Some Remarks on Flocks

L. Bader, C.M. O'Keefe, Tim Penttila

    Research output: Contribution to journalArticle

    8 Citations (Scopus)

    Abstract

    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/ + m/ - 1 of the cone 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 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.
    Original languageEnglish
    Pages (from-to)329-343
    JournalJournal of the Australian Mathematical Society
    Volume76
    Issue number3
    DOIs
    Publication statusPublished - 2004

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    Flock
    Partial
    Cone
    Congruent
    Modulo
    Odd
    Symmetric group
    Vertex of a graph

    Cite this

    Bader, L. ; O'Keefe, C.M. ; Penttila, Tim. / Some Remarks on Flocks. In: Journal of the Australian Mathematical Society. 2004 ; Vol. 76, No. 3. pp. 329-343.
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    abstract = "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/ + m/ - 1 of the cone 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 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.",
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    Bader, L, O'Keefe, CM & Penttila, T 2004, 'Some Remarks on Flocks' Journal of the Australian Mathematical Society, vol. 76, no. 3, pp. 329-343. https://doi.org/10.1017/S1446788700009897

    Some Remarks on Flocks. / Bader, L.; O'Keefe, C.M.; Penttila, Tim.

    In: Journal of the Australian Mathematical Society, Vol. 76, No. 3, 2004, p. 329-343.

    Research output: Contribution to journalArticle

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    AB - 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/ + m/ - 1 of the cone 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 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.

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