Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus
Current research builds labelings for geometrically uniform codes on the double torus through tiling groups. At least one labeling group was provided for all of the 11 regular tessellations on the double torus, derived from triangular Fuchsian groups, as well as extensions of these labeling groups t...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/3/449 |
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| author | Eduardo Michel Vieira Gomes Edson Donizete de Carvalho Carlos Alexandre Ribeiro Martins Waldir Silva Soares Eduardo Brandani da Silva |
| author_facet | Eduardo Michel Vieira Gomes Edson Donizete de Carvalho Carlos Alexandre Ribeiro Martins Waldir Silva Soares Eduardo Brandani da Silva |
| author_sort | Eduardo Michel Vieira Gomes |
| collection | DOAJ |
| description | Current research builds labelings for geometrically uniform codes on the double torus through tiling groups. At least one labeling group was provided for all of the 11 regular tessellations on the double torus, derived from triangular Fuchsian groups, as well as extensions of these labeling groups to generate new codes. An important consequence is that such techniques can be used to label geometrically uniform codes on surfaces with greater genera. Furthermore, partitioning chains are constructed into geometrically uniform codes using soluble groups as labeling, which in some cases results in an Ungerboeck partitioning for the surface. As a result of these constructions, it is demonstrated that, as in Euclidean spaces, modulation and encoding can be combined in a single step in hyperbolic space. |
| first_indexed | 2024-03-09T12:26:08Z |
| format | Article |
| id | doaj.art-cbc394f0afab479d8da0d137562f7535 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T12:26:08Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-cbc394f0afab479d8da0d137562f75352023-11-30T22:34:46ZengMDPI AGSymmetry2073-89942022-02-0114344910.3390/sym14030449Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double TorusEduardo Michel Vieira Gomes0Edson Donizete de Carvalho1Carlos Alexandre Ribeiro Martins2Waldir Silva Soares3Eduardo Brandani da Silva4Department of Mathematics, Campus de Francisco Beltrão, Universidade Técnica Federal do Paraná, UTFPR, Linha Santa Bárbara s/n, Francisco Beltrão 85601-970, BrazilDepartment of Mathematics, Câmpus de Ilha Solteira, Universidade Estadual Paulista, UNESP, Av. Brasil Sul, 56, Ilha Solteira 15385-000, BrazilDepartment of Mathematics, Campus de Pato BrancoUTFPR, Universidade Técnica Federal do Paraná, UTFPR, Via do Conhecimento, s/n-KM 01-Fraron, Pato Branco 85503-390, BrazilDepartment of Mathematics, Campus de Pato BrancoUTFPR, Universidade Técnica Federal do Paraná, UTFPR, Via do Conhecimento, s/n-KM 01-Fraron, Pato Branco 85503-390, BrazilDepartment of Mathematics, Universidade Estadual de Maringá, UEM, Av. Colombo 5790, Maringá 87020-900, BrazilCurrent research builds labelings for geometrically uniform codes on the double torus through tiling groups. At least one labeling group was provided for all of the 11 regular tessellations on the double torus, derived from triangular Fuchsian groups, as well as extensions of these labeling groups to generate new codes. An important consequence is that such techniques can be used to label geometrically uniform codes on surfaces with greater genera. Furthermore, partitioning chains are constructed into geometrically uniform codes using soluble groups as labeling, which in some cases results in an Ungerboeck partitioning for the surface. As a result of these constructions, it is demonstrated that, as in Euclidean spaces, modulation and encoding can be combined in a single step in hyperbolic space.https://www.mdpi.com/2073-8994/14/3/449geometrically uniform codeshyperbolic geometryUngerboeck partitioningdouble torusFuchsian groupssignal constellations |
| spellingShingle | Eduardo Michel Vieira Gomes Edson Donizete de Carvalho Carlos Alexandre Ribeiro Martins Waldir Silva Soares Eduardo Brandani da Silva Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus Symmetry geometrically uniform codes hyperbolic geometry Ungerboeck partitioning double torus Fuchsian groups signal constellations |
| title | Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus |
| title_full | Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus |
| title_fullStr | Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus |
| title_full_unstemmed | Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus |
| title_short | Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus |
| title_sort | hyperbolic geometrically uniform codes and ungerboeck partitioning on the double torus |
| topic | geometrically uniform codes hyperbolic geometry Ungerboeck partitioning double torus Fuchsian groups signal constellations |
| url | https://www.mdpi.com/2073-8994/14/3/449 |
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