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 Fq F q equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t - (n,k,λ ) ( 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 λ members of Y . Nontrivial examples are currently only known for t≤ 2 t ≤ 2 . We show that t - (n,k,λ ) ( n , k , λ ) designs in polar spaces exist for all t and q provided that k>21/2t 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.