On BEL-configurations and finite semifields

In this talk we will explain the Knuth orbit of a semifield S: a set of six isotopism classes of semifields associated to S (from [2]), and discuss ways of extending it, using techniques from finite geometry. A BEL-configuration in Frn q is a triple (D, U,W ), where D is a Desarguesian spread, and U and W are subspaces of dimension n and rn− n, resp., such that no element of D intesects both U and W nontrivially. The concept was introduced in [1], and generalized to linear sets in [3]. It was shown that such a configuration can be used to construct a semifield of order qn, with center containing Fq. It was also shown that every semifield can be constructed in this way. A BEL-configuration for r = 2 allows a new operation on semifields, known as switching, from which a single BEL-configuration gives up to two Knuth orbits of semifields. In collaboration with John Sheekey we obtained a new operation which extends this, generating up to four Knuth orbits from a single BEL-configuration.