高次の座を用いた代数幾何符号の構成On Construction of Algebraic Geometric Codes

with High Degree Places(Update : 2001.02.02)

戒田高康　 (Takayasu KAIDA)

八代工業高等専門学校 情報電子工学科Dept. of Information and Electronic Engineering,

Yatsushiro National College of Technology

Outline of the Presentation

• Background and preparations• Code with high degree places by Xing, et.al.• Function type code and residue type code

with high degree places• Relation between Xing’s code and proposed code• Decoding for proposed code• An example over an elliptic function field• Conclusion and future works

Background

• Algebraic geometric (AG) code– Its code length is restricted by Hasse-Weil bound

• AG code with high degree places– C.Xing, H.Neidereiter and Y.K.Lam[2,3], 1999.– T.Kaida, K.Imamura and T.Moriuchi[5,6], Not only function type but also residue type, 1995~2000

• Relation of codes proposed in [1,2,3]– One construction is new and interested[4], 1999

Preparations

[7]H.Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, 1993

mqP

P

q

FFmPPFKFP

KKF

qGFK

deg , of field residue the: ,/over place a:

fieldconstant full aover fieldfunction algebraic variableonean :/

)(F

Code Proposed by Xing, et.al.

)}(|)({),,(

by defined is code The

where))),(()),...,(((

)(:

,...,2,1for code ],,[

tomisomorphislinear -an : :

supp supp with divisors: ,

,...,2,1for deg with placesdistinct :,...,,

111

1

21

GLfKfGDC

nnPfPff

KGL

riKCdkn

KCF

DGGPD

riPkPPP

nX

r

iirr

n

niiii

iPi

r

ii

iir

i

Parameters of the code

XSd

GkrSX

dgGGlk

dknGDCkG

Sii

Sii

X

r

ii

min and

deg},...,1{ where

, ,1deg)(

withcode ],,[ is ),,( then deg If1

Function Type Code

djKfVfVfVfPf

fffPfKF

KFVVVVPdKFP

j

dd

dd

P

Pd

,2,1for ,)(

such that(1) ),,,()(:

over of basis a :},,,{ deg with /over place a:

:1 Definition

2211

21

21

　

Function Type Code

(2) )}(L|))(,,)(,)({()ˆ,,(

by defined is code of typeFunction set basis a:},,,{ˆ

,,2,1for over of basis a:

Øsuppsupp ,

such that /over divisors:, ,,2,1for deg with

/over placesdistinct :,,,:2 Definition

21

)()2(1

)(

1

21

GfPfPfPfVGDC

VVVV

riKFV

DGPD

KFGDriPd

KFrPPP

rL

r)(

Pi

r

ii

ii

r

i

　 　　

　

Residue Type Code

)(Res)( ,,2,1for

(3) },,,,{)(Res by defined is map sidueRe

,,2,1for 1deg, such that '/'over placesdistinct :,,,

fieldextension th - the:' and '' such that fieldfunction algebraic:'/'

over of basis a:},,,{ deg with /over place a: /over module aldifferenti a:

: Definition

1

21

21

21

d

iPijj

dP

ii

d

Pd

iPVedj

eee

diPPPKFrPPP

dKFKFKF

KFVVVVPdKFP

KF

３

Residue Type Code

)4)}((|)(Res,),({(Res)ˆ,,(

by defined is code typesidueRe2&1 Definition as same e th

setting are ˆset basis and , divisors

:5 Definition to from map theis deg with

/over of map residue The :４ Lemma

1DGVGDC

VGD

KΩPd

KFP

rPP

d

Duality of Function and Residue Code

)()(

by defined is of conorm The

'such that '/'over ' places allover runs sum the where

,')( by defined is of conorm The

/over place a: '':'/' , of field exitensionan :'

:6 Definition

/'/'

'/'

PConnDCon

PnD

PPKFP

PPConP

KFPFKFKFKK

FFPFF

P

PPFF

Duality of Function and Residue Code

'over )','( codeAG alconvention is }}1{,},1{},1{{ˆ with )ˆ,','(

' ,'

'' ,' ),,,,(:(Proof))ˆ,,( code, typeresidue theof code

dual theis ),ˆ,,( code, pefuntion ty The

:7 Theorem

/'1 1

/'

21

KGDCVVGDC

(G)ConGP(D)ConD

FKFFKdddLCNdVGDC

VGDC

L

L

FF

r

i

d

jijFF

qr

L

i

d

　

Duality of Function and Residue Code

　 　　 　

　 　　 　

r

idi

dii

d

idi

ii

i

T

TT

PVPV

PVPVT

ri

iii

i

0

0matrixsingular -non

)()(

)()(

,,2,1For (Proof)

