Lower bounds for the cardinality of minimal blocking sets in projective spaces
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.
scientific article; zbMATH DE number 1974982