Gr\"obner bases and the second generalized Hamming weight of a linear code

Abstract

It is known that for binary codes one can use Gr\"obner bases to obtain a subset of codewords of minimal support that can be used to determine the second generalized Hamming weight of the code. In this paper we establish conditions on a nonbinary code under which the same property holds. We also construct a family of codes over any nonbinary finite field where the property does not hold. Furthermore, we prove that whenever the subset obtained via Gr\"obner basis suffices to determine the second generalized Hamming weight, this invariant can also be recovered from the degrees of the syzygies of a minimal free resolution.