The second minimum size of a finite subspace partition

Abstract Let $$V=V(d,q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>=</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> denote the vector space of dimension d over $${\mathbb F}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> . A subspace partition $$\mathcal {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> 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 $$\mathcal {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> . 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 $$\delta (d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\sigma _q(d,t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of any subspace partition of V for which the largest subspace has dimension t , with $$1\le t&lt;d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> . In this paper, we build on the previous results and unveil additional structural information of subspace partitions. We determine the second minimum size $$\delta '(d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>δ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> over all (nontrivial) subspace partitions of V and furthermore, for $$d\equiv r \pmod {t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≡</mml:mo> <mml:mi>r</mml:mi> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mo>mod</mml:mo> <mml:mspace/> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$0\le r&lt;t&lt;d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>r</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> , we prove the exact value of the second minimum size $$\sigma _q'(d,t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mi>q</mml:mi> <mml:mo>′</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </j