Let \(V=V(d,q)\) denote the vector space of dimension d over \({\mathbb F}_q\) . A subspace partition \(\mathcal {P}\) of V, also known as a vector space partition, is a collection of nonempty subspaces of V such that each nonzero vector of V is in exactly one subspace of \(\mathcal {P}\) . Motivated by applications of minimum blocking sets and maximal partial t-spreads, Beutelspacher (Geom Dedic 9:425–449, 1980) determined in a lemma the minimum possible size \(\delta (d)\) over all (nontrivial) subspace partitions of V. In Heden et al. (Des Codes Cryptogr 64:265–274, 2012) and Năstase and Sissokho (Linear Algebra Appl 435:1213–1221, 2011), we extended Beutelspacher’s Lemma by determining the (first) minimum size \(\sigma _q(d,t)\) of any subspace partition of V for which the largest subspace has dimension t, with \(1\le t<d\) . In this paper, we build on the previous results and unveil additional structural information of subspace partitions. We determine the second minimum size \(\delta '(d)\) over all (nontrivial) subspace partitions of V and furthermore, for \(d\equiv r \pmod {t}\) and \(0\le r<t<d\) , we prove the exact value of the second minimum size \(\sigma _q'(d,t)\) of any subspace partition of V for which the largest subspace has dimension t and when at least one of the following holds: (i) \(r=0\) , (ii) \(t+r\) is even, (iii) \(d<2t\) or (iv) the partition has only subspaces of two different dimensions. Finally, applications to the supertail of a subspace partition and the size of maximal partial spreads are given.
