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

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)

Multi-Sidon spaces over finite fields

Sidon spaces have been introduced by Bachoc, Serra and Z\'emor in 2017 in connection with the linear analogue of Vosper's Theorem. In this paper, we propose a generalization of this notion to sets of subspaces, which we call multi-Sidon space. We analyze their structures, provide examples and introduce a notion of equivalnce among them. Making use of these results, we study a class of linear sets in PG$(r-1,q^n)$ determined by $r$ points and we investigate multi-orbit cyclic subspace codes

Translation dual of a semifield

In this paper we obtain a new description of the translation dual of a semifield introduced in [G. Lunardon, Translation ovoids, J. Geom. 76 (2003) 200–215]. Using such a description we are able to prove that a semifield and its translation dual have nuclei of the same order. Combining the Knuth cubical array and the translation dual, we give an alternate description of the chain of twelve semifields in the table of [S. Ball, G.L. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (2007) 117–129]

The Generalized Translation Dual of a Semifield

In this paper, elaborating on the link between semifields of dimension n over their left nucleus and s-linear sets of rank en disjoint from the secant variety $\Omega({\mathcal S}_{n,n})$ of the Segre variety ${\mathcal S}_{n,n}$ of $PG(n2−1,q)$, $q=se$, we extend some operations on semifield whose definition relies on dualising the relevant linear set

The Generalized Translation Dual of a Semifield

In this paper, elaborating on the link between semifields of dimension n over their left nucleus and Fs-linear sets of rank en disjoint from the secant variety Ω (Sn,n) of the Segre variety Sn,nof PG(n2- 1 , q) , q= se, we extend some operations on semifield whose definition relies on dualising the relevant linear set