On the spectrum of minimal blocking sets in \(\text{PG}(2,q)\)
type: Item
instance of: scholarly article
title: On the spectrum of minimal blocking sets in \(\text{PG}(2,q)\)
zbMATH Open document ID: 1032.51011
author: András Gács
author: Zsuzsa Weiner
author: Tamás Szőnyi
published in: Journal of Geometry
publication date: 2003-08-31T00:00:00Z
review text: A blocking set \(B\) in PG\((2,q)\) is a set of points intersecting every line in at least one point. A blocking set \(B\) is called trivial when it contains a line and it is called minimal when none of its proper subsets still is a blocking set. This article gives a survey on the spectrum of minimal non-trivial blocking sets in PG\((2,q)\). This is the problem of determining the integers \(k\) for which minimal blocking sets of size \(k\) in PG\((2,q)\) exist. The authors have collected the results according to the intervals: (1) \(k <3(q+1)/2\) corresponding to the sizes of small minimal blocking sets, (2) \(3(q+1)/2 \leq k \leq 2q-1\), (3) \(k>2q-1\), and (4) large values of \(k\). In the final subsection, minimal non-trivial blocking sets in planes of small order are discussed. Regarding the different cases, there is hope that small minimal non-trivial blocking sets can be characterized. Regarding (2), there seems to be some structure in the minimal non-trivial blocking sets of these sizes, particularly when \(k\) is only slightly larger than \(3(q+1)/2\). There are in case (3) many blocking sets of size between \(2q-1\) and \(3q-3\), but no examples of minimal non-trivial blocking sets of size \(cq\) are known, where \(c\) is a constant larger than 3. There are several constructions giving minimal blocking sets of size \(cq \log q\). Very few constructions give larger blocking sets; a new construction is presented in this article. In the second section, two general construction methods for minimal blocking sets are discussed. They can be used to construct blocking sets of cardinalities belonging to distinct intervals mentioned above.
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 1972492
zbMATH Keywords: blocking sets
zbMATH Keywords: spectrum
zbMATH Keywords: minimal non-trivial blocking sets
reviewed by: Leo Storme
MaRDI profile type: Publication
full work available at URL: s00022-003-1702-2
OpenAlex ID: W158588660
DOI: 10.1007/S00022-003-1702-2
description: scientific article; zbMATH DE number 1972492