This paper presents the conditions under which a coded digital communication system’s performance may be improved using better signal sets design matched to groups. This improvement is directly related to the determination of the uniformization region of a hyperelliptic curve by using a Fuchsian differential equation and, consequently, the determination of the Fuchsian group and subgroup generators. The hyperelliptic curves of interest are given by \(y^2=z^n \pm 1\) , where \(n\in I\!\!N\) and \(n=2g+1\) or \(n=2g+2\) , and satisfy the following conditions: all the roots are distinct, they are symmetrically arranged (regular polygon) in the Poincaré disk, the juxtaposition of two such polygons gives the uniformization region. These conditions imply that the uniformization regions are regular polygons and that the associate Fuchsian group leads to regular tessellations from which arithmetic Fuchsian groups may be found. This work aims to answer and establish the conditions for the following questions: Can Fuchsian subgroups corresponding to two distinct hyperelliptic curves with the same degree and genus be isomorphic in \({\mathbb {D}}^2\) as well as in \({\mathbb {H}}^2\) ? What are the conditions under which an isometry acting on isomorphic Fuchsian subgroups in \({\mathbb {D}}^2\) remain isomorphic in \({\mathbb {H}}^2\) ? To show the isomorphism, we use the concept of conjugation. The first question’s positive response is achieved for hyperelliptic curves given by \(y^2 = z^n \pm 1\) . Finally, from the results established in \({\mathbb {D}}^2\) and \({\mathbb {H}}^2\) , if the Fuchsian subgroups are isomorphic by conjugation in \({\mathbb {D}}^2\) , they remain isomorphic in \({\mathbb {H}}^2\) by the action of an appropriate isometry as established in Theorem 3 and its generalization in Theorem 4.
