Hyperovals and Fano configurations

LetF be any field, finite or infinite, of characteristic 2. Put =PG(2,F). The classification of hyperovals is a difficult open problem. In this note we study the structure of the translation hyperovals and the hyperconics. We determine which quadrangles in a hyperconic have a given line as Fano line. The hyperconics and the translation hyperovals are similar with respect to containing quadrangles with certain Fano lines. We give two axioms satisfied by both. Remarkably, any hyperoval satisfying these must either be a hyperconic or else a translation hyperoval. It would be of great interest if one could find a way to slightly relax these conditions, or to piece together quadrangles with certain lines as Fano lines to obtain a new hyperoval.We are greatful for the help of Aiden Bruen. This research was funded in part by an NSERC Research Grant of Prof. A.A. Bruen, The University of Western Ontario, to whom we are greatful for the support.     

Hyperovals and Hadamard designs

In this paper, we investigatec-sets in 2-designs, with particular regard to sets of type (0,n) in projective planes. In particular, we associate a Hadamard design to a hyperoval of a projective plane of even orderq and we investigate some properties of its lines. This gives information on the order of the projective plane.     

Difference Sets and Hyperovals

We construct three infinite families of cyclic difference sets, using monomial hyperovals in a desarguesian projective plane of even order. These difference sets give rise to cyclic Hadamard designs, which have the same parameters as the designs of points and hyperplanes of a projective geometry over the field with two elements. Moreover, they are substructures of the Hadamard design that one can associate with a hyperoval in a projective plane of even order.     

α-Flocks and Hyperovals

A generalization of the concept of a flock of a quadratic cone in PG(3,q), q even, where the base of the cone is replaced by a translation oval, was introduced in [4] and is the focus of this work. The related idea of a q-clan is also generalized and studied with a particular emphasis on the connections with hyperovals. Several examples are given leading to new proofs of the existence of known hyperovals, unifying much of what has been done in this area. Finally, a proof that the Cherowitzo hyperovals do form an infinite family is also included.     

Hyperovals in Steiner systems

In this paper, we show that the basic necessary condition for the existence of a (k; 0, 2)-set in an S(2, 4, v) is also sufficient. It solves a problem posed by de Resmini [6] and we also prove some asymptotic results concerning the existence of hyperovals in Steiner systems with large block size. The results are generally applicable to designs with maximal arcs.     

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.     

Hyperovals in nets of small degree

The existence of an r‐net of order n with a hyperoval is proved for r = 5 when n ≥ 63; for r = 7 when n ≥ 84; for r = 9 when n ≥ 59,573 and for r = 15 when n ≥ 1, 873,273. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 322–334, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10018     

Ovals and Hyperovals in Desarguesian Nets

We determine the Desarguesian planes which hold r-nets with ovals and those which hold r-nets with hyperovals for every r7.     

Hyperovals with a transitive collineation group

We investigate the structure of a collineation group G leaving invariant a hyperoval (n+2 — arc) of a finite projective plane of even order n. The main result is that n=2, 4 or 16 when G acts transitively on and 4¦|G|. The case n=16 is investigated in some details.     

k-Arcs,Hyperovals, Partial Flocks and Flocks

Some recent results on k-arcs and hyperovals of PG(2,q),on partial flocks and flocks of quadratic cones of PG(3,q),and on line spreads in PG(3,q) are surveyed. Also,there is an appendix on how to use Veronese varieties as toolsin proving theorems.     

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

A pseudo‐hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo‐hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that is flag‐transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).     

Hyperovals in the known projective planes of order 16

We construct by computer all of the hyperovals in the 22 known projective planes of order 16. Our most interesting result is that four of the planes contain no hyperovals, thus providing counterexamples to the old conjecture that every finite projective plane contains an oval. © 1996 John Wiley & Sons, Inc.     

Characterization of Translation Planes by Orbit Lengths

The Desarguesian, Hall, and Hering translation planes of order q² are characterized as exactly those translation planes of odd order with spreads in PG (3,q) that admit a linear collineation group with infinite orbits one of length q+1 and i of length (q-q) /i for i=1 or 2.     

A Characterization of Orthogonality

Let X be a real inner product space of (finite or infinite) dimension greater than one. We proved (see Theorem 7, Chapter 1 of our book [1]) that if T is a separable translation group of X, and d an appropriate distance function of X which is supposed to be invariant under T and the orthogonal group O of X, then there are, up to isomorphism, exactly two solutions of geometries (X,G(T,O)), G the group generated by T ∪ O, namely euclidean and hyperbolic geometry over X. With the same geometrical definition for both geometries of arbitrary (finite or infinite) dimension > 1 we will characterize in this note the notion of orthogonality.     

A Characterization of Quasi-copulas

The notion of quasi-copula was introduced by C. Alsina, R. B. Nelsen, and B. Schweizer (Statist. Probab. Lett.(1993), 85–89) and was used by these authors and others to characterize operations on distribution functions that can or cannot be derived from operations on random variables. In this paper, the concept of quasi-copula is characterized in simpler operational terms and the result is used to show that absolutely continuous quasi-copulas are not necessarily copulas, thereby answering in the negative an open question of the above mentioned authors.     

On Automorphisms of Some d-Dimensional Dual Hyperovals in PG(d(d + 3)/2, 2)

In [3], a new d-dimensional dual hyperoval S in PG(d(d + 3)/2, 2) for d ≥ 3 was constructed based on Veronesean dual hyperoval. In this note, we determine the automorphism group of the dual hyperoval S. Received: January 26, 2007. Final Version received: January 7, 2008.