Uniformization of hyperelliptic curves as a systematic approach to establishing decision regions of hyperbolic signal sets

Abstract

In this paper, we present a new approach to designing hyperbolic signal sets matched to groups using Whittaker’s proposal in the uniformization of hyperelliptic curves via Fuchsian differential equations (FDEs). This approach provides a systematic procedure of establishing the decision region (fundamental polygon) of a hyperbolic signal as well as the associated Fuchsian group, and consists of the following four steps: (1) Obtaining the genus, g, by embedding a discrete memoryless channel (DMC) on a Riemann surface; (2) Selecting a set of $$(2g+1)$$ or $$(2g+2)$$ symmetric points in the Poincare disk to establish the hyperelliptic curve; (3) From the FDE via algebraic manipulations to arrive at the hypergeometric differential equation (HDE) to obtain the solutions; and (4) Quotients of the FDE linearly independent solutions give rise to the generators of the associated Fuchsian group whose subgroup provides the uniformizing region, implying the determination of the decision region (Voronoi region) of a digital signal. Hence, the following results are achieved: (1) from the solutions of the FDE, the Fuchsian group and subgroup generators are established, and consequently, the desired signal set matched to the Fuchsian subgroup may be constructed; (2) a relation between the parameters of the $$\{p,q\}$$ tessellation and the degree of the hyperelliptic curve is established. Knowing g, related to the hyperelliptic curve degree, and p, the number of sides of the fundamental polygon derived from Whittaker’s uniformizing procedure, the value of q is obtained from the Euler characteristic leading to one of the $$\{4g,4g\}$$ or $$\{4g+2, 2g+1\}$$ or $$\{12g-6,3\}$$ tessellation. These tessellations are essential for their rich geometric and algebraic structures, both required in classical and quantum coding theory applications.