2025/05/07 by Lucas Surmann, Surmann, Lucas
Computer Science · Mathematics · #Coding theory and cryptography #Advanced Differential Equations and Dynamical Systems #Chaos-based Image/Signal Encryption
paper · pdf · doi:10.48550/arxiv.2505.04563
In this article we count tame SL3 - and SL4 -frieze patterns with width w over a finite field K , as well as some tame SLk -frieze patterns for higher k . Let n = w + k + 1 . We consider the sets Ck(n) of tuples of n points in the projective space ℙk-1(K) , such that k consecutive points are always independent (the first and last point in the tuple are considered to be consecutive). First assume gcd(k,n) = 1 . In this case, we prove that the problem of counting tame SLk -frieze patterns can be reduced to counting Ck(n) . We also show that | Ck(n) | is essentially already known as long as k and n are coprime, and we derive the number of tame SLk -frieze-patterns in that case. In the case gcd(k,n) ≠ 1 , we define certain subsets Ck^*(n) and show that it is sufficient to count these sets. Afterwards, we count Ck^*(n) in the cases k = 3 and k = 4 and thus the number of tame SL3 - and SL4 -frieze patterns for any width w .