The complete characterization of the minimum size supertail in a subspace partition

Let q be a prime power and let n be a positive integer. Let V=V(n,q) denote the vector space of dimension n over Fq. A subspace partitionP of V is a collection of subspaces of V with the property that each nonzero vector is in exactly one of the subspaces in P. Suppose that d1,…,dk are the different dimensions, in increasing order, that occur in the subspace partition P. For any integer s, with 2≤s≤k, the ds-supertailS of P is the collection of all subspaces X∈P such that dim⁡X