Small point sets that meet all generators of \(W(2n+1,q)\)
type: Item
instance of: scholarly article
title: Small point sets that meet all generators of \(W(2n+1,q)\)
zbMATH Open document ID: 1052.51007
author: Klaus Metsch
published in: Designs, Codes and Cryptography
publication date: 2004-08-19T00:00:00Z
review text: A set \(O\) of points meeting exactly once every generator of a polar space \(P\) is called an ovoid of \(P\). Clearly, \(O\), if it exists, is a blocking set of minimum cardinality for the set of all generators of \(O\). It has been shown by Thas that several polar spaces, namely \(Q^{-}(2n+1,q)\) with \(n\geq 2\), \(W(2n+1,q)\) with \(q\) even and \(n\geq 2\) or with \(q\) odd and \(n\geq 1\) and \(H(2n,q)\) with \(n\geq 2\), do not admit ovoids. It is then meaningful to ask what is the minimum size of a blocking set for the generators of these spaces. In this paper, the author proves that the size of such a blocking set \(B\) for \(W(2n+1,q)\) with \(q\) even is \( q^{n-1}(q^2+1)\) and characterises exactly the sets meeting this bound as those consisting of the points of a cone with a \((n-2)\)-dimensional vertex over an ovoid of \(W(3,q)\).
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 2093320
zbMATH Keywords: polar spaces
zbMATH Keywords: blocking sets
reviewed by: Luca Giuzzi
MaRDI profile type: Publication
full work available at URL: b:desi.0000015888.27935.e2
OpenAlex ID: W2088457083
DOI: 10.1023/B:DESI.0000015888.27935.E2
description: scientific article; zbMATH DE number 2093320