 00 00 00 00 00 00 00

000 000 100 010 001 000 000 000 100 100

00 00 00 00 00 10 01 00 00 00

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟. 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ 0⎠ 0

rH T = (0100 0100),

'

C 256 16 ( )

*! ei 1 %

#

ei +

*! ' 2

1

2 e = e6 + e10 .

c = r + e = (0001 0001 1111)

i 1 2 3 4 5 6

eH T 0111 1010 1011 0110 1110 1111 0001 1111 1000 0000 0100 0000

e 7 8 9 10 11 12

eH T 0010 0000 0001 0000 0000 1000 0000 0100 0000 0010 0000 0001

Ì 1

2 r

k ≤ n − d + 1 = 8 k ≤ n − log2 (120) 6 k ≤ 5 !"#$ k ≥ 4 212−4 = 256 > V2 (11, 3) = 232,

212−5 = 128 % [12, 4, 6]# [12, 5]# " 4

g(x) (x10 −1) 4

C F3

10 10 − 4 = 6

g(x)(x − 1) = x5 − 1;

C 2 d ≤ 2 &' C

1 "" " 10 C [10, 5, 2]

( m m(x) = x + x2 ;

) m c(x) = m(x)g(x) = x11 + x9 + x7 + x5 + x3 + x,

c = (0 1 0 1 0 1 0 1 0 1 0 1).

h(x) = (x12 − 1)/g(x) = x3 + x2 + x + 1,

r

r(x) = x + x2 + x5 + x7 + x9 + x10 .

r(x) h(x) s(x) = x2 +1

r

w(s(x)) = 2

e(x) = s(x) !

r(x) − s(x) = 1 + x + x5 + x7 + x9 + x10 .

g(x) x9 − 1 F2 [x]

" [9, 7]# ! F2 $ g(x) % F2 [x]

& ' "

! ( C2

)

g(x)

F4 [x] t

g(x) = (x + ω)(x + ω 2 ),

ω

F4

C4

(x + ω) (x + ω 2 )

C h(x) = (x15 − 1)/g(x) = x11 + x8 + x7 + x5 + x3 + x2 + x + 1.

*

1 = (x8 + x2 + 1)g(x) + xh(x).

C (x8 + x2 + 1)g(x) = x12 + x9 + x8 + x6 + x4 + x3 + x2 + x + 1.

+ 63

g(x) C 6 (x63 − 1)

% F2 x63 − 1

x + 1, x3 + x2 + 1 x6 + x4 + x2 + x + 1 x6 + x5 + x2 + x + 1 x6 + x5 + x4 + x2 + 1.

x2 + x + 1, x6 + x + 1 x6 + x4 + x3 + x + 1 x6 + x5 + x3 + x2 + 1

x3 + x + 1 x6 + x3 + 1 x6 + x5 + 1 x6 + x5 + x4 + x + 1

[63, 57] 9 6

(x63 − 1)

(x + 1)(x2 + x + 1)(x3 + x + 1),

(x + 1)(x2 + x + 1)(x3 + x2 + 1),

(x3 + x + 1)(x3 + x2 + 1). 12

α 31

x5 + x2 + 1 = 0.

g(x) F2

g(x) = (x + 1)(x5 + x4 + x3 + x2 + 1)(x5 + x4 + x3 + x + 1).

C !

D′ = {1, α3 , α11 }.

α3 F2 ! x5 + x4 + x3 + x2 + 1

α11 x5 +x4 +x3 +x+1"

C #

g(x)$

g(x) 11 ! F32 %

&

D = {1, α3 , α6 , α11 , α12 , α13 , α17 , α21 , α22 , α24 , α26 }. % '

g(x) =

ω∈D

(x − ω)

g(x) !

D !

11000001011 10000010111 101111 1111000001 11100000101

1 2 8 10 11

− − (3, 4, 5, 6, 7) (9) −

Ì

v

v

5

5

xi 2 xi (1 + x) 0 ≤ i ≤

13

1 5

0 x0 x1 x2 x3 x4 x5 x6 x7 x8

000000 100000 010000 001000 000100 000010 000001 100100 010010 001001

x0 (x + 1) x1 (x + 1) x2 (x + 1) x3 (x + 1) x4 (x + 1) x5 (x + 1) x6 (x + 1) x7 (x + 1) x8 (x + 1)

110000 011000 001100 000110 000011 100101 110010 011001 101100

r

r(x) = 1 + x + x2 + x3 + x6 . r(x) g(x) s(x) = x2 + x 011000 !

x2 (x + 1)

c(x) = r(x) + x4 (x + 1) = 1 + x3 + x6 .

