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 $V(rn,q)$. In this paper we investigate this construction in greater detail, and determine an explicit multiplication for the semifield associated with a BEL-configuration in $V(rn,q)$, extending the results from \cite{BEL2007}, where this was obtained only for $r=n$. Given a BEL-configuration with associated semifields spread $\mathcal{S}$, we also show how to find a BEL-configuration corresponding to the dual spread $\mathcal{S}^d$. Furthermore, we study the effect of polarities in $V(rn,q)$ on BEL-configurations, leading to a characterisation of BEL-configurations associated to symplectic semifields. We give precise conditions for when two BEL-configurations in $V(n^2,q)$ 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 $V(2n,q)$ introduced in \cite{BEL2007}, which, together with the transpose operation, leads to a group of order $8$ acting on BEL-configurations

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

On isotopisms and strong isotopisms of commutative presemifields

In this paper we prove that the $P(q,\ell)$ ($q$ odd prime power and $\ell>1$ 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 $q\equiv 1(mod\,4)$. Consequently, for each $q\equiv -1(mod\,4)$ there exist isotopic commutative presemifields of order $q^{2\ell}$ ($\ell>1$ odd) defining CCZ--inequivalent planar DO polynomials.Comment: References updated, pag. 5 corrected Multiplication of commutative LMPTB semifield

Finite semifields and nonsingular tensors

In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455-464, 1981)

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