476 research outputs found
Exceptional planar polynomials
Planar functions are special functions from a finite field to itself that
give rise to finite projective planes and other combinatorial objects. We
consider polynomials over a finite field that induce planar functions on
infinitely many extensions of ; we call such polynomials exceptional planar.
Exceptional planar monomials have been recently classified. In this paper we
establish a partial classification of exceptional planar polynomials. This
includes results for the classical planar functions on finite fields of odd
characteristic and for the recently proposed planar functions on finite fields
of characteristic two
Planar functions over fields of characteristic two
Classical planar functions are functions from a finite field to itself and
give rise to finite projective planes. They exist however only for fields of
odd characteristic. We study their natural counterparts in characteristic two,
which we also call planar functions. They again give rise to finite projective
planes, as recently shown by the second author. We give a characterisation of
planar functions in characteristic two in terms of codes over .
We then specialise to planar monomial functions and present
constructions and partial results towards their classification. In particular,
we show that is the only odd exponent for which is planar
(for some nonzero ) over infinitely many fields. The proof techniques
involve methods from algebraic geometry.Comment: 23 pages, minor corrections and simplifications compared to the first
versio
A new family of semifields with 2 parameters
A new family of commutative semifields with two parameters is presented. Its
left and middle nucleus are both determined. Furthermore, we prove that for any
different pairs of parameters, these semifields are not isotopic. It is also
shown that, for some special parameters, one semifield in this family can lead
to two inequivalent planar functions. Finally, using similar construction, new
APN functions are given
On homogeneous planar functions
Let be an odd prime and \F_q be the finite field with elements.
A planar function f:\F_q\rightarrow\F_q is called homogenous if for all \lambda\in\F_p and x\in\F_q, where is some
fixed positive integer. We characterize as the unique homogenous planar
function over \F_{p^2} up to equivalence.Comment: Introduction modified to: 1. give the correct definition of
equivalence, 2. add some references. Other part unaltere
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
Binary Cyclic Codes from Explicit Polynomials over \gf(2^m)
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, monomials and
trinomials over finite fields with even characteristic are employed to
construct a number of families of binary cyclic codes. Lower bounds on the
minimum weight of some families of the cyclic codes are developed. The minimum
weights of other families of the codes constructed in this paper are
determined. The dimensions of the codes are flexible. Some of the codes
presented in this paper are optimal or almost optimal in the sense that they
meet some bounds on linear codes. Open problems regarding binary cyclic codes
from monomials and trinomials are also presented.Comment: arXiv admin note: substantial text overlap with arXiv:1206.4687,
arXiv:1206.437
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