46 research outputs found

    On isotopisms and strong isotopisms of commutative presemifields

    Full text link
    In this paper we prove that the P(q,ℓ)P(q,\ell) (qq odd prime power and ℓ>1\ell>1 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 q≡1(mod 4)q\equiv 1(mod\,4). Consequently, for each q≡−1(mod 4)q\equiv -1(mod\,4) there exist isotopic commutative presemifields of order q2ℓq^{2\ell} (ℓ>1\ell>1 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

    Get PDF
    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

    Full text link
    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

    Full text link
    In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras sln(C)sl_n(\mathbb{C}) 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 sln(C)sl_n(\mathbb{C}). 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

    Full text link
    In this paper we investigate partial spreads of H(2n−1,q2)H(2n-1,q^2) 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 H(2n−1,q2)H(2n-1,q^2). We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of H(3,q2)H(3,q^2) for a range of sizes

    On BEL-configurations and finite semifields

    Full text link
    The BEL-construction for finite semifields was introduced in \cite{BEL2007}; a geometric method for constructing semifield spreads, using so-called BEL-configurations in V(rn,q)V(rn,q). In this paper we investigate this construction in greater detail, and determine an explicit multiplication for the semifield associated with a BEL-configuration in V(rn,q)V(rn,q), extending the results from \cite{BEL2007}, where this was obtained only for r=nr=n. Given a BEL-configuration with associated semifields spread S\mathcal{S}, we also show how to find a BEL-configuration corresponding to the dual spread Sd\mathcal{S}^d. Furthermore, we study the effect of polarities in V(rn,q)V(rn,q) on BEL-configurations, leading to a characterisation of BEL-configurations associated to symplectic semifields. We give precise conditions for when two BEL-configurations in V(n2,q)V(n^2,q) 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 V(2n,q)V(2n,q) introduced in \cite{BEL2007}, which, together with the transpose operation, leads to a group of order 88 acting on BEL-configurations

    Semifields, relative difference sets, and bent functions

    Full text link
    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 Z4\mathbb{Z}_4-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

    Full text link
    We show that every (2n,2n,2n,1)(2^n,2^n,2^n,1)-relative difference set DD in Z4n\Z_4^n relative to Z2n\Z_2^n can be represented by a polynomial f(x)\in \F_{2^n}[x], where f(x+a)+f(x)+xaf(x+a)+f(x)+xa is a permutation for each nonzero aa. We call such an ff a planar function on \F_{2^n}. The projective plane Π\Pi obtained from DD in the way of Ganley and Spence \cite{ganley_relative_1975} is coordinatized, and we obtain necessary and sufficient conditions of Π\Pi to be a presemifield plane. We also prove that a function ff on \F_{2^n} with exactly two elements in its image set and f(0)=0f(0)=0 is planar, if and only if, f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y) for any x,y\in\F_{2^n}
    corecore