On vector space partitions and uniformly resolvable designs

Let Vn(q) denote a vector space of dimension n over the field with q elements. A set P of subspaces of Vn(q) is a partition of Vn(q) if every nonzero vector in Vn(q) is contained in exactly one subspace in P. A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of Vn(q) containing ai subspaces of dimension ni for 1 ≤ i ≤ k induces a uniformly resolvable design on qn points with ai parallel classes with block size qni , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph Kqn into ai Kqni factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on qn points where corresponding partitions of Vn(q) do not exist. ∗Part of this research was done while the author was visiting Illinois State University.