Nontrivial t-designs in polar spaces exist for all t

Abstract A finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over Fq F q equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t - (n, k, ) (n, k, λ) 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 λ members of Y. Nontrivial examples are currently only known for t 2 t ≤ 2. We show that t - (n, k, ) (n, k, λ) designs in polar spaces exist for all t and q provided that k>212t k > 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.