Classification of flocks of the quadratic cone in PG(3,64)

Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3,q) for q ≤ 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3,64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2,q), q even, and uses the prior classification of hyperovals in PG(2, 64)

A Tensor Product Action on q-Clan Generalized . . .

The Fundamental Theorem of q-clan geometry implies (among other things) that all automorphisms of the generalized quadrangle GQ(C) of order (q 2 ; q) associated with a q-clan C which fix a special pair of points are automorphisms of the elation group of the quadrangle. When q = 2 e , with a slight modification of the usual representation, we describe these automorphisms in terms of tensor products of pairs of matrices in GL(2; q). The resulting efficiency in computation allows a simplified description of the automorphisms of GQ(C). We apply the general theory to give an improved description of the induced stabilizers of the ovals in PG(2; q) that are associated with the new Subiaco q-clans introduced and studied in [2], [7], [1] and [8]