Abstract
The maximum number m 2(n, q) of points in PG(n, q), n⩾2, such that no three are collinear is known precisely for (n, q)=(n,2), (2,q), (3,q), (4, 3), (5,3). In this paper an improved upper bound of order q n−1−1/2q n−2 is obtained for q even when n⩾4 and q>2. A necessary preliminary is an improved upper bound for m′2(3, q), the maximum size of a k-cap not contained in an ovoid. It is shown that \(m'_2 (3,q){\text{ }} \leqslant q^2 - \tfrac{1}{2}{\text{q}} - \tfrac{1}{2}\sqrt {\text{q}} {\text{ + 2}}\) and that m′2(3, 4)=14.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beutelspacher, A., ‘Blocking Sets and Partial Spreads in Finite Projective Spaces’, Geom. Dedicata 9(1980), 425–449.
Hill, R., ‘Caps and Codes’, Discrete Math. 22(1978), 111–137.
Hirschfeld, J. W. P., Projective Geometries over Finite Fields, Oxford Univ. Press, 1979.
—, ‘Caps in Elliptic Quadrics’, Ann. Discrete Math. 18(1983), 449–466.
—, ‘Maximum Sets in Finite Projective Spaces’, Surveys in Combinatorics, London Math. Soc. Lecture Note Series 82, Cambridge Univ. Press, 1983, pp. 55–76.
--, Finite Projective Spaces of Three Dimensions, Oxford Univ. Press, 1985.
Huber, M., ‘A Characterization of Baer Cones in Finite Projective Spaces’, Geom. Dedicata 18(1985), 197–211.
Segre, B., ‘Le Geometrie di Galois’, Ann. Mat. Pura Appl. 48(1959), 1–97.
—, ‘Introduction to Galois Geometries’, Atti Accad. Naz. Lincei Mem. 8(1967), 133–236.
Thas, J. A., ‘Complete Arcs and Algebraic Curves in PG(2, q)’, J. Algebra (to appear).
Bruen, A. A., ‘Blocking Sets and Skew Subspaces of Projective Space’, Canad. J. Math. 23(1980), 628–630.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hirschfeld, J.W.P., Thas, J.A. Linear independence in finite spaces. Geom Dedicata 23, 15–31 (1987). https://doi.org/10.1007/BF00147388
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00147388