46 research outputs found
On isotopisms and strong isotopisms of commutative presemifields
In this paper we prove that the ( odd prime power and
odd) commutative semifields constructed by Bierbrauer in \cite{BierbrauerSub}
are isotopic to some commutative presemifields constructed by Budaghyan and
Helleseth in \cite{BuHe2008}. Also, we show that they are strongly isotopic if
and only if . Consequently, for each
there exist isotopic commutative presemifields of order (
odd) defining CCZ--inequivalent planar DO polynomials.Comment: References updated, pag. 5 corrected Multiplication of commutative
LMPTB semifield
On symplectic semifield spreads of PG(5,q2), q odd
We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG ( 5 , q2), for q2> 2 .38odd, whose associated semifield has center containing Fq. Equivalently, we classify, up to isotopy, commutative semifields of order q6, for q2> 2 .38odd, with middle nucleus containing q2Fq2and center containing q Fq
MUBs inequivalence and affine planes
There are fairly large families of unitarily inequivalent complete sets of
N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The
number of such sets is not bounded above by any polynomial as a function of N.
While it is standard that there is a superficial similarity between complete
sets of MUBs and finite affine planes, there is an intimate relationship
between these large families and affine planes. This note briefly summarizes
"old" results that do not appear to be well-known concerning known families of
complete sets of MUBs and their associated planes.Comment: This is the version of this paper appearing in J. Mathematical
Physics 53, 032204 (2012) except for format changes due to the journal's
style policie
Symplectic spreads, planar functions and mutually unbiased bases
In this paper we give explicit descriptions of complete sets of mutually
unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras
obtained from commutative and symplectic semifields, and
from some other non-semifield symplectic spreads. Relations between various
constructions are also studied. We show that the automorphism group of a
complete set of MUBs is isomorphic to the automorphism group of the
corresponding orthogonal decomposition of the Lie algebra .
In the case of symplectic spreads this automorphism group is determined by the
automorphism group of the spread. By using the new notion of pseudo-planar
functions over fields of characteristic two we give new explicit constructions
of complete sets of MUBs.Comment: 20 page
Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces
In this paper we investigate partial spreads of through the
related notion of partial spread sets of hermitian matrices, and the more
general notion of constant rank-distance sets. We prove a tight upper bound on
the maximum size of a linear constant rank-distance set of hermitian matrices
over finite fields, and as a consequence prove the maximality of extensions of
symplectic semifield spreads as partial spreads of . We prove
upper bounds for constant rank-distance sets for even rank, construct large
examples of these, and construct maximal partial spreads of for a
range of sizes
On BEL-configurations and finite semifields
The BEL-construction for finite semifields was introduced in \cite{BEL2007};
a geometric method for constructing semifield spreads, using so-called
BEL-configurations in . In this paper we investigate this construction
in greater detail, and determine an explicit multiplication for the semifield
associated with a BEL-configuration in , extending the results from
\cite{BEL2007}, where this was obtained only for . Given a
BEL-configuration with associated semifields spread , we also show
how to find a BEL-configuration corresponding to the dual spread
. Furthermore, we study the effect of polarities in on
BEL-configurations, leading to a characterisation of BEL-configurations
associated to symplectic semifields.
We give precise conditions for when two BEL-configurations in
define isotopic semifields. We define operations which preserve the BEL
property, and show how non-isotopic semifields can be equivalent under this
operation. We also define an extension of the ```switching'' operation on
BEL-configurations in introduced in \cite{BEL2007}, which, together
with the transpose operation, leads to a group of order acting on
BEL-configurations
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}