The sets closest to ovoids in \(Q^-(2n+1,q)\)
type: Item
instance of: scholarly article
title: The sets closest to ovoids in \(Q^-(2n+1,q)\)
zbMATH Open document ID: 0927.51016
author: Klaus Metsch
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 1999-12-06T00:00:00Z
review text: An ovoid of a polar space is a set of points with the property that every maximal subspace of the polar space contains exactly one point of the ovoid. It is known by a result of \textit{J. A. Thas} [Geom. Dedicata 10, 135-144 (1981; Zbl 0458.51010)] that the elliptic quadric \(Q^-(2n+1,q)\) does not have an ovoid when \(n \geq 2\). The author reproves and improves this result of Thas by characterizing the smallest subsets of an elliptic quadric \(Q^-(2n+1,q)\) which intersect every maximal subspace of \(Q^-(2n+1,q)\) in at least one point. He proves that these sets \(B\) have cardinality \(q^{n+1}+q^{n-1}\) with equality if and only if \(B=(U^\perp \setminus U) \cap Q^-(2n+1,q)\) for a subspace \(U\) of dimension \(n-2\) with \(U \subset Q^-(2n+1,q)\). This problem can be generalized to the problem of finding the smallest subsets \(B\) of an elliptic quadric \(Q^-(2n+1,q)\) which intersect all the \(s\)-dimensional subspaces, \(s \leq n-1\), contained in \(Q^-(2n+1,q)\) in at least one point. These sets have been characterized completely by the author in an other article: `A Bose-Burton theorem for elliptic polar spaces' which will appear in Des. Codes Cryptography. The work performed in this article and in the second article are part of a global project of the author to characterize the smallest subsets of polar spaces intersecting all the \(s\)-dimensional subspaces contained in these polar spaces.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 1266821
zbMATH Keywords: ovoid
zbMATH Keywords: elliptic quadric
zbMATH Keywords: polar space
zbMATH Keywords: blocking set
reviewed by: Leo Storme
MaRDI profile type: Publication
Recommended article: A Bose-Burton theorem for elliptic polar spaces
Recommended article: Partial ovoids in classical finite polar spaces
Recommended article: A modular equality for m$m$‐ovoids of elliptic quadrics
Recommended article: Substructures in finite classical polar spaces
Recommended article: Small point sets that meet all generators of \(Q(2n,p)\), \(p>3\) prime
Recommended article: The Hermitian variety \(H(5,4)\) has no ovoid
Recommended article: Translation generalized quadrangles of order \((s,s ^{2})\), \(s\) even, and eggs
Recommended article: On the smallest minimal blocking sets of \(Q(2n,q)\), for \(q\) an odd prime
Recommended article: Small point sets that meet all generators of \(W(2n+1,q)\)
Recommended article: Some \(p\)-ranks related to orthogonal spaces
description: scientific article; zbMATH DE number 1266821