16 research outputs found
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
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
Inner Automorphisms of Finite Semifields
EnUnlike finite fields, finite semifields possess inner automorphisms. A further surprise is that even noncommutative semifields possess inner automorphisms. We compute inner automorphisms and automorphism groups for semifields quadratic over the nucleus, the Hughes-Kleinfeld semifields and the Dickson commutative semifields
Division algebras that generalize Dickson semifields
summary:We generalize Knuth's construction of Case I semifields quadratic over a weak nucleus, also known as generalized Dickson semifields, by doubling of central simple algebras. We thus obtain division algebras of dimension by doubling central division algebras of degree . Results on isomorphisms and automorphisms of these algebras are obtained in certain cases
On the nuclei of a finite semifield
In this paper we collect and improve the techniques for calculating the
nuclei of a semifield and we use these tools to determine the order of the
nuclei and of the center of some commutative presemifields of odd
characteristic recently constructed
On isotopisms of commutative presemifields and CCZ-equivalence of functions
A function from \textbf{F} to itself is planar if for any \textbf{F} the function is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet. We show that the second one in fact coincides with CCZ-equivalence, while using the first one we generalize one of the known families of PN functions. In particular, we prove that, for any odd prime and any positive integers and , the indicators of the graphs of functions and from \textbf{F} to \textbf{F} are CCZ-equivalent if and only if and are CCZ-equivalent.
We also prove that, for any odd prime , CCZ-equivalence of functions from \textbf{F} to \textbf{F}, is strictly more general than EA-equivalence when and is greater or equal to the smallest positive divisor of different from 1
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