"

n = 35,

b = 3,

n − 2b + 1 − m = 27,

m = 3 C (x5 − 1)f (x),

f (x) 3 n0 = 7 f (x) F8 f (x) = x3 + x + 1.

C g(x) = x8 + x6 + x5 + x3 + x + 1.

4

r R(x) r

r(x) r! r(x) = 1 + x2 + x6 + x7 + x8 + x9 + x10 + x11 + x13 + x14

R(x) =

14

r(ω i )xi .

i=0

" #

R(x) = ω 10 x5 + ω 5 x7 + ω 5 x10 + ω 10 x11 + ω 5 x13 + ω 10 x14 .

# R(x) $ R(x) = ω 10 x5 (x − ω 3 )(x − ω 5 )5 (x − ω 11 )(x − ω 12 )(x − ω 14 );

# 5 15 % R & ' ()#

R(x) n − w(r) = 15 − 10 = 5

" * 31#

31 % ω % F32 % x5 + x2 + 1 = 0.

" #

+

i Ci 1 {1, 2, 4, 8, 16} m1 (x) = x5 + x2 + 1 3 {3, 6, 12, 17, 24} m3 (x) = x5 + x4 + x3 + x2 + 1 {5, 9, 10, 18, 20} m5 (x)x5 + x4 + x2 + x + 1 5 7 {7, 14, 19, 25, 28} m7 (x) = x5 + x3 + x2 + x + 1 11 {11, 13, 21, 22, 26} m11 (x) = x5 + x4 + x3 + x + 1 m15 (x) = x5 + x3 + 1 15 {15, 23, 27, 29, 30}

Ì 2 31 ω, ω 2 , . . . , ω 10 .

ÍÒ

2

31

g(x) = m1 (x)m3 (x)m5 (x)m7 (x) = x20 + x18 + x17 + x13 + x10 + x9 + x7 + x6 + x4 + x2 + 1.

g(x) 11

11 g(x) ℄ "

℄ "# $% & '

BCH5 (24, 5) α ∈ F25 #

$ x2 − x + 2 = 0.

% 24&

$ '

α1 , α2 , . . . , α4 .

( %

)

5

24 g(x) = m1 (x)m2 (x)m3 (x)m4 (x) = x8 + x7 − x5 + 2x3 + x − 1.

24 − 8 = 16

m1 (x) α5

i 1 2 3 4 6 7 8

Ci {1, 5} {2, 10} {3, 15} {4, 20} {6} {7, 11} {8, 16}

m1 (x) = x2 − x + 2 m2 (x) = x2 + 3x − 1 m3 (x) = x2 + 3 m4 (x) = x2 − x + 1 m6 (x) = x + 3 m7 (x) = x2 + 3x + 3 m8 (x) = x2 + x + 1

i 9 12 13 14 18 19

Ci {9, 21} m9 (x) = x2 + 2 {12} m12 (x) = x + 1 {13, 17} m13 (x) = x2 + x + 2 {14, 22} m14 (x) = x2 + x2 − 1 {18} m18 (x) = x + 2 {19, 23} m19 (x) = x2 + 2x + 3

Ì

5 24

g(x) BCH5 (24, 6) BCH5 (24, 5) = BCH5 (24, 6) d ≥ 6 g(x) 6 11 t = 5 r(x) r r(x) = 1 + x + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x11 + x13 + x15 + x18 + x21 + x22 + x24 + x25 + x26 + x27 .

S(x) r(x) S(x) = ω 17 x8 + ω 18 x7 + ω 10 x6 + ω 24 x4 + ω 5 x3 + ω 12 x2 + ω 6 x + 1.

Euclid(x10 , S(x), 5, 4) v(x) = ω 28 x3 + ω 29 x2 + ω 6 x + 1,

r(x) = ω 29 x2 + 1.

v(x) ω 2 , ω4 , ω28 r29 r27 r3

F2

c = (111 1 11 111101 0101001 0011011 1 00 1 0).

!"℄ $ % &

' ()

& r r(x) ℄ ℄ !℄ "#!"$!"%!"&!""!"!'!%!! !(!'!#!$!%!&!"!!)"

