Partitions of finite vector spaces into subspaces

摘要

Let Vn(q) denote a vector space of dimension n over the field with q elements. A set 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 . Suppose there exists a partition of Vn(q) into xi subspaces of dimension ni, 1 i k. Then x1, , xk satisfy the Diophantine equation . 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.