On the type(s) of minimum size subspace partitions

Let V = V ( k t + r , q ) be a vector space of dimension k t + r over the finite field with q elements. Let σ q ( k t + 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 n d i the number of subspaces of dimension d i that occur in P and we say [ d 1 n d 1 , … , d m n d m ] 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.