New Bounds for Partial Spreads of $H(2d-1, q^2)$ and Partial Ovoids of the Ree-Tits Octagon

TL;DR

This paper establishes new upper bounds for partial spreads in Hermitian polar spaces and partial ovoids in the Ree-Tits octagon, improving previous bounds and demonstrating the non-existence of ovoids in certain cases.

Contribution

It provides the first bounds for partial spreads of $ ext{H}(2d-1, q^2)$ with even $d$, and tight bounds for partial ovoids of the Ree-Tits octagon, advancing understanding of their combinatorial limits.

Findings

Partial spread size of $ ext{H}(3, q^2)$ is at most $(rac{2p^3+p}{3})^t+1$.

Partial ovoids of the Ree-Tits octagon are at most $26^t+1$ in size.

Ree-Tits octagon $ ext{O}(2^t)$ does not admit an ovoid.

Abstract

Two results are obtained that give upper bounds on partial spreads and partial ovoids respectively. The first result is that the size of a partial spread of the Hermitian polar space $\mathsf{H}(3, q^2)$ is at most $\left(\frac{2p^3+p}{3} \right)^t+1$, where $q=p^t$, $p$ is a prime. For fixed $p$ this bound is in $o(q^3)$, which is asymptotically better than the previous best known bound of $(q^3+q+2)/2$. Similar bounds for partial spreads of $\mathsf{H}(2d-1, q^2)$, $d$ even, are given. The second result is that the size of a partial ovoid of the Ree-Tits octagon $\mathsf{O}(2^t)$ is at most $26^t+1$. This bound, in particular, shows that the Ree-Tits octagon $\mathsf{O}(2^t)$ does not have an ovoid.