Gröbner bases and cocyclic Hadamard matrices

Hadamard ideals were introduced in 2006 as a set of nonlin-ear polynomial equations whose zeros are uniquely related toHadamard matrices with one or two circulant cores of a given or-der. Based on this idea, the cocyclic Hadamard test enables us todescribe a polynomial ideal that characterizes the s...

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Autores: Álvarez Solano, Víctor, Armario Sampalo, José Andrés, Falcón Ganfornina, Raúl Manuel, Frau García, María Dolores, Gudiel Rodríguez, Félix
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2018
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/89718
Acceso en línea:https://hdl.handle.net/11441/89718
https://doi.org/10.1016/j.jsc.2017.09.001
Access Level:acceso abierto
Palabra clave:Hadamard matrices
Basis of cocycles
Polynomial ring
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spelling Gröbner bases and cocyclic Hadamard matricesÁlvarez Solano, VíctorArmario Sampalo, José AndrésFalcón Ganfornina, Raúl ManuelFrau García, María DoloresGudiel Rodríguez, FélixHadamard matricesBasis of cocyclesPolynomial ringHadamard ideals were introduced in 2006 as a set of nonlin-ear polynomial equations whose zeros are uniquely related toHadamard matrices with one or two circulant cores of a given or-der. Based on this idea, the cocyclic Hadamard test enables us todescribe a polynomial ideal that characterizes the set of cocyclicHadamard matrices over a fixed finite group Gof order 4t. Nev-ertheless, the complexity of the computation of the reduced Gröb-ner basis of this ideal is 2O(t2), which is excessive even for very small orders. In order to improve the efficiency of this polynomialmethod, we take advantage of some recent results on the innerstructure of a cocyclic matrix to describe an alternative polyno-mial ideal that also characterizes the aforementioned set of cocyclicHadamard matrices over G. The complexity of the computation de-creases in this way to 2O(t). Particularly, we design two specific procedures for looking for Zt×Z22-cocyclic Hadamard matrices and D4t-cocyclic Hadamard matrices, so that larger cocyclic Hadamard matrices (up to t≤39) are explicitly obtained.Junta de Andalucía FQM-016ElsevierMatemática Aplicada I2018info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/89718https://doi.org/10.1016/j.jsc.2017.09.001reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésJournal of Symbolic Computation, 89 (November-December 2018), 26-40.FQM-016https://www.sciencedirect.com/science/article/pii/S0747717117301098info:eu-repo/semantics/openAccessoai:idus.us.es:11441/897182026-06-17T12:51:07Z
dc.title.none.fl_str_mv Gröbner bases and cocyclic Hadamard matrices
title Gröbner bases and cocyclic Hadamard matrices
spellingShingle Gröbner bases and cocyclic Hadamard matrices
Álvarez Solano, Víctor
Hadamard matrices
Basis of cocycles
Polynomial ring
title_short Gröbner bases and cocyclic Hadamard matrices
title_full Gröbner bases and cocyclic Hadamard matrices
title_fullStr Gröbner bases and cocyclic Hadamard matrices
title_full_unstemmed Gröbner bases and cocyclic Hadamard matrices
title_sort Gröbner bases and cocyclic Hadamard matrices
dc.creator.none.fl_str_mv Álvarez Solano, Víctor
Armario Sampalo, José Andrés
Falcón Ganfornina, Raúl Manuel
Frau García, María Dolores
Gudiel Rodríguez, Félix
author Álvarez Solano, Víctor
author_facet Álvarez Solano, Víctor
Armario Sampalo, José Andrés
Falcón Ganfornina, Raúl Manuel
Frau García, María Dolores
Gudiel Rodríguez, Félix
author_role author
author2 Armario Sampalo, José Andrés
Falcón Ganfornina, Raúl Manuel
Frau García, María Dolores
Gudiel Rodríguez, Félix
author2_role author
author
author
author
dc.contributor.none.fl_str_mv Matemática Aplicada I
dc.subject.none.fl_str_mv Hadamard matrices
Basis of cocycles
Polynomial ring
topic Hadamard matrices
Basis of cocycles
Polynomial ring
description Hadamard ideals were introduced in 2006 as a set of nonlin-ear polynomial equations whose zeros are uniquely related toHadamard matrices with one or two circulant cores of a given or-der. Based on this idea, the cocyclic Hadamard test enables us todescribe a polynomial ideal that characterizes the set of cocyclicHadamard matrices over a fixed finite group Gof order 4t. Nev-ertheless, the complexity of the computation of the reduced Gröb-ner basis of this ideal is 2O(t2), which is excessive even for very small orders. In order to improve the efficiency of this polynomialmethod, we take advantage of some recent results on the innerstructure of a cocyclic matrix to describe an alternative polyno-mial ideal that also characterizes the aforementioned set of cocyclicHadamard matrices over G. The complexity of the computation de-creases in this way to 2O(t). Particularly, we design two specific procedures for looking for Zt×Z22-cocyclic Hadamard matrices and D4t-cocyclic Hadamard matrices, so that larger cocyclic Hadamard matrices (up to t≤39) are explicitly obtained.
publishDate 2018
dc.date.none.fl_str_mv 2018
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/89718
https://doi.org/10.1016/j.jsc.2017.09.001
url https://hdl.handle.net/11441/89718
https://doi.org/10.1016/j.jsc.2017.09.001
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of Symbolic Computation, 89 (November-December 2018), 26-40.
FQM-016
https://www.sciencedirect.com/science/article/pii/S0747717117301098
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.301603

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