Hyperovals of $H(3,q^2)$ when $q$ is even

TL;DR

This paper investigates the action of a specific group on a geometric structure over finite fields, constructs an infinite family of hyperovals with particular symmetry properties, and demonstrates these hyperovals are new compared to existing ones.

Contribution

It introduces a new infinite family of hyperovals in ${ m H}(3,q^2)$ for even q, with automorphism groups containing a specific projective special linear group, and proves their novelty.

Findings

Constructed an infinite family of hyperovals of size 2(q^3 - q)

Hyperovals have automorphism groups containing $PSL(2,q)$

Proved these hyperovals are not isomorphic to known examples

Abstract

For even $q$, a group $G$ isomorphic to $PSL(2,q)$ stabilizes a Baer conic inside a symplectic subquadrangle ${\cal W}(3,q)$ of ${\cal H}(3,q^2)$. In this paper the action of $G$ on points and lines of ${\cal H}(3,q^2)$ is investigated. A construction is given of an infinite family of hyperovals of size $2(q^3-q)$ of ${\cal H}(3,q^2)$, with each hyperoval having the property that its automorphism group contains $G$. Finally it is shown that the hyperovals constructed are not isomorphic to known hyperovals.