The classification of the smallest nontrivial blocking sets in \(PG(n,2)\)
A blocking set with respect to \(t\)-spaces in an \(n\)-dimensional projective space is a set of points intersecting every \(t\)-dimensional subspace. It is called trivial if it contains an \((n-t)\)-dimensional subspace. The smallest non-trivial blocking sets with respect to \(t\)-spaces in finite desarguesian projective spaces \(PG(n,q)\) with \(q>2\) were characterized by \textit{A. Beutelspacher} [Geom. Dedicata 9, 425--449 (1980; Zbl 0377.50007)] as cones with vertex an \((n-t-2)\)-dimensional space \(\pi\) and base a non-trivial blocking set of minimal size in a plane skew to \(\pi\). In this paper, a similar result is obtained for the case \(q=2\), \(n\geq 3\), \(1\leq t\leq n-2\). It is proved that the smallest non-trivial blocking sets are cones with vertex an \((n-t-3)\)-space \(\pi\) and base the set of points on the edges of a tetrahedron in a solid skew to \(\pi\). The authors also show that the smallest non-trivial blocking sets with respect to hyperplanes in \(PG(n,2)\), \(n\geq 3\), are sets of \(5\) points in a \(3\)-dimensional space, no four of which are coplanar.
scientific article; zbMATH DE number 5078231
scientific article