A System of Equations for Describing Cocyclic Hadamard Matrices

Given a basis equation image for 2-cocycles equation image over a group G of order equation image, we describe a nonlinear system of 4t-1 equations and k indeterminates equation image over equation image, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that...

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Autores: Álvarez Solano, Víctor, Armario Sampalo, José Andrés, Frau García, María Dolores, Real Jurado, Pedro
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2008
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/114826
Acceso en línea:https://hdl.handle.net/11441/114826
https://doi.org/10.1002/jcd.20191
Access Level:acceso abierto
Palabra clave:Hadamard matrix
Cocyclic matrix
Coboundary matrix
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spelling A System of Equations for Describing Cocyclic Hadamard MatricesÁlvarez Solano, VíctorArmario Sampalo, José AndrésFrau García, María DoloresReal Jurado, PedroHadamard matrixCocyclic matrixCoboundary matrixGiven a basis equation image for 2-cocycles equation image over a group G of order equation image, we describe a nonlinear system of 4t-1 equations and k indeterminates equation image over equation image, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that (equation image) is a solution of the system if and only if the 2-cocycle equation image gives rise to a cocyclic Hadamard matrix equation image. Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in equation image which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups equation image and equation image. A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups equation image, we have found that it suffices to check t instead of the 4t rows of equation image, to decide the Hadamard character of the matrix (for a special class of cocycles f).Junta de Andalucía FQM-296WileyMatemática Aplicada IJunta de Andalucía2008info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/114826https://doi.org/10.1002/jcd.20191reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésJournal of Combinatorial Designs, 16 (4), 276-290.FQM-296https://onlinelibrary.wiley.com/doi/abs/10.1002/jcd.20191info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1148262026-06-17T12:51:07Z
dc.title.none.fl_str_mv A System of Equations for Describing Cocyclic Hadamard Matrices
title A System of Equations for Describing Cocyclic Hadamard Matrices
spellingShingle A System of Equations for Describing Cocyclic Hadamard Matrices
Álvarez Solano, Víctor
Hadamard matrix
Cocyclic matrix
Coboundary matrix
title_short A System of Equations for Describing Cocyclic Hadamard Matrices
title_full A System of Equations for Describing Cocyclic Hadamard Matrices
title_fullStr A System of Equations for Describing Cocyclic Hadamard Matrices
title_full_unstemmed A System of Equations for Describing Cocyclic Hadamard Matrices
title_sort A System of Equations for Describing Cocyclic Hadamard Matrices
dc.creator.none.fl_str_mv Álvarez Solano, Víctor
Armario Sampalo, José Andrés
Frau García, María Dolores
Real Jurado, Pedro
author Álvarez Solano, Víctor
author_facet Álvarez Solano, Víctor
Armario Sampalo, José Andrés
Frau García, María Dolores
Real Jurado, Pedro
author_role author
author2 Armario Sampalo, José Andrés
Frau García, María Dolores
Real Jurado, Pedro
author2_role author
author
author
dc.contributor.none.fl_str_mv Matemática Aplicada I
Junta de Andalucía
dc.subject.none.fl_str_mv Hadamard matrix
Cocyclic matrix
Coboundary matrix
topic Hadamard matrix
Cocyclic matrix
Coboundary matrix
description Given a basis equation image for 2-cocycles equation image over a group G of order equation image, we describe a nonlinear system of 4t-1 equations and k indeterminates equation image over equation image, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that (equation image) is a solution of the system if and only if the 2-cocycle equation image gives rise to a cocyclic Hadamard matrix equation image. Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in equation image which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups equation image and equation image. A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups equation image, we have found that it suffices to check t instead of the 4t rows of equation image, to decide the Hadamard character of the matrix (for a special class of cocycles f).
publishDate 2008
dc.date.none.fl_str_mv 2008
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/11441/114826
https://doi.org/10.1002/jcd.20191
url https://hdl.handle.net/11441/114826
https://doi.org/10.1002/jcd.20191
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of Combinatorial Designs, 16 (4), 276-290.
FQM-296
https://onlinelibrary.wiley.com/doi/abs/10.1002/jcd.20191
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 Wiley
publisher.none.fl_str_mv Wiley
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 15,301629

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