Free Resolutions and Generalized Hamming Weights of binary linear codes

TL;DR

This paper investigates how free resolutions of monomial ideals relate to the generalized Hamming weights of binary linear codes, providing methods to compute and bound these weights using algebraic invariants.

Contribution

It introduces a novel approach linking free resolutions to GHWs, enabling computation of the first two weights and bounds for the others via algebraic structures.

Findings

First and second GHWs can be computed from graded free resolutions.

Remaining GHWs are bounded by Betti numbers.

Provides algebraic tools for analyzing binary codes.

Abstract

In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.