℄ !"#$% !"$#% !" &% !"!''% !""(% !"!!% !"&% ! σ ω )

+ ) ! !"!$(% !"!,!% !"&% ! !"!,!% ! ℄ σ(x)

- -+./0 + ℄ !"! !"' !"!$ ℄ 1- 1 22) 3% !, !# ( ℄ r r3 r29 r27

F16 11 ω F16

x4 + x + 1 = 0 F2 ! BRS (15, 11) g(x) =

/10

i=1 (x − 1 5

ωi) =

x10 + ω 2 x9 + ω 3 x8 + ω 9 x7 + ω 6 x6 + ω 4x + ω 2 x4 + ωx3 + ω 6 x2 + ωx + ω 10 .

" ! RS (15, 5)

!! m m(x) = 1 + ωx + ω 2 x2 + ωx4 .

# t ti = m(ω i ) t = (0 ω 12 ω 9 ω 2 ω 11 ω 2 ω 12 ω 7 ω 7 ω 9 ω 8 1 0 ω 11 ω 8 ).

" g(x) 14

t(x) = i=0 ti t(x)/g(x) = ω 8 x4 + ω 14 x3 + ω 6 x2 + ω 2 x.

BRS (15, 11)

RS (15, 5)

c = (c0 . . . c14 ) c(x)

i

ci v i

vi =

1 c(ω 15−i ). 15

v = (0 ω 4 ω 2 ω 4 ω 11 0 0 0 0 0 0 0 0 0 0).

t = (0 ω 4 ω 2 ω 4 ω 11 ).

R k RS (8, 6) F9

3 3/4 RS (80, 60) F81

21

n

2 ∈ F11

t = 2 " #

5!

$ % & ' ('

' '

N (x)

E(x)

deg E(x) = 2! E(x) ! deg N (x) ≤ 7! N (2i ) − yi E(2i ) = 0

"((

E(x) = E0 + E1 x + x2 ,

N (x) =

7

Ni xi .

i=0

10

10

N (x) = 5x7 + 4x6 + 2x5 + 9x4 + 2x3 + 3x + 7 E(x) = x2 + 7

p(x) = N (x)/E(x) = 1 + 2x + 3x2 + 4x4 + 5x5 ,

)

c = (4 10 8 0 6 1 1 4 0 9 4),

ÙÒ Ú

2 r

2

! "

#

$%&℄ ( !

Fn+1 ) P k " *

℄ " #$ ℄% " %℄& '&%

) "

( )

$*+ , +-. / + $* $ + ℄% " 0 ℄ " '&% ℄ " '&%&,%℄ / 1

$ -. ℄% ",%℄& '%&

+-.℄

, * #() , ()$ ()$ ( )

$*+ ,

# + 1 * + ()$ ℄ ()$ 2℄

+ !

N (x) E(x) 3 4 - 5

℄ ℄! " # $!℄ ℄! " " #℄ % ' ( )* + ,- ( )* + . + ℄/ . + $℄ 0 1)*+ )''

2'3 4 . + )* + 5677 7 4 8 4 10 8 9 6 :!;

7* (<:=:><:?@℄!; ": ; "$ :!; ; "A :!; ; "? ; ": ; ": ; "$ :!; ; "> ; "@ ℄ 7* B% :>$( ; ": ; ; "= :!; ; "$ ; "< ℄ 7* )# B: ; "$ ; "C ; "A :!; ; "? ; ": ; ": ; "$ :!; ; "> ; "$ ℄ 7* (D . 7 E74E 7* )1 ) < : = : > < : ? < ℄

74

½¼º½ C = BCH(15, 7) g(x) = x10 + x8 + x5 + x4 + x2 + x + 1.

1 7 14

C ′ C 1 7 14 4

1!

⎛ ⎞ 111 ⎜0 1 1 ⎜ G=⎜ ⎜0 0 1 ⎝0 0 0 000

01 10 11 11 01

10 11 01 10 11

010 001 100 110 011

10 01 10 01 00

C ⎛

11 ⎜0 1 ⎜ G′ = ⎜ ⎜0 1 ⎝0 0 00

01 10 11 11 01

101 110 010 101 111

01 10 01 00 10

000 0 0 0⎟ ⎟ 1 0 0⎟ ⎟ 0 1 0⎠ 101

⎞ 000 1 0 0⎟ ⎟ 0 1 0⎟ ⎟ 1 0 1⎠ 010

C ′ "

