An infinite family of hyperovals of $Q^+(5,q)$, $q$ even

TL;DR

This paper introduces an infinite family of hyperovals on the Klein quadric $Q^+(5,q)$ for even q, using ovoids of symplectic quadrangles, and solves the isomorphism problem for these hyperovals.

Contribution

It presents a novel construction of hyperovals on $Q^+(5,q)$ for even q and provides criteria for their isomorphism.

Findings

Constructed an infinite family of hyperovals on $Q^+(5,q)$ for even q.

Determined necessary and sufficient conditions for hyperoval isomorphism.

Connected hyperovals to ovoids of symplectic generalized quadrangles.

Abstract

We construct an infinite family of hyperovals on the Klein quadric $Q^+(5,q)$, $q$ even. The construction makes use of ovoids of the symplectic generalized quadrangle $W(q)$ that is associated with an elliptic quadric which arises as solid intersection with $Q^+(5,q)$. We also solve the isomorphism problem: we determine necessary and sufficient conditions for two hyperovals arising from the construction to be isomorphic.