Weight distribution of cyclic codes defined by quadratic forms and related curves

Autores
Podesta, Ricardo Alberto; Videla Guzman, Denis Eduardo
Año de publicación
2021
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We consider cyclic codes CL associated to quadratic trace forms inm variables (Formula Presented) determined by a family L of q-linearized polynomials R over Fqm, and three related codes CL,0, CL,1, and CL,2. We describe the spectra for all these codes when L is an even rank family, in terms of the distribution of ranks of the forms QR in the family L, and we also computethe complete weight enumerator for CL. In particular, considering the family L = ‹xql›, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes Cl, Cl,0,Cl,1, and Cl,2 over Fq with zeros(Formula Presented) respectively,where q = ps with p prime, α is a generator of F*qm, and m/(m,l)is even. Finally, we give simple necessary and sufficient conditions for Artin–Schreier curves yp−y = xR(x)+βx, p prime, associated to polynomials R ∈ L to be optimal. We then obtain several maximal and minimal such curves inthe case (Formula Presented).
Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
Fil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
Materia
CYCLIC CODES
OPTIMAL CURVES
QUADRATIC FORMS
WEIGHT DISTRIBUTION
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/172738
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/172738
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Weight distribution of cyclic codes defined by quadratic forms and related curvesPodesta, Ricardo AlbertoVidela Guzman, Denis EduardoCYCLIC CODESOPTIMAL CURVESQUADRATIC FORMSWEIGHT DISTRIBUTIONhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We consider cyclic codes CL associated to quadratic trace forms inm variables (Formula Presented) determined by a family L of q-linearized polynomials R over Fqm, and three related codes CL,0, CL,1, and CL,2. We describe the spectra for all these codes when L is an even rank family, in terms of the distribution of ranks of the forms QR in the family L, and we also computethe complete weight enumerator for CL. In particular, considering the family L = ‹xql›, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes Cl, Cl,0,Cl,1, and Cl,2 over Fq with zeros(Formula Presented) respectively,where q = ps with p prime, α is a generator of F*qm, and m/(m,l)is even. Finally, we give simple necessary and sufficient conditions for Artin–Schreier curves yp−y = xR(x)+βx, p prime, associated to polynomials R ∈ L to be optimal. We then obtain several maximal and minimal such curves inthe case (Formula Presented).Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaUnión Matemática Argentina2021-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/172738Podesta, Ricardo Alberto; Videla Guzman, Denis Eduardo; Weight distribution of cyclic codes defined by quadratic forms and related curves; Unión Matemática Argentina; Revista de la Unión Matemática Argentina; 62; 1; 6-2021; 219-2420041-69321669-9637CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://inmabb.criba.edu.ar/revuma/revuma.php?p=doi/v62n1a15info:eu-repo/semantics/altIdentifier/doi/10.33044/revuma.1840info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:43:52Zoai:ri.conicet.gov.ar:11336/172738instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 09:43:53.126CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Weight distribution of cyclic codes defined by quadratic forms and related curves
title Weight distribution of cyclic codes defined by quadratic forms and related curves
spellingShingle Weight distribution of cyclic codes defined by quadratic forms and related curves
Podesta, Ricardo Alberto
CYCLIC CODES
OPTIMAL CURVES
QUADRATIC FORMS
WEIGHT DISTRIBUTION
title_short Weight distribution of cyclic codes defined by quadratic forms and related curves
title_full Weight distribution of cyclic codes defined by quadratic forms and related curves
title_fullStr Weight distribution of cyclic codes defined by quadratic forms and related curves
title_full_unstemmed Weight distribution of cyclic codes defined by quadratic forms and related curves
title_sort Weight distribution of cyclic codes defined by quadratic forms and related curves
dc.creator.none.fl_str_mv Podesta, Ricardo Alberto
Videla Guzman, Denis Eduardo
author Podesta, Ricardo Alberto
author_facet Podesta, Ricardo Alberto
Videla Guzman, Denis Eduardo
author_role author
author2 Videla Guzman, Denis Eduardo
author2_role author
dc.subject.none.fl_str_mv CYCLIC CODES
OPTIMAL CURVES
QUADRATIC FORMS
WEIGHT DISTRIBUTION
topic CYCLIC CODES
OPTIMAL CURVES
QUADRATIC FORMS
WEIGHT DISTRIBUTION
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We consider cyclic codes CL associated to quadratic trace forms inm variables (Formula Presented) determined by a family L of q-linearized polynomials R over Fqm, and three related codes CL,0, CL,1, and CL,2. We describe the spectra for all these codes when L is an even rank family, in terms of the distribution of ranks of the forms QR in the family L, and we also computethe complete weight enumerator for CL. In particular, considering the family L = ‹xql›, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes Cl, Cl,0,Cl,1, and Cl,2 over Fq with zeros(Formula Presented) respectively,where q = ps with p prime, α is a generator of F*qm, and m/(m,l)is even. Finally, we give simple necessary and sufficient conditions for Artin–Schreier curves yp−y = xR(x)+βx, p prime, associated to polynomials R ∈ L to be optimal. We then obtain several maximal and minimal such curves inthe case (Formula Presented).
Fil: Podesta, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
Fil: Videla Guzman, Denis Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
description We consider cyclic codes CL associated to quadratic trace forms inm variables (Formula Presented) determined by a family L of q-linearized polynomials R over Fqm, and three related codes CL,0, CL,1, and CL,2. We describe the spectra for all these codes when L is an even rank family, in terms of the distribution of ranks of the forms QR in the family L, and we also computethe complete weight enumerator for CL. In particular, considering the family L = ‹xql›, with l fixed in N, we give the weight distribution of four parametrized families of cyclic codes Cl, Cl,0,Cl,1, and Cl,2 over Fq with zeros(Formula Presented) respectively,where q = ps with p prime, α is a generator of F*qm, and m/(m,l)is even. Finally, we give simple necessary and sufficient conditions for Artin–Schreier curves yp−y = xR(x)+βx, p prime, associated to polynomials R ∈ L to be optimal. We then obtain several maximal and minimal such curves inthe case (Formula Presented).
publishDate 2021
dc.date.none.fl_str_mv 2021-06
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/172738
Podesta, Ricardo Alberto; Videla Guzman, Denis Eduardo; Weight distribution of cyclic codes defined by quadratic forms and related curves; Unión Matemática Argentina; Revista de la Unión Matemática Argentina; 62; 1; 6-2021; 219-242
0041-6932
1669-9637
CONICET Digital
CONICET
url http://hdl.handle.net/11336/172738
identifier_str_mv Podesta, Ricardo Alberto; Videla Guzman, Denis Eduardo; Weight distribution of cyclic codes defined by quadratic forms and related curves; Unión Matemática Argentina; Revista de la Unión Matemática Argentina; 62; 1; 6-2021; 219-242
0041-6932
1669-9637
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://inmabb.criba.edu.ar/revuma/revuma.php?p=doi/v62n1a15
info:eu-repo/semantics/altIdentifier/doi/10.33044/revuma.1840
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Unión Matemática Argentina
publisher.none.fl_str_mv Unión Matemática Argentina
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1844613380855300096
score 13.070432

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