The Classification of Projective Planes of Order q3 with Cone-Representations in PG (6, q)

The classification of cone-representations of projective planes of orderq³ of index 3 and rank 4 (and so in PG(6,^q)) is completed. Any projective plane with a non-spread representation (being a cone-representation of the second kind) is a dual generalised Desarguesian translation plane, as found by Jha and Johnson, and conversely. Indeed, given any collineation of PG(2,^q) with no fixed points, there exists such a projective plane of order q³ , where q is a prime power, that has the second kind of cone-representation of index 3 and rank 4 in PG(6,q). An associated semifield plane of order q³ is also constructed at most points of the plane. Although Jha and Johnson found this plane before, here we can show directly the geometrical connection between these two kinds of planes.