Classification of 8-dimensional rank two commutative semifields

Abstract

We classify the rank two commutative semifields which are 8-dimensional over their center ${\mathbb{F}}_q$. This is done using computational methods utilizing the connection to linear sets in $\mathrm{PG}(2,q^4)$. We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over ${\mathbb{F}}_3$. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.