Partial Flocks of Non-Singular Quadrics in PG(2 r + 1, q)
Matthew R. Brown
, Christine M. O'Keefe2
and Cristina Tonesi2
2dagger
DOI: 10.1023/B:JACO.0000048518.34713.fc
Abstract
We generalise the definition and many properties of partial flocks of non-singular quadrics in PG(3, q) to partial flocks of non-singular quadrics in PG(2 r + 1, q).
Pages: 359–370
Keywords: flock; partial flock; quadric; exterior set; thas flock
Full Text: PDF
References
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2. W.E. Cherowitzo, T. Penttilla, I. Pinneri, and G.F. Royle, “Flocks and ovals,” Geom. Dedicata 60 (1996), 17-37.
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4. J.C. Fisher and J.A. Thas, “Flocks in PG(3, q),” Math. Z. 169 (1979), 1-11.
5. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Oxford University Press, Oxford, 1979.
6. J.W.P. Hirschfeld and J.A. Thas, “Sets of type (1, n, q + 1) in PG(d, q),” Proc. London Math. Soc. 41(3) (1981), 254-278.
7. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991.
8. J.W.P. Hirschfeld and J.A. Thas, “The characterization of projections of quadrics over finite fields of even order,” J. London Math. Soc. 22(2) (1980), 226-238.
9. A. Klein, “Exterior sets of hyperbolic quadrics,” Bull. Belg. Math. Soc. 7 (2000), 321-331.
10. C.M. O'Keefe and J.A. Thas, “Partial flocks of quadratic cones with a point vertex in PG(n, q), n odd,” J. of Algebraic Combin. 6 (1997), 377-392.
11. L. Storme and J.A. Thas, “k-arcs and partial flocks,” Linear Algebra Appl. 226/228 (1995), 32-45.
12. J.A. Thas, “Flocks of non-singular ruled quadrics in PG(3, q),” Atti Acad. Naz. Lincei Rend. 59 (1975), 83-85.
13. J.A. Thas, “Generalized quadrangles and flocks of cones,” European J. Combin. 8 (1987), 441-452.
14. J.A. Thas, “Flocks, maximal exterior sets, and inversive planes. Finite geometries and combinatorial designs (Lincoln, NE, 1987),” Contemp. Math. 111, Amer. Math. Soc., Providence, RI (1990), 187-218.
15. J.A. Thas, “Maximal exterior sets of hyperbolic quadrics; the complete classification,” J. Comb. Theory Ser. A 56 (1991), 303-308.
16. J.A. Thas, “Flocks and partial flocks of quadrics: A survey,” J. Statist. Plan. Inference 94 (2001), 335-348.
2. W.E. Cherowitzo, T. Penttilla, I. Pinneri, and G.F. Royle, “Flocks and ovals,” Geom. Dedicata 60 (1996), 17-37.
3. F. De Clerck and J.A. Thas, “Exterior sets with respect to the hyperbolic quadric in PG(2n - 1, q),” Finite Geometries, Leture Notes in Pure and Appl. Math. 103 (1985), 83-91.
4. J.C. Fisher and J.A. Thas, “Flocks in PG(3, q),” Math. Z. 169 (1979), 1-11.
5. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Oxford University Press, Oxford, 1979.
6. J.W.P. Hirschfeld and J.A. Thas, “Sets of type (1, n, q + 1) in PG(d, q),” Proc. London Math. Soc. 41(3) (1981), 254-278.
7. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991.
8. J.W.P. Hirschfeld and J.A. Thas, “The characterization of projections of quadrics over finite fields of even order,” J. London Math. Soc. 22(2) (1980), 226-238.
9. A. Klein, “Exterior sets of hyperbolic quadrics,” Bull. Belg. Math. Soc. 7 (2000), 321-331.
10. C.M. O'Keefe and J.A. Thas, “Partial flocks of quadratic cones with a point vertex in PG(n, q), n odd,” J. of Algebraic Combin. 6 (1997), 377-392.
11. L. Storme and J.A. Thas, “k-arcs and partial flocks,” Linear Algebra Appl. 226/228 (1995), 32-45.
12. J.A. Thas, “Flocks of non-singular ruled quadrics in PG(3, q),” Atti Acad. Naz. Lincei Rend. 59 (1975), 83-85.
13. J.A. Thas, “Generalized quadrangles and flocks of cones,” European J. Combin. 8 (1987), 441-452.
14. J.A. Thas, “Flocks, maximal exterior sets, and inversive planes. Finite geometries and combinatorial designs (Lincoln, NE, 1987),” Contemp. Math. 111, Amer. Math. Soc., Providence, RI (1990), 187-218.
15. J.A. Thas, “Maximal exterior sets of hyperbolic quadrics; the complete classification,” J. Comb. Theory Ser. A 56 (1991), 303-308.
16. J.A. Thas, “Flocks and partial flocks of quadrics: A survey,” J. Statist. Plan. Inference 94 (2001), 335-348.