Partitions of finite vector spaces into subspaces

Let V n (q) denote a vector space of dimension n over the field with q elements. A set ${\cal P}$ of subspaces of V n (q) is a partition of V n (q) if every nonzero element of V n (q) is contained in exactly one element of ${\cal P}$ . Suppose there exists a partition of V n (q) into x i subspaces of dimension n i , 1 ≤  i  ≤  k . Then x 1 , …, x k satisfy the Diophantine equation $\sum_{i=1}^{k}{(q^{n_i}-1)x_i}=q^n-1$ . However, not every solution of the Diophantine equation corresponds to a partition of V n (q). In this article, we show that there exists a partition of V n (2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7 x  + 3 y  = 2 n  − 1 and y ≠  1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V n (q) induce uniformly resolvable designs on q n points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008