Nontrivial t-designs in polar spaces exist for all t

A finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over \(\mathbb {F}_q\) ${\mathbb{F}}_q$ equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t- \((n,k,\lambda )\) $(n,k,\lambda )$ design in a finite classical polar space of rank n is a collection Y of totally isotropic k-spaces such that each totally isotropic t-space is contained in exactly \(\lambda \) $\lambda$ members of Y. Nontrivial examples are currently only known for \(t\le 2\) $t\le 2$ . We show that t- \((n,k,\lambda )\) $(n,k,\lambda )$ designs in polar spaces exist for all t and q provided that \(k>\frac{21}{2}t\) $k>\frac{21}{2}t$ and n is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.