On partitions of finite vector spaces of low dimension over

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