Hyperbolic Lattices with Complete Labeling Derived from {4g, 4g} Tessellations

Abstract

Hyperbolic lattices $\mathcal{O}$ are the basic entities used in the design of signal constellations in the hyperbolic plane. Once the identification of the arithmetic Fuchsian group in a quaternion order is made, the next step is to present the codewords of a code over a graph, or a signal constellation (quotient of an order by a proper ideal). However, in order for the algebraic labeling to be complete, it is necessary that the corresponding order be maximal. An order $\mathcal{M}$ in a quaternion algebra $\mathcal{A}$ is called maximal if $\mathcal{M}$ is not contained in any other order in $\mathcal{A}$ (Reiner, Maximal Orders. Academic, London, 1975). The main objective of this work is to describe the maximal orders derived from {4g, 4g} tessellations, for which we have hyperbolic lattices with complete labeling.