Abstract Let $$V=V(d,q)$$ V = V ( d , q ) denote the vector space of dimension d over $${\mathbb F}_q$$ F q . A subspace partition $$\mathcal {P}$$ 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 $$\mathcal {P}$$ 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 $$\delta (d)$$ 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 $$\sigma _q(d,t)$$ σ q ( d , t ) of any subspace partition of V for which the largest subspace has dimension t , with $$1\le t 1 ≤ t < d . In this paper, we build on the previous results and unveil additional structural information of subspace partitions. We determine the second minimum size $$\delta '(d)$$ δ ′ ( d ) over all (nontrivial) subspace partitions of V and furthermore, for $$d\equiv r \pmod {t}$$ d ≡ r ( mod t ) and $$0\le r 0 ≤ r < t < d , we prove the exact value of the second minimum size $$\sigma _q'(d,t)$$ σ q ′ ( d , t )

Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, and let ${\mathcal S}$ denote the set of all nonnegative integer solutions of the equation $x_1a_1+\cdots +x_na_n=y_1b_1+\cdots +y_mb_m$. A solution $(x_1,\ldots,x_n,y_1,\ldots,y_m)$ in ${\mathcal S}$ is called minimal if it cannot be expressed as the sum of two nonzero solutions in ${\mathcal S}$. For each pair $(i,j)$ with $1\leq i\leq n$ and $1\leq j\leq m$, the solution whose only nonzero coordinates are $x_i=b_j$ and $y_j=a_i$ is called a generator. Our main result shows that every minimal solution is a convex combination of the generators and the zero-solution. This proves a conjecture of Henk--Weismantel and, independently, Hosten--Sturmfels.