Flocks and ovals

Abstract

An infinite family of q-clans, called the Subiaco q-clans, is constructed for q = 2 ~. Associated with these q-clans are flocks of quadratic cones, elation generalized quadrangles of order (q2, q), ovals of PG(2, q) and translation planes of order q2 with kernel GF(q). It is also shown that a q-clan, for q = 2 ~, is equivalent to a certain configuration of q + 1 ovals of PG(2, q), called a herd.Mathematics Subject Classification (1991): Primary: 51E21, 51E20, 51E12, 51E15; Secondary: 05B25.

Classification of flocks of the quadratic cone over fields of order at most 29

“…Notation for those we need to refer to will be fixed in each appropriate section. The known infinite families of flocks of the quadratic cone in characteristic 2 are the linear flocks, the Fisher-Thas-Walker flocks [15], [43] for fields of non-square order, the two classes of Payne flocks [30], [31] for fields of nonsquare order, the Subiaco flocks [10], and the Adelaide flocks [9] for fields of square order. No sporadic flocks of the quadratic cone in characteristic 2 are presently known.…”

“…(The links with translation planes are many, with new links via hyperbolic fibrations recently discovered [5], [4].) Since then, it has become clear that complete classification in the Miquelian Laguerre case is extremely difficult, with further constructions appearing in [10], [34], [25], [32], [9], and so attention has turned to small field orders. The previously known classification results are [41] for fields of orders 2, 3, 4 and De Clerck- Gevaert-Thas (1988) [11] for fields of orders 5, 7, 8 (these are computerfree results); Mylle (1991) [27] for the field of order 9, De Clerck-Herssens (1992) [12] for the fields of orders 11, 16, Penttila-Royle (1998) [35] for the fields of orders 13, 17, Brown-O'Keefe-Payne-Penttila-Royle [7] for the field of order 32 (these are computer-based results), see Theorem 2.6.…”

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“…(1) For n = 3 we refer to [2] and [11]. In this case the (k + 1)-arcs of Theorem 8 extend to (k + 2)-arcs by adjoining the point (0, 1, 0).…”

A Result on Spreads of the Generalized QuadrangleT2(O), withOan Oval Arising from a Flock, and Applications

“…Then S(F) has at least q 3 + q 2 subquadrangles S of order q, and, by Payne [9], each of them is a T 2 (O) of Tits, with O an oval of PG (2, q). The hyperovals arising from the ovals O are exactly the elements of the herd of hyperovals defined by the flock F; herds of hyperovals were introduced by Cherowitzo et al in [2].…”