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

Let Vn(q)Vn(q) denote a vector space of dimension nn over the field with qq elements. A set PP of subspaces of Vn(q)Vn(q) is a partition   of Vn(q)Vn(q) if every nonzero vector in Vn(q)Vn(q) is contained in exactly one subspace of PP. If there exists a partition of Vn(q)Vn(q) containing aiai subspaces of dimension nini for 1≤i≤k1≤i≤k, then (ak,ak−1,…,a1)(ak,ak−1,…,a1) must satisfy the Diophantine equation ∑i=1kai(qni−1)=qn−1. In general, however, not every solution of this Diophantine equation corresponds to a partition of Vn(q)Vn(q). In this article, we determine all solutions of the Diophantine equation for which there is a corresponding partition of Vn(2)Vn(2) for n≤7n≤7 and provide a construction of each of the partitions that exist.