Gröbner bases and the second generalized Hamming weight of a linear code
Hernán de Alba, Cecilia Martínez-Reyes
TL;DR
This work extends Gröbner-basis methods for recovering the second generalized Hamming weight $d_2(\mathcal{C})$ from minimal-support codewords to nonbinary codes by introducing the binomial ideal $I(\mathcal{C})$ and the derived set $M_{\mathcal{G}}$. It proves a sufficient condition under which $M_{\mathcal{G}}$ forms a $d_2$-test set, but also shows that for every $q>2$ there exist codes where this fails, clarifying the limitations of the binary analogue. Additionally, it establishes a Betti-number perspective: when $M_{\mathcal{G}}$ is a $d_2$-test set, $d_1(\mathcal{C})$ and $d_2(\mathcal{C})$ can be recovered from the degrees of syzygies in a minimal free resolution of $R/I_{M_{\mathcal{G}}}$. The findings illuminate both the reach and the constraints of Gröbner-basis approaches for nonbinary GHWS and connect these combinatorial invariants to homological algebra via minimal free resolutions.
Abstract
It is known that for binary codes one can use Gröbner 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 Gröbner 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.