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 Poincaré 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.

在本文中，我们提出了一种新的方法，用于根据Whittaker的建议通过Fuchsian微分方程（FDE）均匀化超椭圆曲线，设计与组匹配的双曲信号集。此方法提供了建立双曲信号的决策区域（基本多边形）以及相关的Fuchsian组的系统过程，并且包括以下四个步骤：（1）通过嵌入离散的无记忆通道来获得属g（DMC）在黎曼面上；（2）选择一组\（（（2g + 1）\）或\（（2g + 2）\）在庞加莱圆盘中建立对称点以建立超椭圆曲线；（3）从FDE通过代数运算得到超几何微分方程（HDE）以获得解；（4）FDE线性独立解的商产生了相关的Fuchsian组的生成器，其子组提供了均匀化区域，这意味着确定了数字信号的决策区域（Voronoi区域）。因此，可获得以下结果：（1）从FDE的解中，建立了Fuchsian群和子群发生器，从而可以构造与Fuchsian子群匹配的期望信号集；（2）\（\ {p，q \} \）的参数之间的关系细分并建立超椭圆曲线的程度。知道与超椭圆曲线度有关的g和从Whittaker的均匀化过程得出的基本多边形的边数p，从欧拉特性中得出q的值，得出\（\ {4g，4g \ } \或\（\ {4g + 2，2g + 1 \} \）或\（\ {12g-6,3 \} \）镶嵌。这些镶嵌对于它们丰富的几何和代数结构是必不可少的，这在经典和量子编码理论应用中都是必需的。