32 research outputs found
A class of complete arcs in multiply derived planes
We prove that unital-derived (q^2 - q + 1)-arcs of PG(2, q^2) still yield complete arcs after multiple derivation with respect to disjoint derivation sets on a given line
On regular sets of affine type in finite Desarguesian planes and related codes
In this paper, we consider point sets of finite Desarguesian planes whose
multisets of intersection numbers with lines is the same for all but one
exceptional parallel class of lines. We call such sets regular of affine type.
When the lines of the exceptional parallel class have the same intersection
numbers, then we call these sets regular of pointed type. Classical examples
are e.g. unitals; a detailed study and constructions of such sets with few
intersection numbers is due to Hirschfeld and Sz\H{o}nyi from 1991. We here
provide some general construction methods for regular sets and describe a few
infinite families. The members of one of these families have the size of a
unital and meet affine lines of in one of possible
intersection numbers, each of them congruent to modulo . As a
byproduct, we determine the intersection sizes of the Hermitian curve defined
over with suitable rational curves of degree and
we obtain -divisible codes with non-zero weights. We also
determine the weight enumerator of the codes arising from the general
constructions modulus some -powers.Comment: 16 pages/revised and improved versio
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 -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