Iwan Duursma

(Iwan Duursma )

1

∗

2005.8.22

^!

13

"$#%&'()+*-,+)$. /1023

§1 5

⁴⁵⁶⁷

8

9;:=<>?@

(error correcting code)

^ACB

,

DFEGHI.JK L MNPOQSR1A T

,

^L1TR+UV

>

WYX

(

Z+[-\] ^ _a`]YbC0dceSOCfFV gYhjikal-]FmFn

X

)

QPR8op0dqrFsLtR

.

^u0dv

w B xzy{-|d}~ O€| ‚ƒA md|

.

^|„

,

^)‡†

X

%aˆSOj‰F|+gŠ ‹

X

QYRFA8Œ8m n

.

^Œ‹YŒ

,

QŽ‘-R€’”“+L • –—|ag

, 1

˜+ˆ™1šagSŒ$„”nA›‚FR

.

1 7 5 2 = ⇒ 1 7

^œ

2

uFIL žYŸ¡ -A£¢¥¤d–tR+0O€¦§¨Œ€g

,

^©a%ª0d«

1 + 7 + 5 + 2 = 15

O‰¥Tj¬Fšg­ª n

: 1 7 5 2 | 15 = ⇒ 1 7

^œ

2 | 15

‚1RFA£™1š+gSŒ$„  od%ªB

15 − (1 + 7 + 2) = 5

^{A¯®°LªT8R}

.

^{0-m n}

X

,

±4 ²$³´ªµI¶¸·

¹ªºd»¼½¾ºa¿

,

À8ÁCÂÃ Ä Å²ƒÆj²-Ç

½FÈIÉÊ

51ËÌÍA|ƒn0+–

,

ÎÏ\]dÐFš J-U

.

^ÑÒ

X B

,

^{Q :} o|MNƒOÔÓª,ÖÕ×.Ø3

F 2 k

0 °ÙAŒIg Ú֌Ig­ªT

,

xÛ¢Ü1]MNª0ÔÝYVd¬Fš+~-B

,

m :

iYT$]8Ó1,ÞÕ×.Ø3

F 2 n ß F 2 k

Oà”pdá1â

,

A€|Pn AjL$Ñã¥äC-R

.

u CL+å+æ\]dç è ?@

0CéêPO€ëì1R1A£í”0Fm-n

X

]R

:

î ï

1 q

O€ð%-ìPTFA€‚ªR

.

^{ÓÍ,ÖÕ=.Ø 3}

F q n

0

k

^{í °añÜ Ø3}

C (k ≤ n)

^O

F q ò 0

[n, k]-(

^ç

è

)

^?@

(linear code)

^A|Pn

. n, k

^O8u$ó$

C

^{0 ?@8ô}

(code length),

^í°¨A|Pn

.

—F„

:

,

à”pdá-õUtÍAI0CöF–$éêª0

C

^LtR

.

^ÑÒ

X

bC0-m n

X

9;:÷<>

–$ø”]8ùFR$‹B

?@

w

&Í0d‰ú XIû

RA

X

ŒIg

,

 0tÍAýüþ‡A] RA$Oä-Ž

XIÿ

ŒÖŒamFn

. C

^08Ó

, Õ .

c (

?@

A€|¨n

)

OÜ-L Ú֌doSA T

, 0

^{L]|ܔ0 %PO}

c

⁰

(Hamming)

^1sSA

||

, C

^{0 ?@ ‚aìg}

(0

)

OÐ-šoªA$T$0

,

^-s0SO

C

⁰

(

^-s

)

^A

|¥n

.

(d

^Aj­

)

^h-‚FR1ATB

[n, k, d]-

?@

A£‰

. d

^B

?@

0

9;:=<>! !" X

iYT#%$+ù”R

(

iÍT|&ªb md|

cf. [10, p.106], [11, p.99]).

