On vector space partitions and uniformly resolvable designs

Abstract

Let V n (q) denote a vector space of dimension n over the field with q elements. A set $${\mathcal{P}}$$ of subspaces of V n (q) is a partition of V n (q) if every nonzero vector in V n (q) is contained in exactly one subspace in $${\mathcal{P}}$$ . 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 V n (q) containing a i subspaces of dimension n i for 1 ? i ? k induces a uniformly resolvable design on q n points with a i parallel classes with block size $$q^{n_i}$$ , 1 ? i ? k, and also corresponds to a factorization of the complete graph $$K_{q^n}$$ into $$a_i K_{q^{n_i}}$$ -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 q n points where corresponding partitions of V n (q) do not exist.