TY - GEN
T1 - Algebraic solutions of Newton's identities for cyclic codes
AU - Augot, D.
N1 - Publisher Copyright:
© 1998 IEEE.
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.
UR - https://www.scopus.com/pages/publications/84938876596
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 -