Search

A (normal) bent partition of an n-dimensional vector space V n (p) over the prime field F p , is a partition of V n (p) into an n/2-dimensional subspace U, and subsets A 1 , ... , A K , such that every function f : V n (p) → F p with the following property, is a bent function: The preimage set f - 1 (c) = { x ∈ V n (p) : f (x) = c } contains exactly K/p of the sets A i for every c ∈ F p , and f is also constant on U. The classical examples are bent partitions from spreads or partial spreads, which have been known for a long time. Only recently (Meidl and Pirsic in Des Codes Cryptogr 89:75–89, 2021; Anbar and Meidl in Des Codes Cryptogr 90:1081–1101, 2022), it has been shown that (partial) spreads are not the only partitions with this remarkable property. Bent partitions have been presented, which generalize the Desarguesian spread, but provably do not come from any (partial) spread. In this article we show that also for some classes of semifields we can construct bent partitions, which similarly to finite fields and the Desarguesian spread, can be seen as a generalization of the semifield spread. Our results suggest that there are many partitions, which have similar properties as spreads. [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]

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]

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]

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]

Abstract: A fundamental problem in the theory of curves over finite fields is to determine the sets there is a curve over of genus g with exactly N rational points}. A complete description of is out of reach. So far, mostly bounds for the numbers have been studied. In particular, Elkies et al. proved that there is a constant , such that for any there is some with . This implies that , and solves a long-standing problem by Serre. We extend the result of Elkies et al. substantially: there are constants such that for all , the whole interval is contained in . [Copyright &y& Elsevier]