Proper blocking sets in projective spaces (Q1377801)

A blocking set of a \(d\)-dimensional projective space \({\mathbf P}\) of order \(q\) is a set \(B\) of points of \({\mathbf P}\) such that every hyperplane of \({\mathbf P}\) contains at least one point of \(B\) and no line of \({\mathbf P}\) is contained in \(B\). A blocking set is minimal if it contains no proper subset which is a blocking set. A blocking set \(B\) is called proper if the restriction of \(B\) to any hyperplane \(H\) of \({\mathbf P}\) is not a blocking set in \(H\). This definition is motivated by the main result of the paper: Theorem. If \(B\) is a blocking set of \(\text{PG}(d,q)\), with \(d \geq 2\) and \(q > 3\), such that the cardinality of \(B\) is less than or equal to one more than the minimum cardinality of a blocking set in \(\text{PG}(2,q)\), then there exists a plane of \(\text{PG}(d,q)\) so that the restriction of \(B\) to this plane is a blocking set of the plane. An example of a proper minimal blocking set is given by \(\text{PG}(d,p)\) considered as a subset of \(\text{PG}(d,q)\), with \(q = p^d\) for some prime \(p\). Several other examples are given in the paper.