On partitions of finite vector spaces of low dimension over GF(2)

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 of P. If there exists a partition of Vn(q) containing ai subspaces of dimension ni for 1 ≤ i ≤ k, then (ak, ak−1, . . . , a1) must satisfy the Diophantine equation ∑k i=1 ai(q ni − 1) = q− 1. In general, however, not every solution of this Diophantine equation corresponds to a partition of Vn(q). In this article, we determine all solutions of the Diophantine equation for which there is a corresponding partition of Vn(2) for n ≤ 7.