New multiple hyper-regulus planes (Q928326)

A partial spread \(\mathcal H\) of \(PG(2n-1,q)\) is a hyper-regulus if \(\mathcal H\) has size \(q^{n-1}+\cdots+q+1\) and there is a partial spread \({\mathcal H}^*\) of \(PG(2n-1,q),\) each of whose components intersects each component of \(\mathcal H\) in a point. If \(\mathcal H\) is contained in a spread \(\mathcal S,\) then \({\mathcal S}^*=({\mathcal S}\setminus {\mathcal H})\cup {\mathcal H}*\) is a new spread of \(PG(2n-1,q).\) \textit{C. Culbert} and \textit{G. L. Ebert} [Innov. Incidence Geom. 1, 3--18 (2005; Zbl 1110.51002)] have constructed sets of mutually disjoint hyper-reguli contained in a desarguesian spread of \(PG(5,q),\) which produce new translation planes by multiple replacement. In this paper new classes of translation planes of order \(q^n,\) \(n\leq 3,\) have been constructed by multiple hyper-reguli replacements, which contains the classes of Coulbert and Ebert for \(n=3.\)