A unified construction of finite geometries associated withq-clans in characteristic2

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 PGLð2; qÞ. The calculation of the automorphisms of the associated generalized quadrangles is su‰cient to show that these generalized quadrangles and the associated flocks and translation planes do not belong to any previously known family.

Finite Fourier series and ovals in PG(2, 2h)

“…Let h be a positive integer and write q = 2 \ Since GF(g) -> GF(g), z \-> z 1 + z [3] Finite Fourier series and ovals in PG(2,2'') 23 is a two-to-one mapping there is S e GF(q) with z 2 + z # <$ for all z € GF(q). Hence the polynomial z 2 + z + & is irreducible over GF(q).…”

“…This development comes as a double surprise: first it is somewhat surprising that what was believed to be two families turns out to be just one; second, it is very surprising that there should be such an easy description of these families. The Adelaide hyperovals in particular caused enormous difficulties, with nearly nine years separating their discovery by computer search in 1995 from the proof that they constitute an infinite family [3]. One can easily identify our ovals after having determined the equation their points must satisfy.…”

“…The permutation polynomial for the Adelaide hyperovals, which also come out of our theorem, is considerably more elaborate (see [1] or [3]). With perhaps further infinite families waiting to be discovered whose permutation polynomials are yet more formidable, the time is certainly ripe for an alternative approach.…”

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

“…(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.…”

“…Now isomorph rejection is iterated, so its replacement by a cheaper equivalent improves efficiency even if the proof of equivalence is expensive, as the proof need not be iterated. Thus, rather than perform conjugacy calculations many times, we characterise the known BLT-sets of ß(4, q) 9 for each relevant value of the field order q, by an ad hoc property P(q) of their stabilisers. It should be emphasised that the availability of efficient algorithms for dealing with permutation groups is the reason for our choice of this approach.…”