Minimal blocking sets in \(PG(2,8)\) and maximal partial spreads in \(PG(3,8)\) (Q1431621)

A spread of \(PG(3,q)\) is a set of \(q^2+1\) lines partitioning the point set of \(PG(3,q).\) A partial spread of \(PG(3,q)\) is a set of pairwise disjoint lines of \(PG(3,q)\) not forming a spread. A partial spread is called maximal when it is not contained in a larger partial spread. The largest known maximal partial spreads of \(PG(3,q),\) \(q>3\) have size \(q^2-q+2.\) Let \(\mathcal S\) be a maximal partial spread of \(PG(3,q)\) of size \(q^2+1-\delta.\) A point not lying on a line of \(\mathcal S\) is called a hole of \(\mathcal S.\) If a plane \(\pi\) does not contain any line of \(\mathcal S,\) then it contains \(q + \delta\) holes, which form a non-trivial blocking set of \(\pi\) because of the maximality of \(\mathcal S.\) As the smallest non-trivial blocking set of \(PG(2,8)\) has size \(13,\) a maximal partial spread of \(PG(3,8)\) has size \(q^2+1-\delta\) with \(5 \leq \delta \leq 7.\) By using the properties of the blocking sets of \(PG(2,8),\) the authors prove that \(\delta=7,\) i.e. a maximal partial spread of \(PG(3,8)\) has size \(q^2-q+2=58.\)