On the type(s) of minimum size subspace partitions

Let V=V(kt+r,q) be a vector space of dimension kt+r over the finite field with q elements. Let σq(kt+r,t) denote the minimum size of a subspace partition P of V in which t is the largest dimension of a subspace. We denote by ndi the number of subspaces of dimension di that occur in P and we say [d1nd1,…,dmndm] is the type of P. In this paper, we show that a partition of minimum size has a unique partition type if t+r is an even integer. We also consider the case when t+r is an odd integer, but only give partial results since this case is indeed more intricate.