#

C ′ ⎛ ⎞

110 ⎜1 0 1 ⎜ ⎜1 0 1 ⎜ H′ = ⎜ ⎜0 0 1 ⎜1 0 0 ⎜ ⎝0 1 1 111

11 00 10 10 10 10 00

00 11 00 10 10 00 00

000 000 100 010 001 000 000

00 0 0⎟ ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟. 0 0⎟ ⎟ 1 0⎠ 01

r′ = (01 111 01 110 00),

r $ r′

H ′ s′ = (101 111 0);

% s′ H ′

C ′

e = (00 010 00 000 00).

r′′ = r + (000 010 000 000 000) = (0?1 101 0?1 110 00?).

r′′

#

f = (0f1 0 000 0f70 000 00f14)

(f + r′′ )H T = 0, ⎛

1 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 H =⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

011 101 010 101 010 001 000 000 000 000

01 10 01 00 00 00 00 00 00 00

00 10 11 01 10 01 10 01 00 00

000 000 000 100 110 011 101 010 101 010 C

⎧ ⎪ ⎨f 1 = 0 f7 + 1 = 0 ⎪ ⎩ f14 + 1 = 0,

00 00 00 00 00 00 10 11 01 10

⎞ 00 0 0⎟ ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟ 1 0⎠ 11

r = (001 101 011 110 001).

½¼º¾

I0 = {2, 6, 9}.

σ0 (x) = (1 − α2 x)(1 − α6 x)(1 − α9 x) = α2 x3 + α9 x2 + αx + 1.

r′ (x) r

0

r(x) = α13 + x + α10 x3 + α12 x4 + α6 x5 + α5 x7 + α13 x8 + αx10 + α8 x11 + α7 x12 + α2 x13 + α9 x14 .

S(x) = 1 + α10 x + α6 x2 + α5 x3 + x4 + α12 x5 + α6 x6 + α9 x7

r = 15 − 7

S0 (x)

S(x)σ0 (x)

xr

S0 (x) = α14 x7 + x5 + α4 x4 + α9 x3 + α3 x2 + α8 x + 1.

µ = ⌊(8−3)/2⌋ = 2 ν = ⌈(8+3)/2⌉−1 = 5

Euclid(xr , S0 (x), µ, ν) σ1 (x) = x2 + α4 x + α, ω(x) = α13 x5 + α4 x4 + α14 x3 + α13 x2 + α14 x + α.

σ1 (x) 1, α.

0 14 e

0, 2, 6, 9, 14 e = (α13 0 α4 0 0 0 α9 0 0 α3 0 0 0 0 α4).

c = r − e = (0 1 α4 α10 α12 α6 α9 α5 α13 α3 α α8 α7 α2 α14 ).

℄

" #$ %

! " # ℄ %&' (()℄*+,℄ ! # ℄

σ0 (x) = sigma0

-%

. + -./0' *+ * , ) , )#, ), . + -%&' (( *℄*+2 ' % , )," )#, )!,! ) , ) , )!, , & ' S0 (x) = sX0

. + -%3' 4 -%,-. * )), )),) )#, ), )", %%

!"#!" !"#!" #!#" !" #!" ℄ % "!" #!"!" !#" #!" !" ℄ ℄ &' ℄

σ1 (x)

( (&&)*+,& ℄ --,& .)( .& // 0 ℄

! !"!"!#"!" , 1'-2)2' !"!" 3) ' * )& -') - 0! 4 '&' 0% 4 ' ,0 ' 33 .)+&5--,&3) ' # 6 # ℄

½¼º¿ H C ⎛ 1 ⎜0 H=⎜ ⎝0 0

10 11 01 00

10 01 10 11

⎞ 00 0 0⎟ ⎟. 1 0⎠ 01

n

n ! 4

i < j < k < l

i j k l" H 4 # ! (i, j, k, l) (1, 2, 3, 6) (2, 3, 4, 7)

(1, 2, 5, 7) (2, 4, 5, 6)

(1, 3, 4, 5) (3, 5, 6, 7).

(1, 4, 6, 7)

4

7 4

− 7 = 28

(n2 + n + 1, n + 1) D

n n PG (2, n) ! F3n

! Fn " # α Fn "

½½º½

3

3

2

α(n

+n+1)(n+1)

= 1;

$ PG (2, n) % F⋆n F⋆n " & '

3

L = {1} ∪ {a + α : b ∈ Fn };

L

PG (2, n)" ( )

L

α n2 +n+1 * )

(13, 4)

+$,℄

! "

* %

D = {0, 1, 3, 9} . D 2 − (n2 + n + 1, n + 1, 1)

S *

D % 2

(n + 1)2 nn

+n

= 42 × 312 .

