2017/03/23 by Nurdagül Anbar, Almasa Oduzak, Anbar, Nurdagül +9
Computer Science · Engineering · #12E05 #12E20 #14H05 #Algebraic Geometry (math.AG) #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1703.08086
openalex publication_date 2017/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if p>(d2-3d+4)2, then there is no complete mapping polynomial f in \Fp[x] of degree d≥ 2. For arbitrary finite fields \Fq, a similar non-existence result is obtained recently by I\c sık, Topuzo\u glu and Winterhof in terms of the Carlitz rank of f. Cohen, Mullen and Shiue generalized the Chowla-Zassenhaus-Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f+g are both permutation polynomials of degree d≥ 2 over \Fp, with p>(d2-3d+4)2, then the degree k of g satisfies k ≥ 3d/5, unless g is constant. In this article, assuming f and f+g are permutation polynomials in \Fq[x], we give lower bounds for k %=deg(h) in terms of the Carlitz rank of f and q. Our results generalize the above mentioned result of I\c sık et al. We also show for a special class of polynomials f of Carlitz rank n ≥ 1 that if f+xk is a permutation of \Fq, with gcd(k+1, q-1)=1, then k≥ (q-n)/(n+3).