On the size of a cap in \(\text{PG} (n,q)\) with \(q\) even and \(n\geq 3\)
type: Item
instance of: scholarly article
title: On the size of a cap in \(\text{PG} (n,q)\) with \(q\) even and \(n\geq 3\)
zbMATH Open document ID: 0920.51004
DOI: 10.1023/A:1005095418152
author: Jinming Chao
published in: Geometriae Dedicata
publication date: 1999-09-08T00:00:00Z
review text: A \(k\)-cap in \(PG(n,q)\) is a set of \(k\) points such that no three of them are collinear. A \(k\)-cap is said to be complete if it is maximal with respect set-theoretic inclusion. The largest value of \(k\) for which a \(k\)-cap in \(PG(n,q)\) exists is denoted my \(m_2(n,q)\) and the size of the second largest complete \(k\)-cap in \(PG(n,q)\) is denoted by \({m'}_2(n,q)\). In the paper under review an improved upper bound to the value \({m'}_2(3,q)\), \(q\) even, \(q\geq 8\) is given. Precisely, the author proves that \[ {m'}_2(3,q)\leq q^2-q+5. \] The new upper bound on \({m'}_2(3,q)\) is then used to improve the upper bound on \({m'}_2(n,q)\), \(q\) even and \(n\geq 4\), namely \[ {m'}_2(n,q)\leq q^{n-1}-q^{n-2}+6q^{n-3}-4q^{n-4}-2(q^{n-5+\ldots +q+1)+1}. \]
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1270426
zbMATH Keywords: cap
zbMATH Keywords: finite geometry
reviewed by: Antonio Cossidente
MaRDI profile type: Publication
full work available at URL: a:1005095418152
OpenAlex ID: W125579379
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description: scientific article; zbMATH DE number 1270426