Some new upper bounds for the size of partial ovoids in slim generalized polygons and generalized hexagons of order \((s,s^3)\)
type: Item
instance of: scholarly article
title: Some new upper bounds for the size of partial ovoids in slim generalized polygons and generalized hexagons of order \((s,s^3)\)
zbMATH Open document ID: 0971.51001
DOI: 10.1023/A:1026579608364
author: Kris Coolsaet
author: Hendrik Van Maldeghem
published in: Journal of Algebraic Combinatorics
publication date: 2001-10-28T00:00:00Z
review text: A set of mutually opposite points of a generalized polygon \(\Gamma\) is called a partial ovoid. If a partial ovoid \(\mathcal O\) of a generalized polygon \(\Gamma\) is such that every element of \(\Gamma\) is at distance at most \(n/2\) from some elements of \(\mathcal O\), then \(\mathcal O\) is called an ovoid. The authors prove the following result. Let \(\Gamma\) be a generalized polygon containing a partial ovoid \(\mathcal O\). Suppose \(\Gamma\) is one of the following : an orthogonal quadrangle \(Q(5,2)\), a split Cayley hexagon \(H(2)\), a twisted triality hexagon \(T(2,8)\), or the Ree-Tits Octagon \(O(2)\). Then the number \(k\) of points of \(\mathcal O\) satisfies: (i) \(k \leq 7\) if \(\Gamma = Q(5,2)\) or \(H(2)\); (ii) \(k \leq 25\) if \(\Gamma = T(2,8)\); (iii) \(k \leq 27\) if \(\Gamma = O(2)\). Moreover, the authors prove that if \(\Gamma\) is a generalized hexagon of order \((s,s^3)\), for some integer \(s > 1\), then a partial ovoid has at most \(s^5 - s^3 + s -1\) points.
reviewed by: Dimitri Leemans
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 1553306
zbMATH Keywords: generalized polygon
zbMATH Keywords: generalized hexagon
zbMATH Keywords: ovoid
zbMATH Keywords: partial ovoid
zbMATH Keywords: projective embedding
MaRDI profile type: Publication
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full work available at URL: a:1026579608364
OpenAlex ID: W265748154
description: scientific article; zbMATH DE number 1553306