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 , k a t -[ 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.