Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus

Abstract

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.

Constructing and Visualizing Uniform Tilings

“…Escher used these symmetries in many of his sculptures, drawings, and prints. As explained in Gomez et al [1], hyperbolic tilings have applications in communication theory. There are now methods of categorizing, analyzing, and enumerating all uniform tilings of the sphere, plane, and hyperbolic plane.…”

A General Framework for Geometrically Uniform Codes and Signal Sets Matched to Groups

“…Gomes et al [11] presented a systematic way of labeling geometrically uniform codes in hyperbolic environments arising from surfaces. In this case, the authors worked with triangular tilings on the double torus and, despite explicitly providing the labels for the codes, the metric used was the hyperbolic metric coming from the ambient space.…”