Packings and Steiner systems in polar spaces

TL;DR

This paper classifies $t$-Steiner systems in finite classical polar spaces, revealing they only exist in specific cases, and connects these systems to packings within polar spaces.

Contribution

It provides an almost complete classification of $t$-Steiner systems in polar spaces, showing their existence is limited to certain cases, and introduces a broader result on packings.

Findings

Nontrivial $t$-Steiner systems only exist for specific parameters.

Most $t$-Steiner systems are classified as non-existent outside corner cases.

The classification is derived from a general packing result in polar spaces.

Abstract

A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $n$-spaces such that each totally isotropic $t$-space is contained in exactly one member of $Y$. Nontrivial examples are known only for $t=1$ and $t=n-1$. We give an almost complete classification of such $t$-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces.