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.
Highlights
It is prominent that the finite presentation 〈p, q; p2 q3 1〉 is known as the modular group PSL(2, Z) generated by the linear fractional transformations p: χ ⟶ (− 1/χ) and q: χ ⟶ ((χ − 1)/χ)
In 1978, Professor Graham Higman propounds an unfamiliar type of a graph, titled as coset diagram, which presents the action of PGL(2, Z) on PL(Ft), where Ft is a finite field and t shows a prime power
Small triangles are proposed for the cycle q3, such that q permutes the vertices of triangles in the opposite direction of rotation of clock and an edge is attached to any two vertices that are interchanged by p
Summary
It is prominent that the finite presentation 〈p, q; p2 q3 1〉 is known as the modular group PSL(2, Z) generated by the linear fractional transformations p: χ ⟶ (− 1/χ) and q: χ ⟶ ((χ − 1)/χ). The number of total pairs of contracted vertices to generate all homomorphic images of Δ is 3(Δ2k1. The number of total pairs of contracted vertices to generate all Δ1 − 1 and their mirror homomorphic images of Δ is (3/2)(Δ21 + 3Δ1 − 4). (i) Ω9l7 : Δ4 − 2l7 > 1 (Figure 17(a)) (ii) Ω9l7 : Δ4 − 2l7 1 (Figure 17(b)) (iii) Ω9l7 : Δ4 − 2l7 < 1 and Δ4 − l7 > r + 1 (Figure 17(c)) (iv) Ω9l7 : Δ4 − 2l7 < 1 and Δ4 − l7 r + 1 (Figure 17(d))
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