A New Class of Translation Planes

Abstract

We define a nest of reguli to be a collection P of reguli in a regular spread S of PG(3, q) such that every line of S is contained in exactly 0 or 2 reguli of P. Let U denote the lines of S contained in the reguli of some nest. If V is a partial spread of PG[3,q) covering the same points as U but having no lines in common with U , then V will be called a replacement set for U. Clearly, (S—U) U V is a spread of PG(3, q) , yielding a (potentially new) translation plane of order q 2 which is 2–dimensional over its kernel. Nests of size (q+3)/2 were first studied (under another name) by Bruen and later by many others. Whether such (q+3)/2- nests exist for q 13 and whether such nests are necessarily reversible are still open questions. In this paper we consider nests of size q. We exhibit an infinite family of ?-nests, one for each odd prime q , and show that each nest is reversible. The translation planes so obtained appear to be new, at least for q ≥ 11.