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Free Resolutions and Generalized Hamming Weights of Binary Linear Codes

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ignacio García-Marco & Irene Márquez-Corbella & Edgar Martínez-Moro & Yuriko Pitones, 2022. "Free Resolutions and Generalized Hamming Weights of Binary Linear Codes," Mathematics, MDPI, vol. 10(12), pages 1-15, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2079-:d:839819
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