Regular packings of \(PG(3,q)\) (Q1272775)

Let \(PG(3,q)\) be the \(3\)-dimensional projective space over the Galois field \(GF(q)\). If \(l\), \(m\), \(n\) are three pairwise skew lines of \(PG(3,q)\), a transversal to \(l\), \(m\), \(n\) is a line meeting each one of them. A regulus of \(PG(3,q)\) is the set of \(q+1\) transversals to three pairwise skew lines. A spread \(S\) of \(PG(3,q)\) is a set of \(q^2+1\) lines partitioning the point-set of \(PG(3,q)\), and \(S\) is said to be regular if each line not in \(S\) meets \(S\) in the lines of a regulus. A packing of \(PG(3,q)\) is a partition of the set of lines of \(PG(3,q)\) by spreads and so has size \(q^2+q+1\). If a packing consists of regular spreads, it is said to be regular. Finally, a packing is cyclic if it admits a cyclic group acting regularly on the regular spreads in the packing. In the nice paper under review, the authors present two cyclic regular packings of \(PG(3,q)\), \(q\equiv 2\) mod \(3\) admitting a group of order \(q^2+q+1\). Their construction is based on the Klein correspondence between lines of \(PG(3,q)\) and points of a Klein quadric of \(PG(5,q)\).