A Bose-Burton theorem for elliptic polar spaces
type: Item
instance of: scholarly article
title: A Bose-Burton theorem for elliptic polar spaces
zbMATH Open document ID: 0946.51004
DOI: 10.1023/A:1026483311192
author: Klaus Metsch
published in: Designs, Codes and Cryptography
publication date: 2000-01-24T00:00:00Z
review text: In [J. Comb. Theory 1, 96-104 (1966; Zbl 0152.18106)] \textit{R. C. Bose} and \textit{R. C. Burton} proved that in a finite projective space \(\text{ PG}(d,q)\) a set \(B\) of points that meets every subspace of a fixed dimension \(c\leq d\) has at least \((q^{d+1-c}- 1)/(q-1)\) points with equality if and only if \(B\) consist of the points of a subspace of dimension \(d-c\). There are various interesting generalizations and variations of this problem. The main result of the paper under review for example states that for \(0\leq c \leq n-1\) the smallest set \(B\) of points in the elliptic quadric \(Q := Q^-(2n+1,q)\) that meets every singular subspace of dimension \(c\) of \(Q\) has cardinality \((q^{n+1} + q^c)(q^{n-c} - 1)/(q-1)\). Equality holds if and only if there is a subspace \(U\) of dimension \(c-1\) of \(Q\) such that \(B = (U^{\perp}\setminus U) \cap Q\), where \(\perp\) is the polarity associated with the quadric.
reviewed by: Andreas E. Schroth
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 1392715
zbMATH Keywords: polar spaces
zbMATH Keywords: blocking sets
MaRDI profile type: Publication
full work available at URL: a:1026483311192
OpenAlex ID: W1608507874
description: scientific article; zbMATH DE number 1392715