Algebraic Solutions of Newton's identities for cyclic codes

International audienceThis 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 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