Caps in \(\mathrm{PG}(n,q)\), \(q\) even, \(n\ge 3\)
type: Item
instance of: scholarly article
title: Caps in \(\mathrm{PG}(n,q)\), \(q\) even, \(n\ge 3\)
zbMATH Open document ID: 0770.51015
DOI: 10.1007/BF01264518
published in: Geometriae Dedicata
publication date: 1993-05-16T00:00:00Z
review text: A \(k\)-cap in \(\mathrm{PG}(n,q)\) is a set of \(k\) points, no three of which are collinear. A \(k\)-cap is called complete if it is not contained in a \((k+1)\)-cap. Let \(m_2(n,q)\) denote the maximum value of \(k\) for which there exists a (complete) \(k\)-cap in \(\mathrm{PG}(n,q)\), and let \(m_2'(n,q)\) denote the second largest value of \(k\) for which there exists a complete \(k\)-cap in \(\mathrm{PG}(n,q)\). The exact value of \(m_2(n,q)\) is only known when \((n,q)=(n,2)\), \((2,q)\), \((3,q)\), (4,3) and (5,3). Much less is known about \(m_2'(n,q)\). A result of \textit{J. W. P. Hirschfeld} and \textit{J. A. Thas} [Geom. Dedicata 23, 15--31 (1987; Zbl 0616.51008)] states that for \(q>2\) an even prime power, \(m_2'(3,q)\le q^2-q/2-\sqrt q/2+2\). By clever counting involving tangent lines the authors are able to show there does not exist a complete \(k\)-cap in \(\mathrm{PG}(3,q)\), \(q\ge 64\) an even prime power, for values of \(k\) in various intervals. Hence they are able to deduce that \(m_2'(3,q)<q^2-q+2\sqrt q+1\). From this result they are then able to show that \(m_2(4,q)<q^ 3-q^2+2q\sqrt q+2q-2\sqrt q+1\) whenever \(q\ge 64\) is an even prime power. An improved upper bound for \(m_2(n,q)\), \(n\ge 5\) and \(q\ge 64\) an even prime power, is also obtained.
Mathematics Subject Classification ID: 51E22
zbMATH DE Number: 166876
zbMATH Keywords: complete \(k\)-cap
author: Leo Storme
author: Tamás Szőnyi
reviewed by: Gary L. Ebert
MaRDI profile type: Publication
cites work: On the sharpness of a theorem of B. Segre
cites work: Q3717726
cites work: Caps and codes
cites work: Q3851217
cites work: Q3694152
cites work: Linear independence in finite spaces
cites work: Q4003762
cites work: A family of complete arcs in finite projective planes
cites work: Q5582819
cites work: Elementary proofs of two fundamental theorems of B. Segre without using the Hasse-Weil theorem
cites work: Complete arcs and algebraic curves in \(\text{PG}(2,q)\)
cites work: Arcs in projective planes over prime fields
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description: scientific article; zbMATH DE number 166876