Characterizing the line set of a Baer subspace
type: Item
instance of: scholarly article
title: Characterizing the line set of a Baer subspace
zbMATH Open document ID: 0611.51006
DOI: 10.1016/0097-3165(87)90052-5
author: Albrecht Beutelspacher
author: Dorothea Seeger
published in: Journal of Combinatorial Theory. Series A
publication date: 1987-01-01T00:00:00Z
review text: This is a 3-dimensional generalization of a theorem of \textit{A. Bruen} on blocking sets in finite projective planes [SIAM J. Appl. Math. 21, 380- 392 (1971; Zbl 0252.05014)]. Let \({\mathcal L}\) be a set of lines in \({\mathfrak P}=PG(3,q)\). A blocking set of a projective plane is a set \({\mathcal B}\) of points such that every line contains a point of \({\mathcal B}\) and a point off \({\mathcal B}\). A plane \(\pi\) is said to be a blocking plane if the set of lines in \(\pi\) forms a blocking set of \(\pi\). Let \({\mathcal L}\) now satisfy the properties: (0) There exists at least one blocking plane in \({\mathfrak P}\), (1) Every point of \({\mathfrak P}\) is on at least one line of \({\mathcal L}\), (2) For any two intersecting lines \(\ell\), \(\ell '\in {\mathcal L}\), their span \(<\ell,\ell '>\) is a blocking plane. Then the theorem states that \(| {\mathcal L}| \geq (q+1)(q+\sqrt{q}+1),\) with equality if and only if \({\mathcal L}\) is the line set of a Baer subspace of \({\mathfrak P}\), that is, a subspace of order \(\sqrt{q}\). Thus the examples of minimal cardinality are exactly the line sets of Baer subspaces of \({\mathfrak P}\).
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51A30
zbMATH DE Number: 3988102
zbMATH Keywords: blocking sets in finite projective planes
zbMATH Keywords: blocking plane
zbMATH Keywords: Baer subspace
MaRDI profile type: Publication
full work available at URL: 0097-3165(87)90052-5
OpenAlex ID: W2090454714
cites work: Blocking sets and partial spreads in finite projective spaces
cites work: On Baer subspaces of finite projective spaces
cites work: Q3968056
cites work: A characterization of flat spaces in a finite geometry and the uniqueness of the hamming and the MacDonald codes
cites work: Blocking Sets in Finite Projective Planes
cites work: Blocking Sets and Skew Subspaces of Projective Space
cites work: Sulle \(k\)-calotte di uno spazio lineare finito
cites work: Q5544833
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description: scientific article; zbMATH DE number 3988102