Lattices from abelian extensions and error-correcting codes
Autor(a) principal: | |
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Data de Publicação: | 2021 |
Outros Autores: | , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1216/rmj.2021.51.903 http://hdl.handle.net/11449/233424 |
Resumo: | A construction of laminated lattices of full diversity in odd dimensions d with 3 ≤ d ≤ 15 is presented. The technique, which uses a combination of number fields and error-correcting codes, consists essentially of two steps: In the first, the Abelian number field F of degree d and prime conductor p, where p is a prime congruent to 1 modulo d, is considered. In the second, the lattice is obtained as the canonical embedding (Minkowski homomorphism) of a Z-submodule of OF, the ring of integers of F. The submodule is defined by the parity-check matrices of a Reed–Solomon code over GF(p) and a suitably chosen linear code, typically either binary or over Z/4Z, the ring of integers modulo 4. |
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Lattices from abelian extensions and error-correcting codesAbelian extensionsCyclotomic fieldsError-correcting codesLattice packingQuadratic formsA construction of laminated lattices of full diversity in odd dimensions d with 3 ≤ d ≤ 15 is presented. The technique, which uses a combination of number fields and error-correcting codes, consists essentially of two steps: In the first, the Abelian number field F of degree d and prime conductor p, where p is a prime congruent to 1 modulo d, is considered. In the second, the lattice is obtained as the canonical embedding (Minkowski homomorphism) of a Z-submodule of OF, the ring of integers of F. The submodule is defined by the parity-check matrices of a Reed–Solomon code over GF(p) and a suitably chosen linear code, typically either binary or over Z/4Z, the ring of integers modulo 4.Department of Mathematics and Statistics San Diego State UniversityDepartamento de Matematica Universidade Estadual PaulistaDepartamento de Matematica Universidade Federal do CearaDepartamento de Matematica Universidade Estadual PaulistaSan Diego State UniversityUniversidade Estadual Paulista (UNESP)Universidade Federal do CearaInterlando, J. Carmeloda Nóbrega Neto, Trajano Pires [UNESP]Nunes, José Valter LopesLopes, José Othon Dantas2022-05-01T08:44:37Z2022-05-01T08:44:37Z2021-06-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article903-920http://dx.doi.org/10.1216/rmj.2021.51.903Rocky Mountain Journal of Mathematics, v. 51, n. 3, p. 903-920, 2021.1945-37950035-7596http://hdl.handle.net/11449/23342410.1216/rmj.2021.51.9032-s2.0-85113191469Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengRocky Mountain Journal of Mathematicsinfo:eu-repo/semantics/openAccess2022-05-01T08:44:37Zoai:repositorio.unesp.br:11449/233424Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T20:44:44.602229Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
Lattices from abelian extensions and error-correcting codes |
title |
Lattices from abelian extensions and error-correcting codes |
spellingShingle |
Lattices from abelian extensions and error-correcting codes Interlando, J. Carmelo Abelian extensions Cyclotomic fields Error-correcting codes Lattice packing Quadratic forms |
title_short |
Lattices from abelian extensions and error-correcting codes |
title_full |
Lattices from abelian extensions and error-correcting codes |
title_fullStr |
Lattices from abelian extensions and error-correcting codes |
title_full_unstemmed |
Lattices from abelian extensions and error-correcting codes |
title_sort |
Lattices from abelian extensions and error-correcting codes |
author |
Interlando, J. Carmelo |
author_facet |
Interlando, J. Carmelo da Nóbrega Neto, Trajano Pires [UNESP] Nunes, José Valter Lopes Lopes, José Othon Dantas |
author_role |
author |
author2 |
da Nóbrega Neto, Trajano Pires [UNESP] Nunes, José Valter Lopes Lopes, José Othon Dantas |
author2_role |
author author author |
dc.contributor.none.fl_str_mv |
San Diego State University Universidade Estadual Paulista (UNESP) Universidade Federal do Ceara |
dc.contributor.author.fl_str_mv |
Interlando, J. Carmelo da Nóbrega Neto, Trajano Pires [UNESP] Nunes, José Valter Lopes Lopes, José Othon Dantas |
dc.subject.por.fl_str_mv |
Abelian extensions Cyclotomic fields Error-correcting codes Lattice packing Quadratic forms |
topic |
Abelian extensions Cyclotomic fields Error-correcting codes Lattice packing Quadratic forms |
description |
A construction of laminated lattices of full diversity in odd dimensions d with 3 ≤ d ≤ 15 is presented. The technique, which uses a combination of number fields and error-correcting codes, consists essentially of two steps: In the first, the Abelian number field F of degree d and prime conductor p, where p is a prime congruent to 1 modulo d, is considered. In the second, the lattice is obtained as the canonical embedding (Minkowski homomorphism) of a Z-submodule of OF, the ring of integers of F. The submodule is defined by the parity-check matrices of a Reed–Solomon code over GF(p) and a suitably chosen linear code, typically either binary or over Z/4Z, the ring of integers modulo 4. |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-06-01 2022-05-01T08:44:37Z 2022-05-01T08:44:37Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.1216/rmj.2021.51.903 Rocky Mountain Journal of Mathematics, v. 51, n. 3, p. 903-920, 2021. 1945-3795 0035-7596 http://hdl.handle.net/11449/233424 10.1216/rmj.2021.51.903 2-s2.0-85113191469 |
url |
http://dx.doi.org/10.1216/rmj.2021.51.903 http://hdl.handle.net/11449/233424 |
identifier_str_mv |
Rocky Mountain Journal of Mathematics, v. 51, n. 3, p. 903-920, 2021. 1945-3795 0035-7596 10.1216/rmj.2021.51.903 2-s2.0-85113191469 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Rocky Mountain Journal of Mathematics |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
903-920 |
dc.source.none.fl_str_mv |
Scopus reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
|
_version_ |
1808129241310560256 |