Description of Minimum Weight Codewords of Cyclic Codes by Algebraic Systems

We consider cyclic codes of lengthnover q,nbeing prime toq. For such a cyclic codeC, we describe a system of algebraic equations, denoted by C(w), wherewis a positive integer. The system is constructed from Newtonʹs identities, which are satisfied by the elementary symmetric functions and the (generalized) power sum symmetric functions of the locators of codewords of weightw. The main result is that, in a certain sense, thealgebraicsolutions of C(w) are in one-to-one correspondence with all the codewords ofChaving weight lower thanw. In the particular case wherewis the minimum distance ofC, all minimum weight codewords are described by C(w). Because the system 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 ofGröbner basescan be used. A Gröbner basis of C(w) gives information about the minimum weight codewords.