A unified construction of finite geometries associated with q-clans in characteristic 2

Abstract

Flocks of Laguerre planes, generalized quadrangles, translation planes, ovals, BLTsets, and the deep connections between them, are at the core of a developing theory in the area of geometry over finite fields. Examples are rare in the case of characteristic two, and it is the purpose of this paper to contribute a fifth infinite family. The approach taken leads to a unified construction of this new family with three of the previously known infinite families, namely those satisfying a symmetry hypothesis concerning cyclic subgroups of PGL(2, q). The calculation of the automorphisms of the associated generalized quadrangles is sufficient to show that these generalized quadrangles and the associated flocks and translation planes do not belong to any previously known family.