Singer line orbits in PG(3,q)

Abstract

A complete characterization of reguli that are contained in Singer line orbits is given. The characterization is field theoretic and depends upon modeling PG(3, q ) by the finite field GF( q 4 ), viewed as a four-dimensional vector space over GF( q ). As applications of this characterization one is able to construct various balanced incomplete block designs and group divisible designs, the most interesting one having the parameters of a ( q +1)-fold cover of an inversive plane. A robust method for constructing large families of mutually inequivalent unembeddable translation nets of order q 2 and deficiency q is also given.