14 research outputs found
Semifields in loop theory and in finite geometry
This paper is a relatively short survey the aim of which is to
present the theory of semifields and the related areas of finite
geometry to loop theorists
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
(2^n,2^n,2^n,1)-relative difference sets and their representations
We show that every -relative difference set in
relative to can be represented by a polynomial f(x)\in \F_{2^n}[x],
where is a permutation for each nonzero . We call such an
a planar function on \F_{2^n}. The projective plane obtained from
in the way of Ganley and Spence \cite{ganley_relative_1975} is
coordinatized, and we obtain necessary and sufficient conditions of to be
a presemifield plane. We also prove that a function on \F_{2^n} with
exactly two elements in its image set and is planar, if and only if,
for any x,y\in\F_{2^n}
MRD codes with maximum idealizers
Left and right idealizers are important invariants of linear rank-distance
codes. In the case of maximum rank-distance (MRD for short) codes in
the idealizers have been proved to be isomorphic to
finite fields of size at most . Up to now, the only known MRD codes with
maximum left and right idealizers are generalized Gabidulin codes, which were
first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and
Gabidulin in 2005. In this paper we classify MRD codes in
for with maximum left and right idealizers
and connect them to Moore-type matrices. Apart from generalized Gabidulin
codes, it turns out that there is a further family of rank-distance codes
providing MRD ones with maximum idealizers for , odd and for ,
. These codes are not equivalent to any previously known MRD
code. Moreover, we show that this family of rank-distance codes does not
provide any further examples for .Comment: Reviewers' comments implemented, we changed the titl
MRD codes with maximum idealisers
Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in GF(q)^(n×n) the idealizers have been proved to be isomorphic to finite fields of size at most q^n. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in GF(q)^(n×n) for n9