E8-lattice via the cyclotomic field Q(ξ24)
Autor(a) principal: | |
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Data de Publicação: | 2018 |
Outros Autores: | , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.12732/ijam.v31i1.6 http://hdl.handle.net/11449/221020 |
Resumo: | Lattices can be applied in different areas of research, particularly, they can be applied in information theory and encryption schemes. Signal constellations having lattice structure have been used as a support for signal transmission over the Gaussian and Rayleigh fading channels. The problem to find a good signal constellation for Gaussian channels is associated to the search of lattices which present a good packing density, that is, dense lattices. In this way, we propose an algebraic framework to construct the dense lattice E8 from the principal ideal I = ((1 + ξ3) + ξ3ξ24 + ξ3ξ24 2) of the cyclotomic field Q(ξ24), where ξ3 and ξ24 are the third and 24-th root of unity, respectively. The advantage of obtaining lattices from this method is the identification of the lattice points with the elements of a number field. Consequently, it is possible to utilize some properties of number fields in the study of such lattices. |
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Repositório Institucional da UNESP |
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E8-lattice via the cyclotomic field Q(ξ24)Cyclotomic fieldDense latticeE8-latticePrincipal idealLattices can be applied in different areas of research, particularly, they can be applied in information theory and encryption schemes. Signal constellations having lattice structure have been used as a support for signal transmission over the Gaussian and Rayleigh fading channels. The problem to find a good signal constellation for Gaussian channels is associated to the search of lattices which present a good packing density, that is, dense lattices. In this way, we propose an algebraic framework to construct the dense lattice E8 from the principal ideal I = ((1 + ξ3) + ξ3ξ24 + ξ3ξ24 2) of the cyclotomic field Q(ξ24), where ξ3 and ξ24 are the third and 24-th root of unity, respectively. The advantage of obtaining lattices from this method is the identification of the lattice points with the elements of a number field. Consequently, it is possible to utilize some properties of number fields in the study of such lattices.Department of Communications (DECOM) Campinas State UniversityDepartment of Communications and Electronics Télécom ParisTechDepartment of Mathematics São Paulo State UniversityTelecommunications Engineering São Paulo State UniversityDepartment of Mathematics São Paulo State UniversityTelecommunications Engineering São Paulo State UniversityUniversidade Estadual de Campinas (UNICAMP)Télécom ParisTechUniversidade Estadual Paulista (UNESP)Trinca Watanabe, C. C.Belfiore, J. C.De Carvalho, E. D. [UNESP]Vieira Filho, J. [UNESP]2022-04-28T19:08:43Z2022-04-28T19:08:43Z2018-01-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article63-72http://dx.doi.org/10.12732/ijam.v31i1.6International Journal of Applied Mathematics, v. 31, n. 1, p. 63-72, 2018.1314-80601311-1728http://hdl.handle.net/11449/22102010.12732/ijam.v31i1.62-s2.0-85042432002Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengInternational Journal of Applied Mathematicsinfo:eu-repo/semantics/openAccess2022-04-28T19:08:43Zoai:repositorio.unesp.br:11449/221020Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462022-04-28T19:08:43Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
E8-lattice via the cyclotomic field Q(ξ24) |
title |
E8-lattice via the cyclotomic field Q(ξ24) |
spellingShingle |
E8-lattice via the cyclotomic field Q(ξ24) Trinca Watanabe, C. C. Cyclotomic field Dense lattice E8-lattice Principal ideal |
title_short |
E8-lattice via the cyclotomic field Q(ξ24) |
title_full |
E8-lattice via the cyclotomic field Q(ξ24) |
title_fullStr |
E8-lattice via the cyclotomic field Q(ξ24) |
title_full_unstemmed |
E8-lattice via the cyclotomic field Q(ξ24) |
title_sort |
E8-lattice via the cyclotomic field Q(ξ24) |
author |
Trinca Watanabe, C. C. |
author_facet |
Trinca Watanabe, C. C. Belfiore, J. C. De Carvalho, E. D. [UNESP] Vieira Filho, J. [UNESP] |
author_role |
author |
author2 |
Belfiore, J. C. De Carvalho, E. D. [UNESP] Vieira Filho, J. [UNESP] |
author2_role |
author author author |
dc.contributor.none.fl_str_mv |
Universidade Estadual de Campinas (UNICAMP) Télécom ParisTech Universidade Estadual Paulista (UNESP) |
dc.contributor.author.fl_str_mv |
Trinca Watanabe, C. C. Belfiore, J. C. De Carvalho, E. D. [UNESP] Vieira Filho, J. [UNESP] |
dc.subject.por.fl_str_mv |
Cyclotomic field Dense lattice E8-lattice Principal ideal |
topic |
Cyclotomic field Dense lattice E8-lattice Principal ideal |
description |
Lattices can be applied in different areas of research, particularly, they can be applied in information theory and encryption schemes. Signal constellations having lattice structure have been used as a support for signal transmission over the Gaussian and Rayleigh fading channels. The problem to find a good signal constellation for Gaussian channels is associated to the search of lattices which present a good packing density, that is, dense lattices. In this way, we propose an algebraic framework to construct the dense lattice E8 from the principal ideal I = ((1 + ξ3) + ξ3ξ24 + ξ3ξ24 2) of the cyclotomic field Q(ξ24), where ξ3 and ξ24 are the third and 24-th root of unity, respectively. The advantage of obtaining lattices from this method is the identification of the lattice points with the elements of a number field. Consequently, it is possible to utilize some properties of number fields in the study of such lattices. |
publishDate |
2018 |
dc.date.none.fl_str_mv |
2018-01-01 2022-04-28T19:08:43Z 2022-04-28T19:08:43Z |
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/10.12732/ijam.v31i1.6 International Journal of Applied Mathematics, v. 31, n. 1, p. 63-72, 2018. 1314-8060 1311-1728 http://hdl.handle.net/11449/221020 10.12732/ijam.v31i1.6 2-s2.0-85042432002 |
url |
http://dx.doi.org/10.12732/ijam.v31i1.6 http://hdl.handle.net/11449/221020 |
identifier_str_mv |
International Journal of Applied Mathematics, v. 31, n. 1, p. 63-72, 2018. 1314-8060 1311-1728 10.12732/ijam.v31i1.6 2-s2.0-85042432002 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
International Journal of Applied Mathematics |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
63-72 |
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_ |
1803046158172422144 |