The structure of the minimum size supertail of a subspace partition

Let denote the vector space of dimension n over the finite field with q elements. A subspace partition of V is a collection of nontrivial subspaces of V such that each nonzero vector of V is in exactly one subspace of . For any integer d, the d -supertail of is the set of subspaces in of dimension less than d, and it is denoted by ST. Let denote the minimum number of subspaces in any subspace partition of V in which the largest subspace has dimension t. It was shown by Heden et al. that , where t is the largest dimension of a subspace in ST. In this paper, we show that if , then the union of all the subspaces in ST constitutes a subspace under certain conditions.