Orthogonal Dual Hyperovals, Symplectic Spreads and Orthogonal Spreads

TL;DR

This paper explores the relationship between orthogonal spreads, symplectic spreads, and dual hyperovals in finite geometries, revealing new constructions and connections in orthogonal and symplectic spaces over finite fields.

Contribution

It introduces a novel construction linking orthogonal spreads in higher-dimensional spaces to dual hyperovals and symplectic spreads, expanding the understanding of finite geometric structures.

Findings

Orthogonal spreads yield many dual hyperovals in related orthogonal spaces.

A construction method parallels symplectic spread formation from orthogonal spreads when q is even.

The work uncovers new relationships between different finite geometric configurations.

Abstract

Orthogonal spreads in orthogonal spaces of type $V^+(2n+2,2)$ produce large numbers of rank $n$ dual hyperovals in orthogonal spaces of type $V^+(2n,2)$. The construction resembles the method for obtaining symplectic spreads in $V(2n,q)$ from orthogonal spreads in $V^+(2n+2,q)$ when $q$ is even.