The second minimum size of a finite subspace partition

Let $$V=V(d,q)$$ V = V ( d , q ) denote the vector space of dimension d over Fq F q . A subspace partition P 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 P 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 δ (d) δ ( 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 σ q(d,t) σ q ( d , t ) of any subspace partition of V for which the largest subspace has dimension t , with 1≤ t<d 1 ≤ 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 δ '(d) δ ′ ( d ) over all (nontrivial) subspace partitions of V and furthermore, for d≡ r t d ≡ r ( mod t ) and 0≤ r<t<d 0 ≤ r < t < d , we prove the exact value of the second minimum size σ q'(d,t) σ q ′ ( d , t ) </j