24 research outputs found
Hyperovals in Hall planes
AbstractIn this paper we construct two classes of translation hyperovals in any Hall plane of even orderq2 ≥ 16. Two hyperovals constructed in the same Hall plane are equivalent under the action of the automorphism group of that Hall plane iff they are in the same class
k-arcs and partial flocks
AbstractUsing the relationship between partial flocks of the quadratic cone K in PG(3, q), q even, and arcs in the plane PG(2, q), new results on partial flocks and short proofs for known theorems on translation generalized quadrangles of order (q2, q) and on ovoids in PG(3, q) are obtained. It is shown that large partial flocks of K containing approximately q conics, q even, are always extendable to a flock, which improves a result by Payne and Thas. Then new and short proofs are given for a theorem of Johnson on translation generalized quadrangles and a theorem of Glynn on ovoids
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)
On Translation Hyperovals in Semifield Planes
In this paper we demonstrate the first example of a finite translation plane
which does not contain a translation hyperoval, disproving a conjecture of
Cherowitzo. The counterexample is a semifield plane, specifically a Generalised
Twisted Field plane, of order . We also relate this non-existence to the
covering radius of two associated rank-metric codes, and the non-existence of
scattered subspaces of maximum dimension with respect to the associated spread