Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus
Eduardo Michel Vieira Gomes,
Edson Donizete de Carvalho,
Carlos Alexandre Ribeiro Martins,
Waldir Silva Soares,
Eduardo Brandani da Silva
Affiliations
Eduardo Michel Vieira Gomes
Department of Mathematics, Campus de Francisco Beltrão, Universidade Técnica Federal do Paraná, UTFPR, Linha Santa Bárbara s/n, Francisco Beltrão 85601-970, Brazil
Edson Donizete de Carvalho
Department of Mathematics, Câmpus de Ilha Solteira, Universidade Estadual Paulista, UNESP, Av. Brasil Sul, 56, Ilha Solteira 15385-000, Brazil
Carlos Alexandre Ribeiro Martins
Department of Mathematics, Campus de Pato BrancoUTFPR, Universidade Técnica Federal do Paraná, UTFPR, Via do Conhecimento, s/n-KM 01-Fraron, Pato Branco 85503-390, Brazil
Waldir Silva Soares
Department of Mathematics, Campus de Pato BrancoUTFPR, Universidade Técnica Federal do Paraná, UTFPR, Via do Conhecimento, s/n-KM 01-Fraron, Pato Branco 85503-390, Brazil
Eduardo Brandani da Silva
Department of Mathematics, Universidade Estadual de Maringá, UEM, Av. Colombo 5790, Maringá 87020-900, Brazil
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.
