Free Resolutions and Generalized Hamming Weights of Binary Linear Codes

Ignacio García-Marco, Irene Márquez-Corbella, Edgar Martínez-Moro and Yuriko Pitones
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Ignacio García-Marco: Departamento Matemáticas, Estadística e I.O. and Instituto de Matemáticas y Aplicaciones (IMAULL), Sección de Matemáticas, Universidad de La Laguna, Apartado de Correos 456, 38200 La Laguna, Spain
Irene Márquez-Corbella: Departamento Matemáticas, Estadística e I.O. and Instituto de Matemáticas y Aplicaciones (IMAULL), Sección de Matemáticas, Universidad de La Laguna, Apartado de Correos 456, 38200 La Laguna, Spain
Edgar Martínez-Moro: Institute of Mathematics, University of Valladolid, 47011 Valladolid, Spain
Yuriko Pitones: Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Mexico City 09310, Mexico

Mathematics, 2022, vol. 10, issue 12, 1-15

Abstract: 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.

Keywords: generalized Hamming weight; graded free resolution; second distance; binary code (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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