2 3 A

% '

F2 F3 /

D

13 0

1 2 t(x) = 1 + x + x3 + x9 .

(x13 − 1) F2 [x] g(x) = 1 + x g(x) 12

g(x) F2l l > 1 12 F3 [x] 13 t(x) (x − 1)

t(x)

g(x) = x6 − x5 − x4 − x3 + x2 − x + 1.

g(x)

½½º¾

7

F3

!

PG (n, q)

"

# !

Fqn+1 $

PG (n, q)

I = {0, 1, . . . , (q n+1 − 1)/(q n − 1) − 1}.

I → I σ: x → x + 1

mod (q n+1 − 1)/(q n − 1)

Π PG (3, 4) % Π σ

PG (3, 4)

& F44 α 1 α α(1 + α)

F4 # #

℄℄ ℄ !"! !$ ℄ %!&'( &) *

D = {0, 1, 2, 8, 12, 20, 23, 25, 26, 28, 30, 41, 42, 50, 59, 66, 72, 73, 76, 78, 82}.

85

D

1

84 5 ' 85 D (84, 21, 5) '

D + i = { d + i mod 85 : d ∈ D},

0 ≤ i ≤ 85 C

PG (3, 4)

t(x) =

xj ;

j∈D

D

D + i

ti (x) = xi t(x)

mod x85 − 1.

½¾º½

G24 G23 0 1 i > 0

Ai (G24 ) = Ai−1 (G23 ) + Ai (G23 ).

!

i 0 8 12 16 24

Ai 1 759 2576 759 1

Ì G24 ½¾º¾

"# $ % G24 &

'()℄

G24

℄ "

+ % # #

759 8

# " $# % & '(# )*+

i i ÏÔÓ×

℄ ! "℄#$%&$$''

() $'

() ℄$' *+

,-. . $ ' /$' - . () ! .&/$$$' .!''

!! !0&&&&&&&&&01$$' & & & & & & & & & ℄

! . 1' ,

!$$' * * * * * * * * * * * * * * * * * * * * * ℄ 125

! 3

" 125

, . ℄ ! , ℄* . 2 .03.0$' ,' !' * 4 + #

!( 5 !$''

!*℄ !*℄6''

!4℄ !4℄6''

!+℄ !+℄6''

! ! !1$$''

' & & & & & & & & ℄

. 1'

c = (1010 0101 1100 1101 1011 1111)

½¿º½ 25 F25 C

25 q > r C Fq [x, y]2 k = 2+2 = 6 2 d !"" d ≥ 15 # $ % & d & C ' f (x, y) ( 2 Fq ( 2q % F2q $ ! ) q − 1 < t < q + 1 ) * ) t = 2q − 1) " ) 2q ) + , -) q) . ) / 0 0 $ 10 25 − 10 = 15 ' f (x, y) = (x + 1)(x + 2)

% # C % 1 Ai i {q 2 − 2q, q 2 − (2q − 1), q 2 − (q + 1), q 2 − q, q 2 − (q − 1), q 2 − 1, q 2 }.

2 & C

AC (Z) = 1+240Z 15 +1500Z 16 +4000Z 19 +2640Z 20 +6000Z 21 +1000Z 24 +244Z 25.

½¿º¾

H4 (2) [15, 11, 3] RM 2 (1, 4) [15+1, (15−11)+1] = [16, 5] 4−1 2 = 8 ⋆

RM 2 (1, 4) !

RM 2 (1, 4)

15 7 "

11

# C = RM 2 (1, 4) 8$

! j 16 Ai = A16−i % 8 < i < 16 Ai = 0 !&&

Aj = 0

j = 16 − i < 8 ' RM 2 (1, 4) ! j 0

8

AC (Z) = 1 + 30Z 8 + Z 16 .

!

RM 2 (2, 4) = RM 2 (1, 4)⊥ . # &

C⊥

( )*

+ ,

WC ⊥ (X, Y ) =

1 WC (Y − X, Y + X). |C|

WC (X, Y ) = Y 16 + 30X 8Y 8 + X 16 ,

AC ⊥ (Z) = WC ⊥ (Z, 1) = 1 + 140Z 4 + 448Z 6 + 870Z 8 + 448Z 10 + 140Z 1 2 + Z 16 .

-!

