(2^n,2^n,2^n,1)-relative difference sets and their representations

We show that every $(2^n,2^n,2^n,1)$-relative difference set $D$ in $\Z_4^n$ relative to $\Z_2^n$ can be represented by a polynomial f(x)\in \F_{2^n}[x], where $f(x+a)+f(x)+xa$ is a permutation for each nonzero $a$. We call such an $f$ a planar function on \F_{2^n}. The projective plane $\Pi$ obtained from $D$ in the way of Ganley and Spence \cite{ganley_relative_1975} is coordinatized, and we obtain necessary and sufficient conditions of $\Pi$ to be a presemifield plane. We also prove that a function $f$ on \F_{2^n} with exactly two elements in its image set and $f(0)=0$ is planar, if and only if, $f(x+y)=f(x)+f(y)$ for any x,y\in\F_{2^n}

Translation planes of order q2q2 admitting collineation groups of order q3uq3u preserving a parabolic unital

The set of translation planes of order $q2$ that admit collineation groups of order $q3u$, where u is a prime p-primitive divisor of $q2-1$, consists of exactly the Desarguesian plane, assuming that the group does not contain a translation subgroup of order a multiple of $q2$. This applies to show that if the group preserves a parabolic unital then the plane is forced to be Desarguesian

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

Inherited Groups and Kernels of Derived Translation Planes

When an affine plane is converted to another plane by derivation, the point permutations which act as collineations of both planes form the inherited group. The full group can be larger than the inherited group. For finite translation planes in which some of the Baer subplanes involved are not vector spaces over the kernel of the original plane then the full collineation group of the derived plane is the inherited group provided the order of the plane is greater than 16

Proofs of two conjectures on ternary weakly regular bent functions

We study ternary monomial functions of the form f(x)=\Tr_n(ax^d), where x\in \Ff_{3^n} and \Tr_n: \Ff_{3^n}\to \Ff_3 is the absolute trace function. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from \cite{hk1} are weakly regular bent, settling a conjecture of Helleseth and Kholosha \cite{hk1}. We also prove that the Coulter-Matthews bent functions are weakly regular.Comment: 20 page