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)

Sublines of prime order contained in the set of internal points of a conic

In \cite{BALLBLOKHUISLAVRAUW200*} it was shown that if $q \geq 4n^2-8n+2$ then there are no subplanes of order $q$ contained in the set of internal points of a conic in \PG(2,q^n), $q$ odd, $n\geq 3$. In this article we improve this bound in the case where $q$ is prime to $q > 2n^2-(4-2\sqrt{3})n+(3-2\sqrt{3})$, and prove a stronger theorem by considering sublines instead of subplanes. We also explain how one can apply this result to flocks of a quadratic cone in \PG(3,q^n), ovoids of $Q(4,q^n)$, rank two commutative semifields, and eggs in \PG(4n-1,q)