[63, 12, 9] = [7 × 9, 4 × 3, 3 × 3]. &

H3 (2) & D [9, 3, 3] -! x9 − 1 ∈ F2 [x] F26 !

.

α

/

f (x) = x6 + x3 + 1 = 0.

f (α) = 0

f (α2 ) = 0

f (x) 9 3 C = H3 (2) ⊗ BCH2 (9, 3).

! Fq q = 2m F2 " # m m $ % n = 2 (2 − 1)

m

m1 2 3 4 5 n 2 12 56 240 992

255&

'

m = 4

( ) *

C = RS (15, 10)

255 = 240 + 24 − 1

+ ,

F16

C

6& 0

C′ = C

[16, 10, 7] C′

F2

8 F2 ′′ C [256, 160, 7] ′′′ ′′ / ! C

C - . %

α ∈ F16

255

159

7

'

0

C = RS (31, 21)

11

0

% '

[30, 20]

3/2

C

1

. %

F32 5 150, 100 C ′ F2 C ′′ 300 3/2 ′ % C

C ′′ = |C ′ |C ′ |.

½º

!"# $#$% %%# $ & & '( )* +( , ! %%- ! . / 0 1 2 / 3 4 & 5 6

( %77 8 9 % - / :& ; <* ; & = 5 %%7 6 99 9 5 = 8 <* ; & 5 6 ( %%> 0 9 %# # , 2 8 := >#? >-"$ .. %%$ 2 ( * 1 - , 2 + $!"# $#$% %%7 7 , 2 2 < @ * >.

0 )* ; 0 %%$ % , 2 2 < @ ' " : 99 ..> !"$#%$%. %%$ > , 2 2 < @ ; A %%# 9 &

!" * #

$ $ %& 9 7 / 4 ; 8* 52 $>>$ $ 1 = ' 0 )* ; 0 %%- . 4 + B ; = $>> ! 6 8 (9 8 2 = ' ( 0 * + 9 &+:$.!".7.7# %-7 + < 2 ( ' 1 B ( *: #% 0 )* ; 0 %%%

! " # $

%

& ' ( ) * ) ) ++% &*,- .,- - #) * +. " ) / ) 0 1 ,22, %

!"#& ,2 ) * ) 34 %5& ! , 6 ) 7/8 ,, ( 1 " ( 9 :; <' , = 7/8 :; <' ,222 > (4 1 8 ! ,+ # = = 1

1 7 6 ) 4 ) + %&*,,. ! " ,22 $?7

,- # 1 = = % & , 31 120

! "#$%& > 31. 2 ,! 31 120

' $( ! "' $& >

31.,!, , 31 120

' ) $ ! ,22 31., ! , = 3 #

-2 ) * 7/8 :; <' - 9 ; ; 2 3 3.) ( ,- 1 # ) ( %& -+ % ! ! $%&$$'%()& 3 0 ) ( - 1 # ) ( %& -! % ! ! $%&$$''*'& , 1 3 ) ( +2 1 # ) ( %& - % ! !

$%&$$''*+&

> = $ # = 1 # ( = ) 3 3 - % ! ! $%&$$$,''(& + 9 $ 9 @ #

:; <'/ - > > 0;. . ' 1#( ( > > + +

+ ' ,, ,22+ % -" & ! 5 ? > 7 A > 0 5; 1 , ? > # 4 . ( 9/ A # ) 7 #

% . !

(( !&

! " #" $ # %

&' ( $ )* *+

,+ %$ +* - . / 0*+ # - 1 *$ 2 &+* )+*3 4 ( 5.

# - . 1

$ 6+. ) $ ""# # -+ 7 . ! 08 /' 93 " ## 87 )+ : 0 *;$ " ( $+ ( 0.+$ % ).$ $++ + % $+ ( 6+. % &+ "" # '$ 7 0 2<+ + +* =* 7 + >. "" # 3+ '$ 7* 0 ?$ % 8$ 8$+ ?

" !

