Description of minimum weight codewords of cyclic codes by algebraic systems

We consider cyclic codes of length n over double-struck F sign _q, n being prime to q. For such a cyclic code C, we describe a system of algebraic equations, denoted by script capital L sign c(w), where w is a positive integer. The system is constructed from Newton's identities, which are satisfied by the elementary symmetric functions and the (generalized) power sum symmetric functions of the locators of codewords of weight w. The main result is that, in a certain sense, the algebraic solutions of script capital L sign c(w) are in one-to-one correspondence with all the codewords of C having weight lower than w. In the particular case where w is the minimum distance of C, all minimum weight codewords are described by script capital L sign c(w). Because the system script capital L sign c(w) is very large, with many indeterminates, no great insight can be directly obtained, and specific tools are required in order to manipulate the algebraic systems. For this purpose, the theory of Gröbner bases can be used. A Gröbner basis of script capital L sign c(w) gives information about the minimum weight codewords.