1

)(1

)(

)(11

)(1

Duality of Function and Residue Code

□

　 　　 　 　

0)(0)'('

ofmatrix tion transposithe: where

,'

,')ˆ,,( ofmatix generator :

)ˆ,,( ofmatrix generator :

)','( ofmatrix check :')','( ofmatrix generator :'

(Proof)

tL

tLL

t

tL

LL

LL

LL

LL

GGHG

TT

GTH

TGGVGDCG

VGDCG

GDCHGDCG

Relation of CX and CL

XL

XL

iiii

CCGDCVGDC

riCdkk

of case special a is ),,()ˆ,,(

,...,2,1for code ],,[:

※

Decoding for Proposed Codes

• By generator matrix of the dual code• In extension field by

　

　

riPVPV

PVPVT

T

TT

TGH'TGG

iii

i

idi

dii

d

idi

ii

i

r

tL

,,2,1for )()(

)()(

,0

0matrix singular -non where

,or '

)(1

)(

)(11

)(1

1

1

An Example

)(,

))(1(,

)1)(1()(

,,,,,1)(

}},,1{},,1{},,1{},1{},1{{ˆ and 6 ,Let

fieldfunction elliptican :/ 01such that ),( , )2(

:8 Example

2222

22

5

1

23

yxydx

yxxxdx

xxxdxDG

yxyxyxG

yxxxVQGPD

KFxyyyxKFGFK

ii

L

An Example (Ex. 8)

Pi deg LP of K(x,y) LP of K(x) LP of K(y)

Q 1 y/x 1/x 1/yP1 1 x+1, y x+1 y+1

P2 1 x+1, x2+y x+1 y+1

P3 2 x2+x+1, y x2+x+1 y

P4 2 x2+x+1, y+1 x2+x+1 y+1

P5 3 x2+y x3+x+1 y3+y+1

P6 2 x x y2+y+1

P7 3 x2+y+1 x3+x+1 y3+y2+1

Table 1. Places over F/K with deg 3≦ 　 and local parameters (LP) over F, K(x) and K(y)

An Example

01),2(

01),GF(2

6)(' ,'

3 ,2 ,1 since '' ),2('

:8 Example

36

22

5

1/'

1

54321

6

GF

QGConGPD

dddddFKFGFK

iFF

d

jij

i

An Example (Ex.8)Table 2. Places over F’/K’ with deg=1　 and their LP

Pij LP of F’/K’ Pij LP of F’/K’

P11 x+1, y P41 x+, y+1

P21 x+1, y+1 P42 x+2, y+1

P31 x+, y P51 x+, y+ 2

P32 x+2, y P52 x+2, y+4

P53 x+4, y+

An Example

codebinary )3,3,9(:)ˆ,,(

codebinary )1,6,9(:)ˆ,,(

wordscode all From

111000101011011100111010000

,

010010011010011110111111001010010001110101010111010100

:8 Example

VGDC

VGDC

GG

L

L

Conclusion

• The definition of codes with high degree places– by Xing, et.al. (function type)– function type and residue type

• The duality of function and residue type code• Function type is a special case of Xing’s code• An example over an elliptic function field

Future Works

• Residue type of Xing’s code• Theoretical evaluation for the minimum distanc

e of proposed code• Decoding method of proposed code• Relation between proposed code and conventio

nal codes (AG code, sub-field sub-code, concatenated code)

References[1]C.Xing, H.Niederreiter and K.Y.Lam, “Constructions of algebraic-geometry c

odes”, IEEE Trans. Information Theory, vol.45, pp.1186-1193, May 1999. [2] H.Niederreiter, C.Xing and K.Y.Lam, “A new construction of algebraic-geom

etry codes”, Applicable Algebra in Engineering, Communication and Computing, vol.9, pp.373-381, Springer-Verlag, 1999.

[3] C.Xing, H.Niederreiter and K.Y.Lam, “A generalization of algebraic-geometry codes”, IEEE Trans. Information Theory, vol.45, pp.2498-2501, Nov. 1999.

[4]F.Ozbudak and H.Stichtenoth, “Constructing codes from algebraic curves”, IEEE Trans. Information Theory, vol.45, pp.2502-2505, Nov. 1999.

[5] 戒田高康 , 今村恭己 , 森内勉 , “ 高次の座を用いた代数幾何符号に関する考察” , 第 18 回情報理論とその応用シンポジウム予稿集 , pp.231-234, 花巻 , 1995 年 10 月

[6]T.Kaida and K. Imamura, “Residue type of algebraic geometric codes with high degree places”, Proc. Of International Symposium on Information Theory and Its Applications, pp.453-456, Honolulu, Nov. 2000.

[7]H.Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, 1993.