Symplectic 4-dimensional semifields of order $8^4$ and $9^4$

Abstract

We classify symplectic 4-dimensional semifields over $\mathbb{F}_q$, for $q\leq 9$, thereby extending (and confirming) the previously obtained classifications for $q\leq 7$. The classification is obtained by classifying all symplectic semifield subspaces in $\mathrm{PG}(9,q)$ for $q\leq 9$ up to $K$-equivalence, where $K\leq \mathrm{PGL}(10,q)$ is the lift of $\mathrm{PGL}(4,q)$ under the Veronese embedding of $\mathrm{PG}(3,q)$ in $\mathrm{PG}(9,q)$ of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for $q$ even, $q\leq 8$. For $q$ odd, and $q\leq 9$, our results imply that the isotopism class of a symplectic non-associative 4-dimensional semifield over $\mathbb{F}_q$ is contained in the Knuth orbit of a Dickson commutative semifield.