Some non-existence results for distance-$j$ ovoids in small generalized polygons

TL;DR

This paper uses computational methods to prove the non-existence of certain geometric structures called distance-2 ovoids in a specific small generalized polygon, and provides bounds for partial structures in related polygons.

Contribution

It presents the first computer-aided proof of non-existence for distance-2 ovoids in the dual split Cayley hexagon and establishes upper bounds for partial ovoids in related geometries.

Findings

Non-existence of distance-2 ovoids in the dual split Cayley hexagon $ extsf{H}(4)^D$

Upper bounds on partial distance-2 ovoids in $ extsf{H}(2)^D$ and $ extsf{H}(4)^D$

Computer-based proof techniques applied to finite geometric structures

Abstract

We give a computer-based proof for the non-existence of distance-$2$ ovoids in the dual split Cayley hexagon $\mathsf{H}(4)^D$. Furthermore, we give upper bounds on partial distance-$2$ ovoids of $\mathsf{H}(q)^D$ for $q \in \{2, 4\}$.