Blocking sets in Desarguesian affine and projective planes
type: Item
instance of: scholarly article
title: Blocking sets in Desarguesian affine and projective planes
zbMATH Open document ID: 0912.51004
author: Tamás Szőnyi
published in: Finite Fields and their Applications
publication date: 1999-05-16T00:00:00Z
review text: A blocking set in a projective or affine plane is a set of points which intersects every line. A blocking set which contains a line is called trivial, while one having no proper subset which is a blocking set is called minimal (or irreducible). While a considerable amount of work has been done in this area, there are still several interesting open questions concerning the spectrum of sizes of minimal non-trivial blocking sets. In this paper the author is mainly concerned with ``small'' blocking sets, i.e., those minimal non- trivial blocking sets whose size lies in the interval (\(q+1, 3(q+1)/2\)), where \(q\) is the order of the finite field over which the projective or affine plane is defined. For odd \(q\) the projective triangle provides an example of size \(3(q+1)/2\), while for even \(q\) the similar projective triad gives an example of size \((3q+2)/2\). Blokhuis has shown that when \(q\) is an odd prime, there is no smaller example than the projective triangle. The known examples in this range are all of Rédei type, that is, if \(B\) is such a blocking set, then there exists a line \(\ell\) which intersects \(B\) in exactly \(| B |- q\) points. The main result of the paper is that a non-trivial minimal blocking set of size less than \(3(q+1)/2\) in a Desarguesian projective plane of order \(q = p^n\) intersects every line in 1 modulo \(p\) points. It is also shown that the size of such a blocking set must lie in a few relatively short (with respect to \(q\)) subintervals. In particular, when \(q = p^2\), these results imply that a non-trivial minimal blocking set either contains a Baer-subplane or has size at least \(3(q+1)/2\), and this result is sharp. The technique used to obtain these results is based on the Rédei polynomial which encodes the sizes of the intersections of lines with a set of points as multiplicities of its roots. Several investigators have studied this polynomial algebraically to obtain information about blocking sets, especially those o…
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 1199978
zbMATH Keywords: blocking sets
zbMATH Keywords: bounds
zbMATH Keywords: Rédei polynomial
Wikidata QID: Q60306528
reviewed by: William E. Cherowitzo
MaRDI profile type: Publication
full work available at URL: ffta.1996.0176
OpenAlex ID: W2154186200
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DOI: 10.1006/FFTA.1996.0176
description: scientific article; zbMATH DE number 1199978