Partitions of finite vector spaces into subspaces

Abstract

AbstractLet Vn(q) denote a vector space of dimension n over the field with q elements. A set ${\cal P}$ of subspaces of Vn(q) is a partition of Vn(q) if every nonzero element of Vn(q) is contained in exactly one element of ${\cal P}$. Suppose there exists a partition of Vn(q) into xi subspaces of dimension ni, 1 ≤ i ≤ k. Then x1, …, xk 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 Vn(q). In this article, we show that there exists a partition of Vn(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2n − 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of Vn(q) induce uniformly resolvable designs on qn points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008