Free Resolutions and Generalized Hamming Weights of binary linear codes
Information Theory
2022-04-01 v1 Commutative Algebra
math.IT
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.
Keywords
Cite
@article{arxiv.2203.17194,
title = {Free Resolutions and Generalized Hamming Weights of binary linear codes},
author = {Ignacio García-Marco and Irene Márquez-Corbella and Edgar Martínez-Moro and Yuriko Pitones},
journal= {arXiv preprint arXiv:2203.17194},
year = {2022}
}