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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]

Generalized semifield spreads are partitions Γ = { U , A 1 , ... , A p k } of F p m × F p m obtained from (pre)semifields with a certain additional property, which generalize semifield spreads. In particular, a generalized semifield spread is a bent partition, i.e., every function f : F p m × F p m → F p , which is constant on every set of Γ , such that every c ∈ F p has the same number p k - 1 of A i in the preimage set, is a bent function. We show that from generalized semifield spreads one obtains not only p-ary and vectorial bent functions, but also bent functions f : F p m × F p m → B for any abelian group of order p s , s ≤ k . We investigate the effect of (pre)semifield isotopisms on generalized semifield spreads. We observe that isotopisms can destroy the bent partition property, and derive conditions on an isotopism between two (pre)semifields such that the corresponding partitions are equivalent bent partitions. Most notably, we show that with some other class of isotopisms, one can obtain inequivalent bent partitions, hence different classes of bent functions. This is in contrast to the situation for classical semifield spreads. The spreads of two isotopic (pre)semifields are always equivalent. Employing the 2-rank of Boolean functions we confirm that generalizations of the Desarguesian spread bent functions, which we call generalized PS ap functions, are in general not in the Maiorana-McFarland class. The generalized PS ap class contains functions which are not Maiorana-McFarland nor partial spread bent functions for any partial spread. Explicitly we determine the 2-rank of some Maiorana-McFarland functions in the generalized PS ap class in terms of Fibonacci numbers. [ABSTRACT FROM AUTHOR]

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]

Spread and partial spread constructions are the most powerful bent function constructions. A large variety of bent functions from a 2m-dimensional vector space V 2 m (p) over F p into F p can be generated, which are constant on the sets of a partition of V 2 m (p) obtained with the subspaces of the (partial) spread. Moreover, from spreads one obtains not only bent functions between elementary abelian groups, but bent functions from V 2 m (p) to B, where B can be any abelian group of order p k , k ≤ m . As recently shown (Meidl, Pirsic 2021), partitions from spreads are not the only partitions of V 2 m (2) , with these remarkable properties. In this article we present first such partitions—other than (partial) spreads—which we call bent partitions, for V 2 m (p) , p odd. We investigate general properties of bent partitions, like number and cardinality of the subsets of the partition. We show that with bent partitions we can construct bent functions from V 2 m (p) into a cyclic group Z p k . With these results, we obtain the first constructions of bent functions from V 2 m (p) into Z p k , p odd, which provably do not come from (partial) spreads. [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]

For a positive integer k and a linearized polynomial L(X), polynomials of the form P (X) = G (X) k − L (X) ∈ F q n [ X ] are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of F q n , then P(X) cannot be a permutation if gcd (k , q n − 1) > 1 . Further, necessary conditions for P(X) to be a permutation of F q n are given for the case that G(X) is an arbitrary linearized polynomial. The method uses plane curves, which are obtained via the multiplicative and the additive structure of F q n , and their number of rational affine points. [ABSTRACT FROM AUTHOR]

A bent4 function is a Boolean function with a flat spectrum with respect to a certain unitary transform T . It was shown previously that a Boolean function f in an even number of variables is bent4 if and only if f + σ is bent, where σ is a certain quadratic function depending on T . Hence bent4 functions are also called shifted bent functions. Similarly, a Boolean function f in an odd number of variables is bent4 if and only if f + σ is a semibent function satisfying some additional properties. In this article, for the first time, we analyse in detail the effect of the shifts on plateaued functions, on partially bent functions and on the linear structures of Boolean functions. We also discuss constructions of bent and bent4 functions from partially bent functions and study the differential properties of partially bent4 functions, unifying the previous work on partially bent functions. [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 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]

Zhou (J. Combin. Des. 21(12), 563–584, 2013) introduced modified planar functions in order to describe (2n, 2n, 2n, 1) relative difference sets R as a graph of a function on the finite field 픽2n, and pointed out that projections of R are difference sets that can be described by negabent or bent4 functions, which are Boolean functions given in multivariate form. One of the objectives of this paper is to contribute to the understanding of these component functions of modified planar functions. We identify the versions of the Walsh transforms that apply to modified planar functions on 픽2n and their components, obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function, and point out some further properties of these components. Finally we introduce vectorial bent4 functions (over finite fields), which correspond to relative difference sets in certain groups. We give conditions under which Maiorana-McFarland functions are such functions, hence give rise to relative difference sets in ℤ2n/2×ℤ4n/2. [ABSTRACT FROM AUTHOR]

The Walsh transform $$\widehat{Q}$$ of a quadratic function $$Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p$$ satisfies $$|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$$ for all $$b\in {\mathbb F}_{p^n}$$ , where $$0\le s\le n-1$$ is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class $$\mathcal {C}_1$$ is defined for arbitrary n as $$\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}$$ , and the larger class $$\mathcal {C}_2$$ is defined for even n as $$\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}$$ . For an odd prime p, the subclass $$\mathcal {D}$$ of all p-ary quadratic functions is defined as $$\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}$$ . We determine the generating function for the distribution of the parameter s for $$\mathcal {C}_1, \mathcal {C}_2$$ and $$\mathcal {D}$$ . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $$p > 2$$ , the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed-Muller codes corresponding to $$\mathcal {C}_1$$ and $$\mathcal {C}_2$$ in terms of a generating function. [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]

Small complete arcs and caps in Galois spaces over finite fields $$\mathbb {F}_q$$ with characteristic greater than three are constructed from singular cubic curves. For $$m$$ a divisor of $$q+1$$ or $$q-1$$ , complete plane arcs of size approximately $$q/m$$ are obtained, provided that $$(m,6)=1$$ and $$m<\frac{1}{4}q^{1/4}$$ . If in addition $$m=m_1m_2$$ with $$(m_1,m_2)=1$$ , then complete caps in affine spaces of dimension $$N\equiv 0 \pmod 4$$ with roughly $$\frac{m_1+m_2}{m}q^{N/2}$$ points are described. These results substantially widen the spectrum of $$q$$ s for which complete arcs in $$AG(2,q)$$ of size approximately $$q^{3/4}$$ can be constructed. Complete caps in $$AG(N,q)$$ with roughly $$q^{(4N-1)/8}$$ points are also provided. For infinitely many $$q$$ s, these caps are the smallest known complete caps in $$AG(N,q)$$ , $$N \equiv 0 \pmod 4$$ . [ABSTRACT FROM AUTHOR]

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

Bicovering arcs in Galois affine planes of odd order are a powerful tool for the construction of complete caps in spaces of arbitrarily higher dimensions. The aim of this paper is to investigate whether the arcs contained in elliptic cubic curves are bicovering. As a result, bicovering k-arcs in AG(2, q) of size k≤ q/3 are obtained, provided that q−1 has a prime divisor m with 7< m<(1/8) q. Such arcs produce complete caps of size kq in affine spaces of dimension N≡0(mod4). When q= p with p prime and h≤8, these caps are the smallest known complete caps in AG( N, q), N≡0(mod4). [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]

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]