Number of Distinct Homomorphic Images in Coset Diagrams

Abstract

The representation of the action of $\text{PGL}\left(2,\mathbb{Z}\right)$ on $F_t\cup \left\{\infty \right\}$ in a graphical format is labeled as coset diagram. These finite graphs are acquired by the contraction of the circuits in infinite coset diagrams. A circuit in a coset diagram is a closed path of edges and triangles. If one vertex of the circuit is fixed by ${\left(pq\right)}^{{\Delta }_1}{\left(p{q}^{-1}\right)}^{{\Delta }_2}{\left(pq\right)}^{{\Delta }_3}\dots {\left(p{q}^{-1}\right)}^{{\Delta }_m}\in \text{PSL}\left(2,\mathbb{Z}\right)$ , then this circuit is titled to be a length- $m$ circuit, denoted by $\left({\Delta }_1,{\Delta }_2,{\Delta }_3,\dots ,{\Delta }_m\right)$ . In this manuscript, we consider a circuit $\Delta$ of length 6 as $\left({\Delta }_1,{\Delta }_2,{\Delta }_3,{\Delta }_4,{\Delta }_5,{\Delta }_6\right)$ with vertical axis of symmetry, that is, ${\Delta }_2={\Delta }_6,{\Delta }_3={\Delta }_5$ . Let ${\Gamma }_1$ and ${\Gamma }_2$ be the homomorphic images of $\Delta$ acquired by contracting the vertices $a,u$ and $b,v$ , respectively, then it is not necessary that ${\Gamma }_1$ and ${\Gamma }_2$ are different. In this study, we will find the total number of distinct homomorphic images of $\Delta$ by contracting its all pairs of vertices with the condition ${\Delta }_1>{\Delta }_2>{\Delta }_3>{\Delta }_4.$ The homomorphic images are obtained in this way having versatile applications in coding theory and cryptography. One can attain maximum nonlinearity factor using this in the encryption process.