Algebraic characterization of minimum weight codewords of cyclic codes

We consider primitive cyclic codes of length n over GF(q), where n=q^m-1, and for any such code with defining set I(C), we define a system of algebraic equations, S_I(C)(w), constructed with the Newton identities for the weight w. We prove that algebraic solutions of this system are in correspondence with all codewords of C of weight lower than w. To deal effectively with the system S_I(C)(w), we compute a Groebner basis of this system, which gives pertinent information on minimum weight codewords. A few examples are given