84 research outputs found
A characterization of the classical unital
No full textWe 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
Flock generalized quadrangles and tetradic sets of elliptic quadrics of PG(3, q)
No full textA flock of a quadratic cone of PG(3,q) is a partition of the non-vertex points into plane sections. It was shown by Thas in 1987 that to such flocks correspond generalized quadrangles of order (q2,q), previously constructed algebraically by Kantor (q odd) and Payne (q even). In 1999, Thas gave a geometrical construction of the generalized quadrangle from the flock via a particular set of elliptic quadrics in PG(3,q). In this paper we characterise these sets of elliptic quadrics by a simple property, construct the generalized quadrangle synthetically from the properties of the set and strengthen the main theorem of Thas 1999.S.G. Barwick, Matthew R. Brown and Tim Penttilahttp://www.elsevier.com/wps/find/journaldescription.cws_home/622862/description#descriptio
Sublines and subplanes of PG(2, q(3)) in the Bruck-Bose representation in PG(6, q)
No full textIn 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
An investigation of the tangent splash of a subplane of PG(2,q3)
No full textReceived: 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
The feet of orthogonal Buekenhout-Metz unitals
No full textIn this article we look at the geometric structure of the feet of an orthogonal Buekenhout–Metz unital in PG(2, q2). We show that the feet of each point form a set of type (0, 1, 2, 4). Further, we discuss the structure of any 4-secants, and determine exactly when the feet form an arc.S.G Barwick, W. -A. Jackson, P. Wil
Conics and multiple derivation
No full textLet C̄ be a conic in PG(2,q 2) and suppose we derive with respect to a derivation set or multiple derivation set. This paper looks at whether the conic C̄ gives rise to an inherited arc in the derived plane. Further, we construct two families of conics which give rise to inherited arcs after certain double derivations. Finally we construct a family of complete (q 2+1)-arcs in certain André planes. © 2012 Published by Elsevier B.V. All rights reserved.S.G Barwick and D.J. Marshal
The dual Yoshiara construction gives new extended generalized quadrangles
No full textA Yoshiara family is a set of q+3 planes in PG(5,q),q even, such that for any element of the set the intersection with the remaining q+2 elements forms a hyperoval. In 1998 Yoshiara showed that such a family gives rise to an extended generalized quadrangle of order (q+1,q−1). He also constructed such a family S(〇) from a hyperoval 〇 in PG(2,q). In 2000 Ng and Wild showed that the dual of a Yoshiara family is also a Yoshiara family. They showed that if 〇 has o-polynomial a monomial and 〇 is not regular, then the dual of S(〇) is a new Yoshiara family. This article extends this result and shows that in general the dual of S(〇) is a new Yoshiara family, thus giving new extended generalized quadrangles.S. G. Barwick and Matthew R. Brownhttp://www.elsevier.com/wps/find/journaldescription.cws_home/622824/description#descriptio
Geometric constructions of optimal linear perfect hash families
No full textA 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
Characterising hyperbolic hyperplanes of a non-singular quadric in PG (4, q)
No full textLet 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
Rebellion at Coranderrk
No full textMore than a century ago an Aboriginal community in Victoria campaigned for recognition of their right to occupy and control the small acreage they had farmed for 25 years. Others wanted to develop this tract. Government spokesmen denied that the occupants had inherited any rights to this land and declared that, anyway, they were not really Aborigines. This book is about the rebellion at Coranderrk Aboriginal Station between 1874 and 1886. It describes how Coranderrk families fought to keep their land. To explain why they fought I must begin with the years before, to show what this 'miserable spadeful of ground' meant to them, and how they came to be there. Finally, I sketch what ultimately happened. First published in 1998, 12 years after the death of its author Diane Barwick, Rebellion at Coranderrk was an attempt to rectify some of the injustices of the past two-hundred-plus years in Australia, and to prevent similar occurrences in the future. It remains acutely relevant
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