Lower bounds for the cardinality of minimal blocking sets in projective spaces
type: Item
instance of: scholarly article
title: Lower bounds for the cardinality of minimal blocking sets in projective spaces
zbMATH Open document ID: 1030.51008
DOI: 10.1016/S0012-365X(02)00863-4
author: Martin Bokler
published in: Discrete Mathematics
publication date: 2003-09-04T00:00:00Z
review text: For \(m \leq n\), an \(m\)-blocking set \(B\) in \(PG(n,q)\) is a set of points such that every \((n-m)\)-dimensional subspace meets \(B\). A minimal \(m\)-blocking set \(B\) is an \(m\)-blocking set such that no proper subset of \(B\) still is an \(m\)-blocking set. \textit{R. C. Bose} and \textit{R. C. Burton} [J. Comb. Theory 1, 96-104 (1966; Zbl 0152.18106)] proved that the smallest \(m\)-blocking sets in \(PG(n,q)\) are equal to \(m\)-dimensional subspaces \(PG(m,q)\) of \(PG(n,q)\). \textit{A. Beutelspacher} [Geom. Dedicata 9, 425-449 (1980; Zbl 0377.50007)] and \textit{U. Heim} [Mitt. Math. Semin. Gießen 226, 4-82 (1996; Zbl 0859.51005)] proved that the smallest minimal \(m\)-blocking sets of \(PG(n,q)\), not equal to an \(m\)-dimensional subspace, are cones with an \((m-2)\)-dimensional vertex \(\Omega\) and with base a minimal 1-blocking set \(B\) of smallest cardinality, different from a line, in a plane \(\Pi\) skew to \(\Omega\). The author presents improvements to the above-mentioned results by increasing the upper bound on the size of these \(m\)-blocking sets. Important improvements are obtained for the case \(q\) square and for the case \(q=p^{3h}\), \(p \geq 7\), \(q\) not square.
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1974982
zbMATH Keywords: blocking sets
zbMATH Keywords: Baer cones
reviewed by: Leo Storme
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1974982