Free Resolutions and Generalized Hamming Weights of Binary Linear Codes

In this work, we explore the relationship between the graded free resolution of some monomial ideals and the Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure that is smaller than the set of codewords of minimal support that provides us with some information...

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Bibliographic Details
Published in:Mathematics
Main Authors: Ignacio García-Marco, Irene Márquez-Corbella, Edgar Martínez-Moro, Yuriko Pitones
Format: Article
Language:English
Published: MDPI AG 2022-06-01
Subjects:
generalized Hamming weight
graded free resolution
second distance
binary code
Online Access:https://www.mdpi.com/2227-7390/10/12/2079
Description
Summary:In this work, we explore the relationship between the graded free resolution of some monomial ideals and the Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure that is smaller than the set of codewords of minimal support that provides us with some information about the GHWs. We prove that the first and second generalized Hamming weights of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated with a binomial ideal related with the code. Moreover, the remaining weights are bounded above by the degrees of the syzygies in the resolution.
ISSN:2227-7390

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