q-Analogues of t-designs and their existence

Abstract

A t-[v,k,Λ ] design in a vector space of dimension v over a finite field is a family of k-subspaces such that each t-subspace is contained in precisely Λ elements of this family. They may be considered as a generalization of a spread in a projective space. It is shown that for given t,v,kat-[v,k,Λ ] design exists for all sufficiently large Λ provided the necessary parametric conditions are satisfied. The result is proved by solving a much more general question. Analogues of these results for affine spaces are also proved. We also describe a reciprocity relation for the number of distinct t-[v,k,Λ ] designs in a vector space, for given t, v, and k. This relation is similar to the one obtained by Shrikhande and Singhi for t-(v,k,Λ ) designs and by the authors for orthogonal arrays.