On large partial ovoids of symplectic and Hermitian polar spaces

TL;DR

This paper establishes constructive lower bounds for the sizes of the largest partial ovoids in certain symplectic and Hermitian polar spaces, advancing understanding of their combinatorial structures.

Contribution

It provides new constructive lower bounds on the sizes of large partial ovoids in specific symplectic and Hermitian polar spaces.

Findings

Derived lower bounds for partial ovoid sizes in ${\

${ m W}(3, q)$, ${ m W}(5, q)$, ${ m H}(4, q^2)$, ${ m H}(6, q^2)$, and ${ m H}(8, q^2)$.

Applicable to odd square and even or odd square q, with conditions on divisibility by 3.

Abstract

In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces ${\cal W}(3, q)$, $q$ odd square, $q \not\equiv 0 \pmod{3}$, ${\cal W}(5, q)$ and of the Hermitian polar spaces ${\cal H}(4, q^2)$, $q$ even or $q$ odd square, $q \not\equiv 0 \pmod{3}$, ${\cal H}(6, q^2)$, ${\cal H}(8, q^2)$.