! #@ - 5 :$$ 2 + + + # $

% #@! " " # - :+* & 08 /' 93 @@ + ,<+ 7+*$+ / # A 5 ) : *% '

( ( 4<% 7+*$+8

7* ,* ) 4<% 6+. ) /' 93 + " , :B*+ - )3 4 +* % !$+ $ ! $ ?,!# ! # - :* > ) '

08 /' 93 @ + ,<+ 7+*$+ + A 3 * 0 *$+ C + ( + 2 &+* $ ! $ ! 0$$ (+*$ ( 5. 1$ + # - &33 - > 0'++' ,* 3.! *$ % *.$+ +* .$ + q! , * + %+ 6+. % , *. # #!" 5$ A 1 D $ & ) * ) + ? 7 /9 "" -+ % +* E(85. )* ? - 7 ℄ '+* + + + % +* + . +* +* -% 1 : /+ " ( $ " % ! $ 6+. ) $ + @ 5+* %' . ) 7 * @ -% 1 G+ )H & & 6+ ,<+ 7+*$+ 08 /' 93 + 1H $ % ($ 7+*$+ 0 +. ) -? - 1' A ,. ) $

/(0( A+ ) 1+. % ?+++ % , *. ) ""

!"#$% & ' ' '

( ) % * + ,-((( . -) ./ "'$ # & '0 1' 222 "3% 22$℄ 2"!$%#'' #' #

5 6 "5$ # " $ 7 ./ / . / 1 '

/8 -8 ) . / "$ ' " $ 7 9 ) / (.: ;8 / '< "

$ ! = >>

98 8 # "

$ 1 ( ./ / ) 8 ) -8 ) . / ' 2 1 ( 8 < )

5 6 ## < 1 ( ./ 8 8? &8 / ) & . / 67 - @? A 6 1 #&'! # " !"! # $% $ 1 ( (8 1 1 / :? 18 7 1 B / @? 8 8 / ) /

B C8 -.# "#$%'!2&' '22 2 /8 ? /

.D / 8 E E F @? G A ' 2' 1 : ; >

B @? G A ## 2# 5 6 '' 2 8? !" # $ % " & ' # @9E9 HI ) (D 1 B "$ #' " !" &$'$( $ 2 8) / E F '! D 3 1 ) / '2 2! 8) 8 ' !< ) ' E F D ' - 8 / / 8 "'2 $ 8 /

B/ 8 ( 9 2 : 6? ( 6/ / - . ) ( ( '! " $ 22 H 6 > 8 < ) ) & ./ 6 HD) 5 6 @? G A '< 2< - E 1 E 6 * & <"#$% & 8 ' 2 1 8 '# ) & & ) 5 6 '<<

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Collana Unitext - La Matematica per il 3+2 a cura di F. Brezzi C. Ciliberto B. Codenotti M. Pulvirenti A. Quarteroni G. Rinaldi W.J. Runggaldier

Volumi pubblicati A. Bernasconi, B. Codenotti Introduzione alla complessità computazionale 1998, X+260 pp. ISBN 88-470-0020-3 A. Bernasconi, B. Codenotti, G. Resta Metodi matematici in complessità computazionale 1999, X+364 pp, ISBN 88-470-0060-2 E. Salinelli, F. Tomarelli Modelli dinamici discreti 2002, XII+354 pp, ISBN 88-470-0187-0 S. Bosch Algebra 2003, VIII+380 pp, ISBN 88-470-0221-4 S. Graffi, M. Degli Esposti Fisica matematica discreta 2003, X+248 pp, ISBN 88-470-0212-5 S. Margarita, E. Salinelli MultiMath - Matematica Multimediale per l’Università 2004, XX+270 pp, ISBN 88-470-0228-1

A. Quarteroni, R. Sacco, F. Saleri Matematica numerica (2a Ed.) 2000, XIV+448 pp, ISBN 88-470-0077-7 2002, 2004 ristampa riveduta e corretta (1a edizione 1998, ISBN 88-470-0010-6) A partire dal 2004, i volumi della serie sono contrassegnati da un numero di identificazione. I volumi indicati in grigio si riferiscono a edizioni non più in commercio. 13. A. Quarteroni, F. Saleri Introduzione al Calcolo Scientifico (2a Ed.) 2004, X+262 pp, ISBN 88-470-0256-7 (1a edizione 2002, ISBN 88-470-0149-8) 14. S. Salsa Equazioni a derivate parziali - Metodi, modelli e applicazioni 2004, XII+426 pp, ISBN 88-470-0259-1 15. G. Riccardi Calcolo differenziale ed integrale 2004, XII+314 pp, ISBN 88-470-0285-0 16. M. Impedovo Matematica generale con il calcolatore 2005, X+526 pp, ISBN 88-470-0258-3 17. L. Formaggia, F. Saleri, A. Veneziani Applicazioni ed esercizi di modellistica numerica per problemi differenziali 2005, VIII+396 pp, ISBN 88-470-0257-5 18. S. Salsa, G. Verzini Equazioni a derivate parziali - Complementi ed esercizi 2005, VIII+406 pp, ISBN 88-470-0260-5 19. C. Canuto, A. Tabacco Analisi Matematica I (2a Ed.) 2005, XII+448 pp, ISBN 88-470-0337-7 (1a edizione, 2003, XII+376 pp, ISBN 88-470-0220-6) 20. F. Biagini, M. Campanino Elementi di Probabilitá e Statistica 2006, XII+236 pp, ISBN 88-470-0330-X

