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

Algebraic and Geometric Characterizations Related to the Quantization Problem of the $C_{2,8}$ Channel