Semifields, relative difference sets, and bent functions

Recently, the interest in semifields has increased due to the discovery of several new families and progress in the classification problem. Commutative semifields play an important role since they are equivalent to certain planar functions (in the case of odd characteristic) and to modified planar functions in even characteristic. Similarly, commutative semifields are equivalent to relative difference sets. The goal of this survey is to describe the connection between these concepts. Moreover, we shall discuss power mappings that are planar and consider component functions of planar mappings, which may be also viewed as projections of relative difference sets. It turns out that the component functions in the even characteristic case are related to negabent functions as well as to $\mathbb{Z}_4$-valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite fields", 09-13 December 2013, Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

Wada Dessins associated with Finite Projective Spaces and Frobenius Compatibility

\textit{Dessins d'enfants} (hypermaps) are useful to describe algebraic properties of the Riemann surfaces they are embedded in. In general, it is not easy to describe algebraic properties of the surface of the embedding starting from the combinatorial properties of an embedded dessin. However, this task becomes easier if the dessin has a large automorphism group. In this paper we consider a special type of dessins, so-called \textit{Wada dessins}. Their underlying graph illustrates the incidence structure of finite projective spaces \PR{m}{n}. Usually, the automorphism group of these dessins is a cyclic \textit{Singer group} $\Sigma_\ell$ permuting transitively the vertices. However, in some cases, a second group of automorphisms $\Phi_f$ exists. It is a cyclic group generated by the \textit{Frobenius automorphism}. We show under what conditions $\Phi_f$ is a group of automorphisms acting freely on the edges of the considered dessins.Comment: 23 page

Direction problems in affine spaces

This paper is a survey paper on old and recent results on direction problems in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October 5, 2012, Brussels, Belgiu

Singer quadrangles

[no abstract available

Combinatorial problems in finite geometry and lacunary polynomials

We describe some combinatorial problems in finite projective planes and indicate how R\'edei's theory of lacunary polynomials can be applied to them