^'

X

-s)(*KYOjéê ‚-R

. c ∈ C

0)FsO

wt (c)

^{L ڂ}

. A i := ] { c ∈ C ; wt (c) = i }

^{Aj­+A T}

, W C (x, y) :=

n

X

i=0

A i x n−i y i

O

C

^0),.-/01

(weight enumerator)

^A32!4

.

5

1 C = { 00, 11 } ⊂ F 2

2 . 0PAT W C (x, y) = x 2 + y 2 . C O 2 —687g C 0 = C ⊕ C = { 0000, 0011, 1100, 1111 } O39PR1A W C 0 (x, y) = x 4 + 2x 2 y 2 + y 4 = W C (x, y) 2 . 0-mFn

X

,

^Fs

(*KPOÐFš R AjL

,

^{?@ 0)1s}

X

$‚1R|;:|<:a]1Aj–(*KÍ0=a}-Lù‹-RªA€|¥n>

?@

Õ –tR

.

?@ w

&Aæ

X

$‚”RB;C ‰‡AŒg

[10], [11], [13] (

oU͌I-B&ÍAõƒbED‰

), [15]

^]Sb

.

F èG

K AI0$6ªB

[9],

^{H%IJ ?@}

X

—|gB

[16]

^]Yb–tR

.

∗

KMLONQPRKTSU%V

,

^WTXY

§2 Zeta

îï

?@

0

zeta

^$Ô%ªB

, Duursma [5]

L1p8gBSäIo

.

u$B-s(*K‹ƒŽ PäC-R

. C

O

F q ò 0 [n, k, d] ?@ , ua0-s)(*KYO W C (x, y) A‘‚-R . „8oLB d h C ⊥ (tªA›Lé

ê

)

^0Ùh

2

^{ò A›‚FR}

. C

⁰

zeta

$$%”BÔí”0-m-n

X

éê¥äCFR

:

î ï

2 ([6], p.58) C

X

Œg

,

^í%

n − d

^-0tR(*K

P (T ) ∈ Q[T ]

^–oU

1

^—

Œg

,

P (T )

(1 − T )(1 − qT ) (y(1 − T ) + xT ) n = · · · + W C (x, y) − x n

q − 1 T n−d + · · ·

– ‚-R

. P (T )

^O

C

⁰

zeta

^/0!1

, Z(T ) := P (T )/ { (1 − T )(1 − qT ) }

^O

C

⁰

zeta

A32!4

.

$0CéêªB :

Ž|a–

,

^ªB

T

^{0 w KF]08L}

,

ìÍT% "!FO-| ‚

.

^u0

T n−d

0$#a%

X

C

0PäM(*K –ãFRm n

X

]R

,

A|¥n$AjLtR

.

^„8o

, P (T )

^{0$ A å&%'}

h)(֌g¥Ž›‹LB]|+–

,

^I

X

—|agB

[1], [2], [12]

^]YbO+*,

.

í X

, C

^0.-/

(dual code) C ⊥ O C ⊥ := { u ∈ F q n

; u · v = 0, ∀ v ∈ C }

L éê‚FR

.

^oU1Œ

, u = (u 1 , u 2 , · · · , u n ), v = (v 1 , v 2 , · · · , v n ) ∈ F q n X

Œg

, u · v = u 1 v 1 + u 2 v 2 + · · · + u n v n .

„ao

, C ⊥ = C

^{0ÍA T}

C

O"01.-/

(self-dual)

^LtR”A€|ƒn

.

„2

C, C ⊥ 0 zeta (*K P (T ), P ⊥ (T )083

X B

P ⊥ (T ) = P 1 qT

q g T g+g ⊥

AI|ƒnR$&#-–tR

([6, p.59]).

^oUªŒ

, g := n + 1 − k − d, g ⊥ B C ⊥ 0

3

‚FR54

.

⁶

X B

MacWilliams

^078K

W C ⊥ (x, y) = 1

]C W C (x + (q − 1)y, x − y) ([13, p.146, Th. 13])

^{–§ |PŽ FR}

.

