Algebraic solutions of Newton's identities for cyclic codes

D. Augot

doi:10.1109/ITW.1998.706411

Algebraic solutions of Newton's identities for cyclic codes

D. Augot

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

This paper consider the use of Newton's identities for establishing properties of cyclic codes. The main tool is to consider these identities as equations, and to look for the properties of the solutions. First these equations have been considered as necessary conditions for establishing non-existence properties of cyclic codes, such as the non-existence of codewords of a given weight. The properties of these equations are studied, and the properties of the solution to the algebraic system are given. The main theorem is that codewords in a Hamming sphere around a given word can be characterized by algebraic conditions. This theorem enables one to describe the minimum codewords of a given cyclic codes, by algebraic conditions. The equations are solved using the Buchberger's algorithm for computing a Groebner basis. Examples are also given with alternant codes, and with a non-linear code.

Original language	English
Title of host publication	1998 Information Theory Workshop, ITW 1998
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	49
Number of pages	1
ISBN (Electronic)	0780344081, 9780780344082
DOIs	https://doi.org/10.1109/ITW.1998.706411
Publication status	Published - 1 Jan 1998
Event	1998 Information Theory Workshop, ITW 1998 - Killarney, Ireland Duration: 22 Jun 1998 → 26 Jun 1998

Publication series

Name	1998 Information Theory Workshop, ITW 1998

Conference

Conference	1998 Information Theory Workshop, ITW 1998
Country/Territory	Ireland
City	Killarney
Period	22/06/98 → 26/06/98

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10.1109/ITW.1998.706411

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title = "Algebraic solutions of Newton's identities for cyclic codes",

abstract = "This paper consider the use of Newton's identities for establishing properties of cyclic codes. The main tool is to consider these identities as equations, and to look for the properties of the solutions. First these equations have been considered as necessary conditions for establishing non-existence properties of cyclic codes, such as the non-existence of codewords of a given weight. The properties of these equations are studied, and the properties of the solution to the algebraic system are given. The main theorem is that codewords in a Hamming sphere around a given word can be characterized by algebraic conditions. This theorem enables one to describe the minimum codewords of a given cyclic codes, by algebraic conditions. The equations are solved using the Buchberger's algorithm for computing a Groebner basis. Examples are also given with alternant codes, and with a non-linear code.",

author = "D. Augot",

note = "Publisher Copyright: {\textcopyright} 1998 IEEE.; 1998 Information Theory Workshop, ITW 1998 ; Conference date: 22-06-1998 Through 26-06-1998",

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AU - Augot, D.

PY - 1998/1/1

Y1 - 1998/1/1

N2 - This paper consider the use of Newton's identities for establishing properties of cyclic codes. The main tool is to consider these identities as equations, and to look for the properties of the solutions. First these equations have been considered as necessary conditions for establishing non-existence properties of cyclic codes, such as the non-existence of codewords of a given weight. The properties of these equations are studied, and the properties of the solution to the algebraic system are given. The main theorem is that codewords in a Hamming sphere around a given word can be characterized by algebraic conditions. This theorem enables one to describe the minimum codewords of a given cyclic codes, by algebraic conditions. The equations are solved using the Buchberger's algorithm for computing a Groebner basis. Examples are also given with alternant codes, and with a non-linear code.

AB - This paper consider the use of Newton's identities for establishing properties of cyclic codes. The main tool is to consider these identities as equations, and to look for the properties of the solutions. First these equations have been considered as necessary conditions for establishing non-existence properties of cyclic codes, such as the non-existence of codewords of a given weight. The properties of these equations are studied, and the properties of the solution to the algebraic system are given. The main theorem is that codewords in a Hamming sphere around a given word can be characterized by algebraic conditions. This theorem enables one to describe the minimum codewords of a given cyclic codes, by algebraic conditions. The equations are solved using the Buchberger's algorithm for computing a Groebner basis. Examples are also given with alternant codes, and with a non-linear code.

U2 - 10.1109/ITW.1998.706411

DO - 10.1109/ITW.1998.706411

M3 - Conference contribution

AN - SCOPUS:84938876596

T3 - 1998 Information Theory Workshop, ITW 1998

SP - 49

BT - 1998 Information Theory Workshop, ITW 1998

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 1998 Information Theory Workshop, ITW 1998

Y2 - 22 June 1998 through 26 June 1998

ER -

Algebraic solutions of Newton's identities for cyclic codes

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