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Some remarks on flocks

Part of: Designs and configurations Finite geometry and special incidence structures Geometry Linear incidence geometry

Published online by Cambridge University Press:  09 April 2009

Laura Bader ,
Christine M. O'Keefe  and
Tim Penttila
Laura Bader
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli ‘Federico II’, Complesso di Monte S. Angelo, Via Cintia - Edificio T, I-80126 Napoli, Italy e-mail: laura.bader@dma.unina.it
Christine M. O'Keefe
Affiliation:
CSIRO ICT Centre, GPO Box 664, Canberra 2601, Australia e-mail: christine.okeefe@csiro.au
Tim Penttila
Affiliation:
School of Mathematics and Statistics (M019), The University of Western Australia, 35 Stirling Highway, Crawley 6009 WA, Australia e-mail: penttila@maths.uwa.edu.au
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Abstract

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New proofs are given of the fundamental results of Bader, Lunardon and Thas relating flocks of the quadratic cone in PG(3, q), q odd, and BLT-sets of Q(4, q). We also show that there is a unique BLT-set of H(3, 9). The model of Penttila for Q(4, q), q odd, is extended to Q(2m, q) to construct partial flocks of size qm/2+m/2 – 1 of the cone kin PG(2m – 1, q) with vertex a point and base Q(2m – 2, q), where q is congruent to 1 or 3 modulo 8 and m is even. These partial flocks are larger than the largest previously known for m > 2. Also, the example of O'Keefe and Thas of a partial flock of k in PG(5, 3) of size 6 is generalised to a partial flock of the cone k of PG(2pn – 1, p) of size 2pn, for any prime p congruent to 1 or 3 modulo 8, with the corresponding partial BLT-set of Q(2pn, p) admitting the symmetric group of degree 2pn + 1.

Keywords

flockgeneralised quadrangleBLT-setparabolic quadricovoid

MSC classification

Secondary: 51E12: Generalized quadrangles, generalized polygons 51A40: Translation planes and spreads 51A50: Polar geometry, symplectic spaces, orthogonal spaces 51E20: Combinatorial structures in finite projective spaces 05B25: Finite geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 329 - 344
Copyright
Copyright © Australian Mathematical Society 2004

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