DSpace Community:http://hdl.handle.net/2440/2992015-09-01T00:15:04Z2015-09-01T00:15:04ZCharacterising pointsets in PG(4,q) that correspond to conicsBarwick, S.G.Jackson, W.A.http://hdl.handle.net/2440/940262015-08-31T23:29:09Z2014-12-31T13:30:00ZTitle: Characterising pointsets in PG(4,q) that correspond to conics
Author: Barwick, S.G.; Jackson, W.A.
Abstract: We consider a non-degenerate conic in PG(2,q2), q odd, that is tangent to ℓ∞ and look at its structure in the Bruck–Bose representation in PG(4,q). We determine which combinatorial properties of this set of points in PG(4,q) are needed to reconstruct the conic in PG(2,q2). That is, we define a set C in PG(4,q) with q2 points that satisfies certain combinatorial properties. We then show that if q≥7, we can use C to construct a regular spread S in the hyperplane at infinity of PG(4,q), and that C corresponds to a conic in the Desarguesian plane P(S)≅PG(2,q2) constructed via the Bruck–Bose correspondence.
Description: Received: 23 November 2014 / Revised: 8 April 2015 / Accepted: 1 May 2015 / Published online: 20 May 20152014-12-31T13:30:00ZA characterization of translation ovals in finite even order planesBarwick, S.G.Jackson, W.A.http://hdl.handle.net/2440/939992015-08-31T06:50:18Z2014-12-31T13:30:00ZTitle: A characterization of translation ovals in finite even order planes
Author: Barwick, S.G.; Jackson, W.A.2014-12-31T13:30:00ZAn investigation of the tangent splash of a subplane of PG(2,q3)Barwick, S.G.Jackson, W.A.http://hdl.handle.net/2440/939982015-08-31T08:45:29Z2014-12-31T13:30:00ZTitle: An investigation of the tangent splash of a subplane of PG(2,q3)
Author: Barwick, S.G.; Jackson, W.A.
Abstract: In 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.
Description: Received: 12 May 2013 / Revised: 23 March 2014 / Accepted: 8 April 2014 / Published online: 3 May 20142014-12-31T13:30:00ZA characterization of the classical unitalBarwick, S.G.http://hdl.handle.net/2440/939972015-08-31T07:18:40Z1993-12-31T13:30:00ZTitle: A characterization of the classical unital
Author: Barwick, S.G.
Abstract: 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].1993-12-31T13:30:00Z