Click here to load reader
Upload
lucian
View
121
Download
2
Embed Size (px)
DESCRIPTION
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. - PowerPoint PPT Presentation
Citation preview
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.