21. S. Leonesi, C. Toffalori Numeri e Crittografia 2006, VIII+178 pp, ISBN 88-470-0331-8 22. A. Quarteroni, F. Saleri Introduzione al Calcolo Scientifico (3a Ed.) 2006, X+306 pp, ISBN 88-470-0480-2 23. S. Leonesi, C. Toffalori Un invito all’Algebra 2006, XVII+432 pp, ISBN 88-470-0313-X 24. W.M. Baldoni, C. Ciliberto, G.M. Piacentini Cattaneo Aritmetica, Crittografia e Codici 2006, XVI+518 pp, ISBN 88-470-0455-1 25. A. Quarteroni Modellistica numerica per problemi differenziali (3a Ed.) 2006, XIV+452 pp, ISBN 88-470-0493-4 (1a edizione 2000, ISBN 88-470-0108-0) (2a edizione 2003, ISBN 88-470-0203-6) 26. M. Abate, F. Tovena Curve e superfici 2006, XIV+394 pp, ISBN 88-470-0535-3 27. L. Giuzzi Codici correttori 2006, XVI+402 pp, ISBN 88-470-0539-6

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Elementi di Probabilità e Statistica (UNITEXT La Matematica per il 3+2)

Modelli dinamici discreti, 2a edizione (UNITEXT La Matematica per il 3+2)

Analisi Matematica I: Teoria ed esercizi con complementi in rete (UNITEXT La Matematica per il 3+2)

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Gruppi: Una introduzione a idee e metodi della Teoria dei Gruppi (UNITEXT La Matematica per il 3+2) Italian

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Matematica numerica (UNITEXT La Matematica per il 3+2)

Geometria Differenziale (UNITEXT La Matematica per il 3+2)

Curve e superfici (UNITEXT La Matematica per il 3+2)

Topologia (UNITEXT La Matematica per il 3+2)

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Numeri e Crittografia (UNITEXT La Matematica per il 3+2)

Esercizi di finanza matematica (UNITEXT La Matematica per il 3+2) Italian

Modellistica Numerica per Problemi Differenziali, 3a edizione (UNITEXT La Matematica per il 3+2)

Un invito all'Algebra (UNITEXT La Matematica per il 3+2) Italian

Teoria delle Equazioni e Teoria di Galois (UNITEXT La Matematica per il 3+2)

Data mining: Metodi e strategie (UNITEXT La Matematica per il 3+2)

Algebra lineare ed equazioni differenziali ordinarie (UNITEXT La Matematica per il 3+2)

Elementi di Probabilità e Statistica (UNITEXT La Matematica per il 3+2)

Modelli dinamici discreti, 2a edizione (UNITEXT La Matematica per il 3+2)

Analisi Matematica I: Teoria ed esercizi con complementi in rete (UNITEXT La Matematica per il 3+2)

Analisi matematica II: Teoria ed esercizi con complementi in rete (UNITEXT La Matematica per il 3+2)

Introduzione alla finanza matematica: Derivati, prezzi e coperture (Unitext La Matematica Per Il 3+2) (Italian Edition)

Analisi Matematica I: Teoria ed esercizi con complementi in rete (UNITEXT La Matematica per il 3+2)

Matematica: si parte!: Nozioni di base ed esercizi per il primo anno di Ingegneria (UNITEXT La Matematica per il 3+2)

Logica matematica per l'informatica

Matematica per l'Ingegneria dell'Informazione

Teoria Spettrale e Meccanica Quantistica: Operatori in Spazi di Hilbert (UNITEXT La Matematica per il 3+2)

Equazioni a derivate parziali: Metodi, modelli e applicazioni, 2a edizione (UNITEXT La Matematica per il 3+2)

Gruppi: Una introduzione a idee e metodi della Teoria dei Gruppi (UNITEXT La Matematica per il 3+2) Italian

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