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We investigate pairs of permutations F,G $F,G$ of Fpn ${{\mathbb{F}}}_{{p}^{n}}$ such that F(x+a)−G(x) $F(x+a)-G(x)$ is a permutation for every a∈Fpn $a\in {{\mathbb{F}}}_{{p}^{n}}$. We show that, in that case, necessarily G(x)=℘(F(x)) $G(x)=\wp (F(x))$ for some complete mapping −℘ $-\wp $ of Fpn ${{\mathbb{F}}}_{{p}^{n}}$, and call the permutation F $F$ a perfect ℘ $\wp $ nonlinear (P℘ $\wp $N) function. If ℘(x)=cx $\wp (x)=cx$, then F $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on Fpn×Fpn ${{\mathbb{F}}}_{{p}^{n}}\times {{\mathbb{F}}}_{{p}^{n}}$ involving ℘ $\wp $, we obtain a quasigroup, and show that the graph of a P℘ $\wp $N function F $F$ is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for P℘ $\wp $N functions, respectively, for the difference sets in the corresponding quasigroup. [ABSTRACT FROM AUTHOR]

We study the automorphisms of a function field of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. More precisely, we show that the order of a nilpotent subgroup G of its automorphism group is bounded by 16 (g – 1) when G is not a p-group. We show that if |G| = 16(g – 1), then g – 1 is a power of 2. Furthermore, we provide an infinite family of function fields attaining the bound. [ABSTRACT FROM AUTHOR]

There are two types of Belyi's Theorems for curves defined over finite fields of characteristic p , namely the Wild and the Tame p -Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi's theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame 2 -Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi's Theorems unlike the known ones that use technical results from Algebraic Geometry. [ABSTRACT FROM AUTHOR]

We consider vectorial maps F (x 1 , ... , x n) = (f 1 (x 1 , ... , x n) , ... , f n (x 1 , ... , x n)) : F q n ↦ F q n , which induce permutations of F q n. We show that the degrees of the components f 1 , f 2 , ... , f n ∈ F q [ x 1 , ... , x n ] are at least 2 when 2 ≤ deg (F) = d < q and d | (q − 1). Our proof uses an absolutely irreducible curve over F q and the number of rational points on it that we relate to the cardinality of the value set of a polynomial. [ABSTRACT FROM AUTHOR]

We consider the polynomials of the form P(x) = xk - γTr(x) over 픽 qn for n ≥ 2. We show that P(x) is not a permutation of 픽qn in the case gcd(k, qn - 1) > 1. Our proof uses an absolutely irreducible curve over 픽qn and the number of rational points on it. [ABSTRACT FROM AUTHOR]

The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if p > ( d 2 − 3 d + 4 ) 2 , then there is no complete mapping polynomial f in F p [ x ] of degree d ≥ 2 . For arbitrary finite fields F q , a similar non-existence result was obtained recently by Işık, Topuzoğlu 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 F p , with p > ( d 2 − 3 d + 4 ) 2 , then the degree k of g satisfies k ≥ 3 d / 5 , unless g is constant. In this article, assuming f and f + g are permutation polynomials in F q [ x ] , we give lower bounds for the Carlitz rank of f in terms of q and k . Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials f of Carlitz rank n ≥ 1 that if f + x k is a permutation over F q , with gcd ⁡ ( k + 1 , q − 1 ) = 1 , then k ≥ ( q − n ) / ( n + 3 ) . [ABSTRACT FROM AUTHOR]

We give a complete characterization of all Galois subfields of the generalized Giulietti–Korchmáros function fields C n / F q 2 n for n ≥ 5 . Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal function fields. [ABSTRACT FROM AUTHOR]

For c ∈ F 2 n , a c -bent 4 function f from the finite field F 2 n to F 2 is a function with a flat spectrum with respect to the unitary transform V f c , which is designed to describe the component functions of modified planar functions. For c = 0 the transform V f c reduces to the conventional Walsh transform, and hence a 0-bent 4 function is bent. In this article we generalize the concept of partially bent functions to the transforms V f c . We show that every quadratic function is partially bent, and hence it is plateaued with respect to any of the transforms V f c . In detail we analyse two quadratic monomials. The first has values as small as possible in its spectra with respect to all transforms V f c , and the second has a flat spectrum for a large number of c . Moreover, we show that every quadratic function is c -bent 4 for at least three distinct c . In the last part we analyse a cubic monomial. We show that it is c -bent 4 only for c = 1 , the function is then called negabent, which shows that non-quadratic functions exhibit a different behaviour. [ABSTRACT FROM AUTHOR]

For an odd prime p and an even integer n with gcd (n,p)=1, we consider quadratic functions from Fpn to Fp of codimension k. For various values of k, we obtain classes of quadratic functions giving rise to maximal and minimal Artin-Schreier curves over Fpn. We completely classify all maximal and minimal curves obtained from quadratic functions of codimension 2 and coefficients in the prime field Fp. [ABSTRACT FROM AUTHOR]

New families of complete caps in finite Galois spaces are obtained. For most pairs [ABSTRACT FROM AUTHOR]

We introduce a method based on Bezout's theorem on intersection points of two projective plane curves, for determining the nonlinearity of some classes of quadratic functions on F 2 2 m . Among those are the functions of Taniguchi (2019) [20] , Carlet (2011) [7] , and Zhou and Pott (2013) [22] , all of which are APN under certain conditions. This approach helps to understand why the majority of the functions in those classes have solely bent and semibent components, which in the case of APN functions is called the classical spectrum. More precisely, we show that all Taniguchi functions have the classical spectrum independent from being APN. We determine the nonlinearity of all functions belonging to Carlet's class and to the class of Zhou and Pott, which also confirms with comparatively simple proofs earlier results on the Walsh spectrum of APN-functions in these classes. Using the Hasse-Weil bound, we show that some simple sufficient conditions for the APN-ness of the Zhou-Pott functions, which are given in the original paper, are also necessary. [ABSTRACT FROM AUTHOR]