Open Access Publications from the University of California

## Published Web Location

https://doi.org/10.5070/C63160424
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.

Mathematics Subject Classifications: 51E23, 05E30, 33C80

Keywords: Association schemes, codes, polar spaces, Steiner systems