On recursive constructions of Z2Z4Z8-linear Hadamard codes

The Z2Z4Z8-additive codes are subgroups of Z α1 2 × Z α2 4 × Z α3 8 . A Z2Z4Z8-linear Hadamard code is a Hadamard code, which is the Gray map image of a Z2Z4Z8-additive code. In this paper, we generalize some known results for Z2Z4-linear Hadamard codes to Z2Z4Z8-linear Hadamard codes with α1 ̸= 0,...

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Autores: Bhunia, Dipak Kumar|||0000-0003-4852-8739, Fernández Córdoba, Cristina|||0000-0001-5880-144X, Villanueva, M|||0000-0001-6179-0833
Tipo de recurso: artículo
Fecha de publicación:2024
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:285314
Acceso en línea:https://ddd.uab.cat/record/285314
https://dx.doi.org/urn:doi:10.3934/amc.2023047
Access Level:acceso abierto
Palabra clave:Hadamard code
Gray map
Z2 Z4 Z8-linear code
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spelling On recursive constructions of Z2Z4Z8-linear Hadamard codesBhunia, Dipak Kumar|||0000-0003-4852-8739Fernández Córdoba, Cristina|||0000-0001-5880-144XVillanueva, M|||0000-0001-6179-0833Hadamard codeGray mapZ2 Z4 Z8-linear codeThe Z2Z4Z8-additive codes are subgroups of Z α1 2 × Z α2 4 × Z α3 8 . A Z2Z4Z8-linear Hadamard code is a Hadamard code, which is the Gray map image of a Z2Z4Z8-additive code. In this paper, we generalize some known results for Z2Z4-linear Hadamard codes to Z2Z4Z8-linear Hadamard codes with α1 ̸= 0, α2 ̸= 0, and α3 ̸= 0. First, we give a recursive construction of Z2Z4Z8- additive Hadamard codes of type (α1, α2, α3;t1, t2, t3) with t1 ≥ 1, t2 ≥ 0, and t3 ≥ 1. It is known that each Z4-linear Hadamard code is equivalent to a Z2Z4-linear Hadamard code with α1 ̸= 0 and α2 ̸= 0. Unlike Z2Z4-linear Hadamard codes, in general, this family of Z2Z4Z8-linear Hadamard codes does not include the family of Z4-linear or Z8-linear Hadamard codes. We show that, for example, for length 211, the constructed nonlinear Z2Z4Z8-linear Hadamard codes are not equivalent to each other, nor to any Z2Z4-linear Hadamard, nor to any previously constructed Z2s -Hadamard code, with s ≥ 2. Finally, we also present other recursive constructions of Z2Z4Z8-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones. 22024-01-0120242024-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/285314https://dx.doi.org/urn:doi:10.3934/amc.2023047reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104664GB-I00Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2022-137924NB-I00Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 RED2022-134306-TAgència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2020/FI-SDUR-00475Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00643open accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2853142026-06-06T12:50:31Z
dc.title.none.fl_str_mv On recursive constructions of Z2Z4Z8-linear Hadamard codes
title On recursive constructions of Z2Z4Z8-linear Hadamard codes
spellingShingle On recursive constructions of Z2Z4Z8-linear Hadamard codes
Bhunia, Dipak Kumar|||0000-0003-4852-8739
Hadamard code
Gray map
Z2 Z4 Z8-linear code
title_short On recursive constructions of Z2Z4Z8-linear Hadamard codes
title_full On recursive constructions of Z2Z4Z8-linear Hadamard codes
title_fullStr On recursive constructions of Z2Z4Z8-linear Hadamard codes
title_full_unstemmed On recursive constructions of Z2Z4Z8-linear Hadamard codes
title_sort On recursive constructions of Z2Z4Z8-linear Hadamard codes
dc.creator.none.fl_str_mv Bhunia, Dipak Kumar|||0000-0003-4852-8739
Fernández Córdoba, Cristina|||0000-0001-5880-144X
Villanueva, M|||0000-0001-6179-0833
author Bhunia, Dipak Kumar|||0000-0003-4852-8739
author_facet Bhunia, Dipak Kumar|||0000-0003-4852-8739
Fernández Córdoba, Cristina|||0000-0001-5880-144X
Villanueva, M|||0000-0001-6179-0833
author_role author
author2 Fernández Córdoba, Cristina|||0000-0001-5880-144X
Villanueva, M|||0000-0001-6179-0833
author2_role author
author
dc.subject.none.fl_str_mv Hadamard code
Gray map
Z2 Z4 Z8-linear code
topic Hadamard code
Gray map
Z2 Z4 Z8-linear code
description The Z2Z4Z8-additive codes are subgroups of Z α1 2 × Z α2 4 × Z α3 8 . A Z2Z4Z8-linear Hadamard code is a Hadamard code, which is the Gray map image of a Z2Z4Z8-additive code. In this paper, we generalize some known results for Z2Z4-linear Hadamard codes to Z2Z4Z8-linear Hadamard codes with α1 ̸= 0, α2 ̸= 0, and α3 ̸= 0. First, we give a recursive construction of Z2Z4Z8- additive Hadamard codes of type (α1, α2, α3;t1, t2, t3) with t1 ≥ 1, t2 ≥ 0, and t3 ≥ 1. It is known that each Z4-linear Hadamard code is equivalent to a Z2Z4-linear Hadamard code with α1 ̸= 0 and α2 ̸= 0. Unlike Z2Z4-linear Hadamard codes, in general, this family of Z2Z4Z8-linear Hadamard codes does not include the family of Z4-linear or Z8-linear Hadamard codes. We show that, for example, for length 211, the constructed nonlinear Z2Z4Z8-linear Hadamard codes are not equivalent to each other, nor to any Z2Z4-linear Hadamard, nor to any previously constructed Z2s -Hadamard code, with s ≥ 2. Finally, we also present other recursive constructions of Z2Z4Z8-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones.
publishDate 2024
dc.date.none.fl_str_mv 2
2024-01-01
2024
2024-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
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dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/285314
https://dx.doi.org/urn:doi:10.3934/amc.2023047
url https://ddd.uab.cat/record/285314
https://dx.doi.org/urn:doi:10.3934/amc.2023047
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104664GB-I00
Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2022-137924NB-I00
Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 RED2022-134306-T
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2020/FI-SDUR-00475
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00643
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
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