A characterization of the classical unital

We define Buekenhout unitals in derivable translation planes of dimension 2 over their kernel and provide a characterization of these unitals. We use this result to improve the characterization of classical unitals given by Lefèvre-Percsy [13] and Faina and Korchmáros [7].S.G. Barwic

A characterization of translation ovals in finite even order planes

In this article we consider a set C of points in PG(4,q), q even, satisfying certain combinatorial properties with respect to the planes of PG(4,q). We show that there is a regular spread in the hyperplane at infinity, such that in the corresponding Bruck-Bose plane PG(2,q2), the points corresponding to C form a translation hyperoval, and conversely. CrownS.G.Barwick, Wen-Ai Jackso

An investigation of the tangent splash of a subplane of PG(2,q3)

Received: 12 May 2013 / Revised: 23 March 2014 / Accepted: 8 April 2014 / Published online: 3 May 2014In PG(2,q3), let π be a subplane of order q that is tangent to ℓ∞. The tangent splash of π is defined to be the set of q2+1 points on ℓ∞ that lie on a line of π. This article investigates properties of the tangent splash. We show that all tangent splashes are projectively equivalent, investigate sublines contained in a tangent splash, and consider the structure of a tangent splash in the Bruck–Bose representation of PG(2,q3) in PG(6,q). We show that a tangent splash of PG(1,q3) is a GF (q)-linear set of rank 3 and size q2+1; this allows us to use results about linear sets from Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89–104, 2010) to obtain properties of tangent splashes.S. G. Barwick, Wen-Ai Jackso

Characterising hyperbolic hyperplanes of a non-singular quadric in PG (4, q)

Let H be a non-empty set of hyperplanes in PG(4,q), q even, such that every point of PG(4,q) lies in either 0, 1/2q³ or 1/2(q³+q²²) hyperplanes of H, and every plane of PG(4,q) lies in 0 or at least 1/2q hyperplanes of H. Then H is the set of all hyperplanes which meet a given non-singular quadric Q(4, q) in a hyperbolic quadric.S.G. Barwick, Alice M.W. Hui, Wen-Ai Jackson, Jeroen Schillewaer

Sublines and subplanes of PG(2, q(3)) in the Bruck-Bose representation in PG(6, q)

In this article we look at the Bruck-Bose representation of PG(2,q 3) in PG(6,q). We look at sublines and subplanes of order q in PG(2,q 3) and describe their representation in PG(6,q). We then show how these results can be generalized to the Bruck-Bose representation of PG(2,q n) in PG(2n,q). © 2011 Elsevier Inc. All rights reserved.S.G. Barwick, Wen-Ai Jackso

Geometric constructions of optimal linear perfect hash families

A linear (qd,q,t)-perfect hash family of size s in a vector space V of order qd over a field F of order q consists of a sequence 1,…,s of linear functions from V to F with the following property: for all t subsets XV there exists i{1,…,s} such that i is injective when restricted to F. A linear (qd,q,t)-perfect hash family of minimal size d(t−1) is said to be optimal. In this paper we use projective geometry techniques to completely determine the values of q for which optimal linear (q3,q,3)-perfect hash families exist and give constructions in these cases. We also give constructions of optimal linear (q2,q,5)-perfect hash families.S.G. Barwick, and Wen-Ai Jackso