A geometric construction of finite semifields

Abstract

A finite semifield is shown to be equivalent to the existence of a particular geometric configuration of subspaces with respect to a Desarguesian spread in a finite dimensional vector space over a finite field. In 1965 Knuth \cite{Knuth1965} showed that each finite semifield generates in total six (not necessarily isotopic) semifields. In certain cases, the geometric interpretation obtained here allows us to construct another six semifields, providing a link between some known examples which are not related by Knuth's operations

On BEL-configurations and finite semifields

“…So for r = n, the approach using the representation of the subspaces as U f and W g , is the same as in the proof of [3,Theorem 4.1]. But the advantage of the viewpoint taken here is that this explicit multiplication can not only be obtained for r = n, but for any r ≥ 2.…”

“…So, as was shown in [3], the switching operation can be applied to rank two semifields. In [11], it was shown that in this special case, switching has a geometric interpretation ("dualising an ovoid").…”

“…The BEL-construction for finite semifields was introduced in [3]; a geometric method for constructing semifield spreads, using so-called BELconfigurations 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 [3], where this was obtained only for r = n. Given a BELconfiguration with associated semifields spread S, we also show how to find a BEL-configuration corresponding to the dual spread S d .…”

“…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 [3], which, together with the transpose operation, leads to a group of order 8 acting on BELconfigurations. …”

Finite semifields and nonsingular tensors

“…In this section we concentrate on a geometric construction of finite semifield spreads. The construction we give here is taken from [47], but the main idea is the slightly less general construction given in [9] (where L is a subspace, i.e. t = 1).…”