Number of Distinct Homomorphic Images in Coset Diagrams

Abstract

The representation of the action of PGL 2 , Z on F t ∪ ∞ 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 p q Δ 1 p q − 1 Δ 2 p q Δ 3 … p q − 1 Δ m ∈ PSL 2 , Z , then this circuit is titled to be a length- m circuit, denoted by Δ 1 , Δ 2 , Δ 3 , … , Δ m . In this manuscript, we consider a circuit Δ of length 6 as Δ 1 , Δ 2 , Δ 3 , Δ 4 , Δ 5 , Δ 6 with vertical axis of symmetry, that is, Δ 2 = Δ 6 , Δ 3 = Δ 5 . Let Γ 1 and Γ 2 be the homomorphic images of Δ acquired by contracting the vertices a , u and b , v , respectively, then it is not necessary that Γ 1 and Γ 2 are different. In this study, we will find the total number of distinct homomorphic images of Δ by contracting its all pairs of vertices with the condition Δ 1 > Δ 2 > Δ 3 > Δ 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.

Counting of Distinct Equivalence Classes of Circuits in PSL2, Z‐Space

“…A closed path of triangles and edges in a coset diagram for the action on PL(F q ) by the modular group is called a circuit [11,12]. For a coset diagram, a circuit is termed as a length-l circuit if its one vertex is fixed by…”

Number of Distinct Fragments in Coset Diagrams for P S L 2 , Z

“…, that is by joining a l 3 3m 1 +1 with x 1 3m 2 to produce Γ 13 (m 1 ,m 2 ,l 3 ) , x 1 * 3n 2 get also connected with a l 3 * 3n 1 +1 . So, the fragments Γ 13 (m 1 ,m 2 ,l 3 ) and Γ 13 (n 1 ,n 2 ,l 3 ) are the mirror image of each other if and only if…”

“…13 . Tis indicates, no fragment in F l 3 13 is the mirror image of others. Hence, all the elements in F l 3 13 are distinct.…”

“…Proof. For a fx l 3 , let us join x 1 3k 14 , the vertex from Ω 2 , with the vertices a l 3 3k (13,l 3 ) +1 from Ω 1 and attain a set of fragments F l 3 13 � Γ 13 (k (13,l 3 ) ,k 14 ,l 3 ) ; 􏼚 16), where the word…”

“…contains words such that if u is any word from D l 3 13 implies (a l 3 3k (13,l 3 ) +1 )u and (x 1 3k 14 )u lies on Ω 1 and Ω 2 , respectively. Tus, the fragment Γ 13 (k (13,l 3 ) ,k 14 ,l 3 ) has |D l 3 13 | � 6 pairs of connecting vertices in Ω 1 and Ω 2 . In other words, for each u ∈ D l 3 13 , the connection of (a l 3 3k (13,l 3 ) +1 )u and (x 1 3k 14 )u gives the same fragment Γ 13 (k (13,l 3 ) ,k 14 ,l 3 ) .…”