On BEL-configurations and finite semifields

Abstract

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

The BEL-rank of finite semifields

“…They are the semifields labelled XIV, XXIV, XXVI and XXVII in [20]. We note that none are related to each other by the semifield operations s and e defined in [3] and [15]. We now give a form for the multiplication arising from such a BEL-configuration for each of these.…”

The isotopism problem of a class of 6-dimensional rank 2 semifields and its solution

“…This operation extends the Knuth orbit and has been recently generalized in [29]. Another recent extension of the Knuth orbit is contained in [26], using the so-called BEL-configurations (see [3], [17], and [18]).…”