Covering sets of spreads in \(PG(3,q)\) (Q5943891)

An \(l\)-local covering set is a collection of spreads in \(PG(3,q)\), each containing the line \(l\), with the property that any skew line to \(l\) is contained in a unique spread of the collection. Using the Klein correspondence, one can associate with a regular spread \(S\) containing \(l\) a collection of \(q^2\) regular spreads containing \(l\) which form an \(l\)-local covering, called the special \(l\)-local covering determined by \(S\). NEWLINENEWLINENEWLINEMore generally, a covering set of spreads in \(PG(3,q)\) is any collection of spreads with the property that any given pair of skew lines in \(PG(3,q)\) is contained in a unique spread of the collection. Covering sets are one of the ingredients in the old proposal of R. H. Bruck to construct a projective plane of order \(q(q+1)\) from the points and lines of \(PG(3,q)\). Previously, the author has shown that there are no covering sets in \(PG(3,2)\) or \(PG(3,3)\). The existence problem for covering sets in \(PG(3,q)\), for \(q\geq 4\), remains open. NEWLINENEWLINENEWLINEIn the paper under review the author uses a computer search to show there does not exist a covering set of spreads in \(PG(3,4)\) which contains a special local covering set as defined above.