93,539 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
Computational search for isotopic semifields and planar functions in characteristic 3
In this thesis, we investigate the possibility of finding new planar functions and corresponding semifields in characteristic 3 by the construction of isotopic semifields from the known families and sporadic instances of planar functions. Using the conditions laid out by Coulter and Henderson, we are able to deduce that a number of the known infinite families can never produce CCZ-inequivalent functions via isotopism. For the remaining families, we computationally investigate the isotopism classes of their instances over finite fields of order 3^n for n ≤ 8. We find previously unknown isotopisms between the semifields corresponding to some of the known planar functions for n = 6 and n = 8. This allows us to refine the known classification of planar functions up to isotopism, and to provide an updated, partial classification up to isotopism over finite fields of order 3^n for n ≤ 8.Masteroppgave i informatikkINF399MAMN-INFMAMN-PRO
A Note on Cyclic Codes from APN Functions
Cyclic codes, as linear block error-correcting codes in coding theory, play a
vital role and have wide applications. Ding in \cite{D} constructed a number of
classes of cyclic codes from almost perfect nonlinear (APN) functions and
planar functions over finite fields and presented ten open problems on cyclic
codes from highly nonlinear functions. In this paper, we consider two open
problems involving the inverse APN functions and the Dobbertin
APN function . From the calculation of
linear spans and the minimal polynomials of two sequences generated by these
two classes of APN functions, the dimensions of the corresponding cyclic codes
are determined and lower bounds on the minimum weight of these cyclic codes are
presented. Actually, we present a framework for the minimal polynomial and
linear span of the sequence defined by ,
where is a primitive element in . These techniques can also be
applied into other open problems in \cite{D}
Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies
We investigate the appearance of spontaneous coherence in the parametric
emission from planar semiconductor microcavities in the strong coupling regime.
Calculations are performed by means of a Quantum Monte Carlo technique based on
the Wigner representation of the coupled exciton and cavity-photon fields. The
numerical results are interpreted in terms of a non-equilibrium phase
transition occurring at the parametric oscillation threshold: below the
threshold, the signal emission is incoherent, and both the first and the
second-order coherence functions have a finite correlation length which becomes
macroscopic as the threshold is approached. Above the threshold, the emission
is instead phase-coherent over the whole two-dimensional sample and intensity
fluctuations are suppressed. Similar calculations for quasi-one-dimensional
microcavities show that in this case the phase-coherence of the signal emission
has a finite extension even above the threshold, while intensity fluctuations
are suppressed
Magnetic remanent states in antiferromagnetically coupled multilayers
In antiferromagnetically coupled multilayers with perpendicular anisotropy
unusual multidomain textures can be stabilized due to a close competition
between long-range demagnetization fields and short-range interlayer exchange
coupling.
In particular, the formation and evolution of specific topologically stable
planar defects within the antiferromagnetic ground state, i.e. wall-like
structures with a ferromagnetic configuration extended over a finite width,
explain configurational hysteresis phenomena recently observed in
[Co/Pt(Pd)]/Ru and [Co/Pt]/NiO multilayers.
Within a phenomenological theory, we have analytically derived the
equilibrium sizes of these "ferroband" defects as functions of the
antiferromagnetic exchange, a bias magnetic field, and geometrical parameters
of the multilayers. In the magnetic phase diagram, the existence region of the
ferrobands mediates between the regions of patterns with sharp
antiferromagnetic domain walls and regular arrays of ferromagnetic stripes.
The theoretical results are supported by magnetic force microscopy images of
the remanent states observed in [Co/Pt]/Ru.Comment: Paper submitted by the Joint European Magnetics Symposia 2008, Dublin
(4 pages, 3 figures
On derivatives of polynomials over finite fields through integration
In this note, using rather elementary technique and the derived formula that relates the coefficients of a polynomial over a finite field and its derivative, we deduce many interesting results related to derivatives of Boolean functions and derivatives of mappings over finite fields. For instance, we easily identify several infinite classes of polynomials which cannot possess linear structures. The same technique can be applied for deducing a nontrivial upper bound on the degree of so-called planar mappings
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