J. Aust. Math. Soc.  76 (2004), 329-343
Some remarks on flocks

Laura Bader
  Dipartimento di Matematica e Applicazioni
  Universit\`a di Napoli `Federico II'
  Complesso di Monte S. Angelo
  Via Cintia - Edificio T
  I-80126 Napoli
Christine M. O'Keefe
  CSIRO ICT Centre
  GPO Box 664
  Canberra 2601 ACT
Tim Penttila
  School of Mathematics and Statistics (M019)
  The University of Western Australia
  35 Stirling Highway
  Crawley 6009 WA

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(2mq) to construct partial flocks of size  qm/2 + m/2 - 1  of the cone $\mathcal{K}$ in 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 $\mathcal{K}$ in PG(5, 3) of size 6 is generalised to a partial flock of the cone $\mathcal{K}$ 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(2pnp) admitting the symmetric group of degree 2pn + 1.
Download the article in PDF format (size 125 Kb)

TeXAdel Scientific Publishing ©  Australian MS