In this paper, we consider the steps to be followed in the analysis and interpretation of the quantization problem related to the $C_{2,8}$ channel, where the Fuchsian differential equations, the generators of the Fuchsian groups, and the tessellations associated with the cases $g=2$ and $g=3$, related to the hyperbolic case, are determined. In order to obtain these results, it is necessary to determine the genus $g$ of each surface on which this channel may be embedded. After that, the procedure is to determine the algebraic structure (Fuchsian group generators) associated with the fundamental region of each surface. To achieve this goal, an associated linear second-order Fuchsian differential equation whose linearly independent solutions provide the generators of this Fuchsian group is devised. In addition, the tessellations associated with each analyzed case are identified. These structures are identified in four situations, divided into two cases $(g=2$ and $g=3)$, obtaining, therefore, both algebraic and geometric characterizations associated with quantizing the $C_{2,8}$ channel.

翻译：本文研究了与$C_{2,8}$信道相关的量子化问题分析与解释所需遵循的步骤，确定了双曲情形下$g=2$和$g=3$两种情况对应的Fuchs型微分方程、Fuchs群生成元以及镶嵌结构。为获得这些结果，需先确定该信道可嵌入的每个曲面的亏格$g$，随后确定与每个曲面基本区域相关的代数结构（Fuchs群生成元）。为此，构造了一个相关的二阶线性Fuchs型微分方程，其线性无关解提供了该Fuchs群的生成元。此外，还识别了每种分析情形对应的镶嵌结构。这些结构在四种情形中得到识别，分为两组（$g=2$和$g=3$），从而获得了与$C_{2,8}$信道量子化相关的代数及几何双重刻画。