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

TL;DR

This paper classifies symplectic 4-dimensional semifields over finite fields with q ≤ 9, extending previous classifications and confirming non-existence results for certain cases, with implications for algebraic structures in finite geometry.

Contribution

The paper extends classification of symplectic 4-dimensional semifields to q ≤ 9 and confirms non-existence for even q ≤ 8, providing new insights into their algebraic structure.

Findings

No non-associative symplectic 4-dimensional semifields for even q ≤ 8.

Classified all such semifields for q ≤ 9.

Symplectic non-associative semifields are in the Knuth orbit of Dickson semifields.

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.