On the generalized Hamming weights of hyperbolic codes

A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides...

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Detalhes bibliográficos
Autores: Márquez Corbella, Irene, Camps-Moreno, Eduardo, Garc´ıa-Marco, Ignacio, L´opez, Hiram H.
Formato: artículo
Fecha de publicación:2024
País:España
Recursos:Universidad de La Laguna (ULL)
Repositorio:RIULL. Repositorio Institucional de la Universidad de La Laguna
OAI Identifier:oai:riull.ull.es:915/40893
Acesso em linha:http://riull.ull.es/xmlui/handle/915/40893
https://doi.org/10.1142/S0219498825500628
Access Level:acceso abierto
Palavra-chave:Reed–Muller
Evaluation codes
Hyperbolic codes
Generalized Hamming weights
Footprint
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spelling On the generalized Hamming weights of hyperbolic codesMárquez Corbella, IreneCamps-Moreno, EduardoGarc´ıa-Marco, IgnacioL´opez, Hiram H.Reed–MullerEvaluation codesHyperbolic codesGeneralized Hamming weightsFootprintA hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code C, we find the largest Reed–Muller code contained in C and the smallest Reed–Muller code containing C. We then prove that similar to Reed–Muller and affine Cartesian codes, the rth generalized Hamming weight and the rth footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the rth footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the rth footprint of a hyperbolic code that, sometimes, are sharp.Grupo de investigación GASIULL (Geometría algebraica: singularidades e invariantes)202520252024info:eu-repo/semantics/articleapplication/pdfhttp://riull.ull.es/xmlui/handle/915/40893https://doi.org/10.1142/S0219498825500628reponame:RIULL. Repositorio Institucional de la Universidad de La Lagunainstname:Universidad de La Laguna (ULL)InglésJournal of Algebra and its Applications, 23(7).Licencia Creative Commons (Reconocimiento-No comercial-Sin obras derivadas 4.0 Internacional)info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/4.0/deed.es_ESoai:riull.ull.es:915/408932026-06-22T13:13:57Z
dc.title.none.fl_str_mv On the generalized Hamming weights of hyperbolic codes
title On the generalized Hamming weights of hyperbolic codes
spellingShingle On the generalized Hamming weights of hyperbolic codes
Márquez Corbella, Irene
Reed–Muller
Evaluation codes
Hyperbolic codes
Generalized Hamming weights
Footprint
title_short On the generalized Hamming weights of hyperbolic codes
title_full On the generalized Hamming weights of hyperbolic codes
title_fullStr On the generalized Hamming weights of hyperbolic codes
title_full_unstemmed On the generalized Hamming weights of hyperbolic codes
title_sort On the generalized Hamming weights of hyperbolic codes
dc.creator.none.fl_str_mv Márquez Corbella, Irene
Camps-Moreno, Eduardo
Garc´ıa-Marco, Ignacio
L´opez, Hiram H.
author Márquez Corbella, Irene
author_facet Márquez Corbella, Irene
Camps-Moreno, Eduardo
Garc´ıa-Marco, Ignacio
L´opez, Hiram H.
author_role author
author2 Camps-Moreno, Eduardo
Garc´ıa-Marco, Ignacio
L´opez, Hiram H.
author2_role author
author
author
dc.contributor.none.fl_str_mv Grupo de investigación GASIULL (Geometría algebraica: singularidades e invariantes)
dc.subject.none.fl_str_mv Reed–Muller
Evaluation codes
Hyperbolic codes
Generalized Hamming weights
Footprint
topic Reed–Muller
Evaluation codes
Hyperbolic codes
Generalized Hamming weights
Footprint
description A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code C, we find the largest Reed–Muller code contained in C and the smallest Reed–Muller code containing C. We then prove that similar to Reed–Muller and affine Cartesian codes, the rth generalized Hamming weight and the rth footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the rth footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the rth footprint of a hyperbolic code that, sometimes, are sharp.
publishDate 2024
dc.date.none.fl_str_mv 2024
2025
2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://riull.ull.es/xmlui/handle/915/40893
https://doi.org/10.1142/S0219498825500628
url http://riull.ull.es/xmlui/handle/915/40893
https://doi.org/10.1142/S0219498825500628
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of Algebra and its Applications, 23(7).
dc.rights.none.fl_str_mv Licencia Creative Commons (Reconocimiento-No comercial-Sin obras derivadas 4.0 Internacional)
info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/4.0/deed.es_ES
rights_invalid_str_mv Licencia Creative Commons (Reconocimiento-No comercial-Sin obras derivadas 4.0 Internacional)
https://creativecommons.org/licenses/by-nc-nd/4.0/deed.es_ES
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:RIULL. Repositorio Institucional de la Universidad de La Laguna
instname:Universidad de La Laguna (ULL)
instname_str Universidad de La Laguna (ULL)
reponame_str RIULL. Repositorio Institucional de la Universidad de La Laguna
collection RIULL. Repositorio Institucional de la Universidad de La Laguna
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,812455

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