Constructing minimal blocking sets using field reduction (Q2808797)

A \textit{blocking set} \(B\) with respect to \((k-1)-\)spaces in \(\mathrm{PG}(n-1,q)\) is a set of points such that every \((k-1)\)-dimensional subspace of \(\mathrm{PG}(n-1,q)\) contains at least a point of \(B.\) If no proper subset of \(B\) is a blocking set, then \(B\) is \textit{minimal}.NEWLINENEWLINELet \(\Pi\) be the Barlotti-Cofman representation of \(\mathrm{PG}(n-1,q^t)\) in \(\mathrm{PG}(nt-1,q)\) via a Desarguesian spread \(\mathcal S\) of a hyperplane \(\mathrm{PG}(nt-1,q)\) of \(\mathrm{PG}(nt,q)\) whose points are both points of \(\mathrm{PG}(nt,q)\) not in \(\mathrm{PG}(nt-1,q)\) and the elements of \(\mathcal S.\) Let \(\Omega\) be an \((nt-kt-2)\)-dimensional subspace of \(\mathrm{PG}(nt,q)\) disjoint with a plane \(\Gamma\) of \(\mathrm{PG}(nt,q).\) If \(\bar{B}\) is a blocking set of \(\Gamma,\) let \(K\) be the cone of \(\langle \Omega, \Gamma\rangle\) with vertex \(\Omega\) and base \(\bar{B}.\) Let \(B={\mathcal B}(K)\) be the set of points of \(\Pi\simeq \mathrm{PG}(n-1,q^t)\) defined by \(K,\) i.e. the points of \(B\) are both points of \(K \setminus \mathrm{PG}(nt-1,q)\) and the elements of the \((t-1)-\)spread contained in \(K.\)NEWLINENEWLINEThe author gives two different constructions such that \(B\) is a minimal blocking set of \(\mathrm{PG}(n-1,q^t),\) the first one generalizing the example constructed in [\textit{F. Mazzocca} and \textit{O. Polverino}, J. Algebraic Combin. 24, No. 1, 61--81 (2006; Zbl 1121.51007)]. Finally, she proves that a small blocking set of the first type is linear if and only if \(\bar{B}\) is linear.