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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 ∑i=1kai(qni−1)=qn−1. In general, however, not...