On the Decoding of Cyclic Codes Using Gröbner Bases

Philippe Loustaunau; Eric V. York

doi:10.1007/s002000050084

On the Decoding of Cyclic Codes Using Gröbner Bases

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s002000050084 ◽

1997 ◽

Vol 8 (6) ◽

pp. 469-483 ◽

Cited By ~ 13

Author(s):

Philippe Loustaunau ◽

Eric V. York

Keyword(s):

Cyclic Codes ◽

Gröbner Bases ◽

Grobner Bases

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Use of Grobner bases to decode binary cyclic codes up to the true minimum distance

IEEE Transactions on Information Theory ◽

10.1109/18.333885 ◽

1994 ◽

Vol 40 (5) ◽

pp. 1654-1661 ◽

Cited By ~ 27

Author(s):

Xuemin Chen ◽

I.S. Reed ◽

T. Helleseth ◽

T.K. Truong

Keyword(s):

Minimum Distance ◽

Cyclic Codes ◽

Gröbner Bases ◽

Grobner Bases

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Strong Gröbner bases and cyclic codes over a finite-chain ring.

Electronic Notes in Discrete Mathematics ◽

10.1016/s1571-0653(04)00175-1 ◽

2001 ◽

Vol 6 ◽

pp. 240-250 ◽

Cited By ~ 5

Author(s):

Graham H. Norton ◽

Ana Sǎlǎgean

Keyword(s):

Cyclic Codes ◽

Gröbner Bases ◽

Grobner Bases ◽

Finite Chain ◽

Chain Ring ◽

Finite Chain Ring

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Cyclic codes and minimal strong Gröbner bases over a principal ideal ring

Finite Fields and Their Applications ◽

10.1016/s1071-5797(03)00003-0 ◽

2003 ◽

Vol 9 (2) ◽

pp. 237-249 ◽

Cited By ~ 23

Author(s):

G.H. Norton ◽

A. Salagean

Keyword(s):

Principal Ideal ◽

Cyclic Codes ◽

Gröbner Bases ◽

Grobner Bases ◽

Principal Ideal Ring ◽

Ideal Ring

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Algebraic-geometric codes and multidimensional cyclic codes: a unified theory and algorithms for decoding using Grobner bases

IEEE Transactions on Information Theory ◽

10.1109/18.476246 ◽

1995 ◽

Vol 41 (6) ◽

pp. 1733-1751 ◽

Cited By ~ 45

Author(s):

K. Saints ◽

C. Heegard

Keyword(s):

Cyclic Codes ◽

Gröbner Bases ◽

Grobner Bases ◽

Unified Theory ◽

Algebraic Geometric Codes ◽

Geometric Codes

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Efficient decoding of (binary) cyclic codes above the correction capacity of the code using grobner bases

IEEE International Symposium on Information Theory, 2003. Proceedings. ◽

10.1109/isit.2003.1228378 ◽

2003 ◽

Cited By ~ 7

Author(s):

D. Augot ◽

M. Bardet ◽

J.-C. Faugere

Keyword(s):

Cyclic Codes ◽

Gröbner Bases ◽

Grobner Bases ◽

Efficient Decoding

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APPLYING BUCHBERGER'S CRITERIA FOR COMPUTING GRÖBNER BASES OVER FINITE-CHAIN RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500345 ◽

2013 ◽

Vol 12 (07) ◽

pp. 1350034 ◽

Cited By ~ 1

Author(s):

AMIR HASHEMI ◽

PARISA ALVANDI

Keyword(s):

Cyclic Codes ◽

Gröbner Bases ◽

Special Kind ◽

Grobner Bases ◽

Discrete Mathematics ◽

Galois Rings ◽

Finite Chain ◽

Chain Ring ◽

Finite Chain Ring ◽

Finite Chain Rings

Norton and Sălăgean [Strong Gröbner bases and cyclic codes over a finite-chain ring, in Proc. Workshop on Coding and Cryptography, Paris, Electronic Notes in Discrete Mathematics, Vol. 6 (Elsevier Science, 2001), pp. 391–401] have presented an algorithm for computing Gröbner bases over finite-chain rings. Byrne and Fitzpatrick [Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput.31 (2001) 565–584] have simultaneously proposed a similar algorithm for computing Gröbner bases over Galois rings (a special kind of finite-chain rings). However, they have not incorporated Buchberger's criteria into their algorithms to avoid unnecessary reductions. In this paper, we propose the adapted version of these criteria for polynomials over finite-chain rings and we show how to apply them on Norton–Sălăgean algorithm. The described algorithm has been implemented in Maple and experimented with a number of examples for the Galois rings.

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