^m+ g9

X

, C

^–;:<$=

]YŽ

, P (T ) = P ⊥ (T )

X m :

,

>

1

P (T ) = P 1 qT

q g T 2 g

–

:

-—

(

H%@?dçÍ0.A@B

zeta

^{AOA )BDC}

G

!).

§3 Riemann

^EF

Ha%G?Cç

(F q ò ) 0HAIB zeta $%”05JA xRiemann KL~ (Weil

X

m  Ig6 1äIo

)

^AB

1

A|ƒn h0LtR

.

^{-Lo :}

,

^{: <= ?@ 0}

zeta

^{$ %}

P (T )

^{hBDC $$%8KSO”o‚}

Aj‹ƒŽ

,

^:<=

?@

X

‚1R

Riemann

^KL”B

,

H%D?dç‡A5B íª0-m-n

X

éK‚FR+08–

Lt.:ªn

:

î ï

3 ([7], p.119) C

^OG:<=

?@

,

^u0

zeta

^(*KPO

P (T )

^Aj‚”R

. P (T )

^{0 M%Í0N}

α

X

Œ€g

,

| α | = 1

√ q

–

:

-—ƒA T

, C

^B

Riemann

KLSO”o8‚¥A€|Pn

.

€B1]R

G

K\ !

,

"$#&%'(*)+

Riemann

,-/.&0214365&'7 8*9;:<*=

>?@1;;&A

.

^B+

,

^(C);

zeta

^D

8

FEC'G% HI&JLKNMPOQRTSNUV+

,

W X

1 Riemann

,*-7.01@37YZP[*\(C)]5+V^;;&M@_N`*a*b c/

.

5d'6NFe >

.

^fdg@D7#%

Duursma

⁺

1

E NhGij*k7.&,-$#%l'

:

W X

2 ([7], p.119)

^m

Extremal

MYZ[*\*(C);+

Riemann

,*-7.01@3Gn;+Po$#'p_

.

fCfe

,

^(l)pq

n

VYZ[*\*( );rpe

,

HCst*u;!@HIJLKN'.

extremal code

^5d'6

(

fdg+vT_P'w xA

).

y

2 [8, 4, 4]

^zCJ

Hamming

^(C)

C 8 ([11, p.112], [15, p.35]

^{

).

^fdg+

F 2 | extremal M

YPZ[\(C)

.

^};~€*a+

W C 8 (x, y) = x 8 + 14x 4 y 4 + y 8 e ([15, p.135]), P (T ) = 1

5 (1 + 2T + 2T 2 ).

P (T )

^P+

α = ( − 1 ± i)/2

^M*@e

, | α | = 1/ √

2 = 1/ √ q, Riemann

^,*-7.021@3

.

YNZP[ \l(l)V*+N‚*ƒ

:

M4„C…T!

4

^E;>

(I

^†

∼ IV

^†

, cf. [4])

^!

,

‡C6ˆpNB*‰T3

extremal code

3Š*%P\]#&%

Riemann

,*-!@‹ŒŽ;EVf 5+

89;:

v_l7‘g%C'

.

^’

1

Duursma

⁺

IV

†$YZ[\(*)„p…2\I#&% m

extremal

^M/‘“

Riemann

,-‹pŽln .&”

#d%C'

([8]).

§4

^{• –}

H—˜™+

,

B‰(C)}T~P€aelMp'Pš›p€*a;œ/

zeta

^€a

P (T )

^!N`ž

eVKG f 5Ÿ; C'C1

.

^¡;l“

W 12 (x, y) = x 12 − 33x 8 y 4 − 33x 4 y 8 + y 12 .

fg +>C¢p£ap¤V¥Ce

, F 2 |

YZ[\(*);4DN¦!l>C

([2, § 5], [14, pp.104-105]).

^‡f

e

q = 2

^5G#%

zeta

^{€a7.§ 25}

P (T ) = 1

15 (2T 2 − 1)(2T 2 + 1)(2T 2 + 2T + 1)

5MIŒ

,

^fdg+

Riemann

^,C-

(

^{3ŠC% }

α

^!

| α | = 1/ √

2)

^.01@3

!

