The structure of the minimum size supertail of a subspace partition

2016/06/01 by E. Nastase, Nastase, E., P. Sissokho +1
Mathematics · #51B25 #51E14 #51E20 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:51B25 #msc:51E14 #msc:51E20

paper · pdf · doi:10.48550/arxiv.1606.00120

15 pages

arxiv created 2016/06/01 · arxiv updated 2016/06/02

Abstract

Let V=V(n,q) denote the vector space of dimension n over the finite field with q elements. A subspace partition \mathcal P of V is a collection of nontrivial subspaces of V such that each nonzero vector of V is in exactly one subspace of \mathcal P. For any integer d, the d-supertail of \mathcal P is the set of subspaces in \mathcal P of dimension less than d, and it is denoted by ST. Let σ_q(n,t) denote the minimum number of subspaces in any subspace partition of V in which the largest subspace has dimension t. It was shown by Heden et al. that |ST|≥ σ_q(d,t), where t is the largest dimension of a subspace in ST. In this paper, we show that if |ST|=σ_q(d,t), then the union of all the subspaces in ST constitutes a subspace under certain conditions.

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