On large maximal partial ovoids of the parabolic quadric $\q(4,q)$

TL;DR

This paper proves the non-existence of certain large maximal partial ovoids in a specific geometric structure, providing an alternative proof and additional combinatorial insights for prime power cases.

Contribution

It offers a new proof for the non-existence of large maximal partial ovoids in $ ext{Q}(4,q)$ when q is a prime power with h > 1, and enhances understanding of known examples for prime q.

Findings

Maximal partial ovoids of size q^2-1 do not exist for q=p^h, h>1.

Provides an alternative proof method using the $T_2(O)$ representation.

Gives additional combinatorial information for known examples when q is prime.

Abstract

We use the representation $T_2(O)$ for $\q(4,q)$ to show that maximal partial ovoids of $\q(4,q)$ of size $q^2-1$, $q=p^h$, $p$ odd prime, $h > 1$, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for $q$ prime.