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Linear independence in finite spaces

Authors:	J. W. P. Hirschfeld J. A. Thas

Affiliation:	(1) Mathematics Division, University of Sussex, BN19QH Brighton, United Kingdom;(2) Seminarie voor Meetkunde en Kombinatoriek, Rijksuniversiteit Gent, Krijgslaan 281, 9000 Gent, Belgium

Abstract:	The maximum number m₂(n, q) of points in PG(n, q), n2, 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 n4 and q>2. A necessary preliminary is an improved upper bound for m₂(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 m₂(3, 4)=14.

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