Nontrivial t-designs in polar spaces exist for all t

Abstract

AbstractA finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over $$\mathbb {F}_q$$ 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 , 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 $$\lambda $$ λ members of Y. Nontrivial examples are currently only known for $$t\le 2$$ t ≤ 2 . We show that t-$$(n,k,\lambda )$$ ( n , k , λ ) designs in polar spaces exist for all t and q provided that $$k>\frac{21}{2}t$$ 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.