^B+FD

8

{a+

P (T ) = − P 1 2T

2 g T 2 g

5'

,

#6?@17.25N

.

^‡#%

,

^‚

8

B‰/N( )T 5œPL

P (T )

^PPi

.Šl%l~25

, Riemann

,l-IœP/`Cž;eVKG;f 5 !C_T

.

^T‘

, extremal

^5d'6

"$ `CžTeK

, Riemann

^,l-GD7#%

,

^{B‰VN(l)T 5œTMf5}

(extremal

^M7‘

Riemann

^,*-‹Ž

?)

^{! KP%*'25I‘} gT! @e >l

(

^{" #$#+}

[2], [3]

^.%&

).

fg ‘'

<=

_6‘

,

^{m (*)(}

zeta

^D

8

n wCxFQ*)*+p+

,

Bp‰, (*)* };~Pp€a,+fPA I‘

-

,

.VŒ0/;'1(2* P¢£*€aL.43

=

P\65+#1 7l!,.P'8 Fe +Mp'p_ 5 8*g;! Ge >l

. (2005

⁹

11

^:

28

^;<,=

)

> ? @ A

[1]

^{B C(D}

,

^{EGF H8I}

:

(C)( !J8KLD

8

58M'KONP ,- RQS

(Iwan Duursma

^T

U

WVGX

),

(C)$5ZY)[ ‚

8:N8G\

,

^{]6^CJ_}

8\

QG`Gabc6db6e

1361 (2004), 91-101.

[2]

^{B8 CfD}

:

†( ) GJ(K$L@D

8

5‡ gM4KGNhP ,l-

(Iwan Duursma

^T

U

V8X

,

ij

1

^{E[ Pz(k}

),

^l[m

8n

ij o

c

n

sabpq

2004

^rsGp

(2005), 31-44,

^’1*+

http://www.math.is.tohoku.ac.jp/~taya/sendaiNC/2004/program.html.

[3] Chinen, K. : Zeta functions for formal weight enumerators and the extremal property, submitted.

[4] Conway, J, H. and Sloane, N. J. A. : Sphere Packings, Lattices and Groups, 3rd Ed., Springer Verlag, 1999.

[5] Duursma, I. : Weight distribution of geometric Goppa codes, Trans. Amer. Math. Soc.

351, No.9 (1999), 3609-3639.

[6] : From weight enumerators to zeta functions, Discrete Appl. Math. 111 (2001), 55-73.

[7] : A Riemann hypothesis analogue for self-dual codes, DIMACS series in Dis- crete Math. and Theoretical Computer Science 56 (2001), 115-124.

[8] : Extremal weight enumerators and ultraspherical polynomials, Discrete Math.

268, No.1-3 (2003), 103-127.

[9] Ebeling, W. : Lattices and Codes, 2nd Ed., Vieweg, 2002.

[10]

^{tu v}

,

^{wx y*I}

:

(C)$5zY*)(

8\

,

^{pŽ=|

, 1993.

[12]

^{EF H8I}

,

^{B C(D}

:

(l)( OJ8KLD 5‡6 MK!NP ,*-

,

^{p m (C)Gb}

‚ln

,

8G\

_

497 (2004), 42 - 47.

[13] MacWilliams, F. J. and Sloane, N. J. A. : The Theory of Error-Correcting Codes, North- Holland, 1977.

[14]

^{s D}

:

⁽⁾

\Gn

5

unitary reflection groups

^¢£ap¤ 54 FD¦+4E*'G%

,

12

^‚

8T:

o

c n

Prs!p

(1996), 96-116.

[15] Pless, V. : Introduction to the Theory of Error-Correcting Codes, 3rd Ed., John Wiley &

Sons, 1998.

[16] Stichtenoth, H. : Algebraic Function Fields and Codes, Springer Verlag, 1993.