Some non-existence results on $m$-ovoids in classical polar spaces

TL;DR

This paper establishes new non-existence results for $m$-ovoids in certain classical polar spaces, refining previous bounds through geometric and combinatorial methods.

Contribution

It introduces an approach based on geometric and combinatorial arguments to improve bounds on the existence of $m$-ovoids in classical polar spaces.

Findings

Improved bounds on $m$-ovoids in $Q^-(2r+1,q)$, $W(2r-1,q)$, and $H(2r,q^2)$ for $r>2$

Extended previous algebraic and combinatorial methods to new geometric approaches

Enhanced understanding of the non-existence conditions for $m$-ovoids in these spaces

Abstract

In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$ for $r>2$. In [4] a lower bound on $m$ for the existence of $m$-ovoids of $H(4,q^2)$ is found by using the connection between $m$-ovoids, two-character sets, and strongly regular graphs. This approach is generalized in [3] for the polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$, $r>2$. In [1] an improvement for the particular case $H(4,q^2)$ is obtained by exploiting the algebraic structure of the collinearity graph, and using the characterization of an $m$-ovoid as an intruiging set. In this paper, we use an approach based on geometrical and combinatorial arguments, inspired by the results from [10], to improve the bounds from [3].