Search

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]

An almost perfect non-linear (APN) function over F2n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {F}_{2^n}$$\end{document} is called exceptional APN if it remains APN over infinitely many extensions of F2n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {F}_{2^n}$$\end{document}. Exceptional APN functions have attracted attention of many researchers in the last decades. While the classification of exceptional APN monomials has been done by Hernando and McGuire, it has been conjectured by Aubry, McGuire and Rodier that up to equivalence, the only exceptional APN functions are the Gold and the Kasami–Welch monomial functions. Since then, many partial results have been on classifying non-exceptional APN polynomials. In this paper, for the classification of the exceptional property of APN functions, we introduce a new method that uses techniques from curves over finite fields. Then, we apply the method with Eisenstein’s irreducibility criterion and Kummer’s theorem to obtain new non-exceptional APN functions. [ABSTRACT FROM AUTHOR]

Bent partitions are partitions of an elementary abelian group, which have similarities to partitions from spreads. In fact, a spread partition is a special case of a bent partition. In particular, bent partitions give rise to a large number of (vectorial) bent functions. Examples of bent partitions, which generalize the Desarguesian spread, have been presented by Anbar, Meidl and Pirsic, 2021, 2022. Bent partitions, which generalize some other classes of (pre)semifield spreads, have been presented by Anbar, Kalaycı, Meidl 2023. In this article, it is shown that these bent partitions induce (p k + 1) -class amorphic associations schemes on F p m × F p m , where k is a divisor of m with special properties. This implies a construction of amorphic association schemes from some classes of (pre)semifields. [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]

A bent function from F 2 n to F 2 , n even, can be transformed into a negabent function, or slightly more general into a bent 4 , also called shifted bent function, by adding a certain quadratic term. If n is odd, then negabent functions similarly correspond to semibent functions with some additional property. Whereas bent functions induce relative difference sets in F 2 n × F 2 , negabent functions induce relative difference sets in F 2 n − 1 × Z 4 . We analyse equivalence of negabent functions respectively of their relative difference sets. We show that equivalent bent functions can correspond to inequivalent negabent functions, hence one can obtain inequivalent relative difference sets in F 2 n − 1 × Z 4 with EA-equivalence. We also show that this is not the case when n is odd. Finally we analyse the class of semibent functions that corresponds to negabent functions and show that though partially bent semibent functions always can be shifted to negabent or bent 4 functions, there are many semibent functions which do not correspond to negabent and bent 4 functions. [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]

In this article we give a Drinfeld modular interpretation for various towers of function fields meeting Zink's bound. [ABSTRACT FROM AUTHOR]

Let p be a prime and n be a positive integer. We consider rational functions fb(X)=X+1/(Xp-X+b)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_b(X)=X+1/(X^p-X+b)$$\end{document} over Fpn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {F}}_{p^n}$$\end{document} with Tr(b)≠0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{Tr}(b)\ne 0$$\end{document}. In Hou and Sze (Finite Fields Appl 68, Paper No. 10175, 2020), it is shown that fb(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_b(X)$$\end{document} is not a permutation for p>3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p>3$$\end{document} and n≥5\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 5$$\end{document}, while it is for p=2,3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p=2,3$$\end{document} and n≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 1$$\end{document}. It is conjectured that fb(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_b(X)$$\end{document} is also not a permutation for p>3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p>3$$\end{document} and n=3,4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n=3,4$$\end{document}, which was recently proved sufficiently large primes in Bartoli and Hou (Finite Fields Appl 76, Paper No. 101904, 2021). In this note, we give a new proof for the fact that fb(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_b(X)$$\end{document} is not a permutation for p>3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p>3$$\end{document} and n≥5\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 5$$\end{document}. With this proof, we also show the existence of many elements b∈Fpn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b\in {\mathbb {F}}_{p^n}$$\end{document} for which fb(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_b(X)$$\end{document} is not a permutation for n=3,4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n=3,4$$\end{document}. [ABSTRACT FROM AUTHOR]

For an odd prime $$p$$ and an even integer $$n$$ with $$\gcd (n,p) > 1$$ , we consider quadratic functions from $${\mathbb F}_{p^{n}}$$ to $${\mathbb F}_p$$ of codimension $$k$$ . For various values of $$k$$ , we obtain classes of quadratic functions giving rise to maximal and minimal Artin-Schreier curves over $${\mathbb F}_{p^{n}}$$ . We completely classify all maximal and minimal curves obtained from quadratic functions of codimension $$2$$ and coefficients in the prime field $${\mathbb F}_p$$ . The results complement our results obtained earlier for the case $$\gcd (n,p) = 1$$ . The arguments are more involved than for the case $$\gcd (n,p) = 1$$ . [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]

Over any quadratic finite field we construct function fields of large genus that have simultaneously many rational places, small p -rank, and many automorphisms. [ABSTRACT FROM AUTHOR]

Abstract: Let be an algebraically closed field of characteristic , and let be a curve over of genus . Assume that and that admits a non-singular plane model. The following result is proven: if has more than automorphisms, then is birationally equivalent to a Hermitian curve. [Copyright &y& Elsevier]

A fundamental problem in the theory of curves over finite fields is to determine the sets M( g):= { N ∈ ℕ / there is a curve over $$ \mathbb{F}_q $$ of genus g with exactly N rational points}. A complete description of M( g) is out of reach. So far, mostly bounds for the numbers N( g):= max M( g) have been studied. In particular, Elkies et al. proved that there is a constant γ > 0 such that for any g ≥ 0 there is some N ∈ M( g) with N ≥ γ g. This implies that lim inf N( g)/ g > 0, and solves a long-standing problem by Serre. We extend the result of Elkies et al. substantially and show that there are constants α, β > 0 such that for all g ≥ 0, the whole interval [0, α g − β] ∩ ℕ is contained in M( g). [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold: Firstly, we generalize well-known formulas for ramification and different exponents in cyclic extensions of function fields over a field K (due to H. Hasse) to extensions E = F( y), where y satisfies an equation of the form f( y) = u · g( y) with polynomials f( y), g( y) ∈ K[ y] and u ∈ F. This result depends essentially on Abhyankar’s Lemma which gives information about ramification in a compositum E = E1 E2 of finite extensions E1, E2 over a function field F. Abhyankar’s Lemma does not hold if both extensions E1/ F and E2/ F are wildly ramified. Our second objective is a generalization of Abhyankar’s Lemma if E1/ F and E2/ F are cyclic extensions of degree p = char( K). This result may be useful for the study of wild towers of function fields over finite fields. [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]