On sets of type \((m,m+q)_2\) in \(\mathrm{PG}(3,q)\)
type: Item
instance of: scholarly article
title: On sets of type \((m,m+q)_2\) in \(\mathrm{PG}(3,q)\)
zbMATH Open document ID: 1395.51007
author: Fulvio Zuanni
published in: Journal of Geometry
publication date: 2017-12-14T00:00:00Z
review text: A set \(S\) of points of \(\mathrm{PG}(3,q)\) is a set of type \((m,m+q)_2\) if every plane of \(\mathrm{PG}(3,q)\) meets \(S\) in \(m\) or \(m+q\) points and there exists planes of both types. The notation \(H_m(q)\) refers to a set of type \((m,m+q)_2\) of size \(m(q+1)\). An \(H_m(q)\) is called \textit{irreducible} if \(m=1\) or if it does not contain an \(H_n(q)\) with \(1 \leq n \leq m-1\). Let \(\mathcal{P}\) be a partial spread of \(\mathrm{PG}(3,q)\) of size \(m\), then the set \(K\) of all points on the lines of \(\mathcal{P}\) is an \(H_m(q)\). The complement of \(K\), i.e. the set of holes with relation to \(\mathcal{P}\) is an \(H_{q^2+1-m}(q)\), not containing a line if and only if \(\mathcal{P}\) is a maximal partial spread. The main results of this paper are classification results of \(H_n(q)\)'s for small values of the parameters \(n\) and \(q\), and it turns out that the classification of maximal partial spreads for these values of \(q\) plays an important role. The main results concerning sets \(H_m(q)\) are summarized in the following theorem. Note that this result classifies \textit{previously known} structures, as is clearly indicated in the paper. Theorem. Up to projective equivalence, the only irreducible sets \(H_4(q)\) exist for \(q \in \{4,5,9\}\), and are {\parindent=0.7cm\begin{itemize}\item[--] two irreducible \(H_4(4)\), each of them the set of holes of one of the 13 maximal partial spreads of \(\mathrm{PG}(3,4)\) size \(13\); \item[--] the unique irreducible \(H_4(5)\), which is the set of holes of one of the maximal partial spreads of \(\mathrm{PG}(3,4)\) size \(22\); \item[--] the unique irreducible \(H_4(9)\) which is the set of points of a Baer subplane. \end{itemize}} Assume that \(q+1 \mid (m+q)(q^2+1)\), then the notation \(K_m\) refers to a set of type \((m,m+q)_2\) of size \(\frac{(m+q)(q^2+1)}{q+1}\). A set \(K_m\) is irreducible if \(m< q+1\) or it does not contain a set \(K_n\) with \(n <m\). The final section of the paper…
reviewed by: Jan De Beule
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E14
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 6818637
zbMATH Keywords: sets of type \((m, n)_2\)
zbMATH Keywords: two-character sets
zbMATH Keywords: two-intersection sets
zbMATH Keywords: two-weight codes
zbMATH Keywords: partial spreads
zbMATH Keywords: mixed partitions
zbMATH Keywords: configurations
zbMATH Keywords: regular graphs
MaRDI profile type: Publication
full work available at URL: s00022-017-0401-3
OpenAlex ID: W2752512413
cites work: Blocking sets and partial spreads in finite projective spaces
cites work: On the size of a maximal partial spread
cites work: Some new results on optimal codes over \(\mathbb F_5\)
cites work: Partial Spreads and Replaceable Nets
cites work: Blocking Sets in Finite Projective Planes
cites work: The Geometry of Two-Weight Codes
cites work: On the non-existence of a projective (75, 4, 12, 5) set in PG(3, 7)
cites work: Q3739795
cites work: Weights of linear codes and strongly regular normed spaces
cites work: Q5698964
cites work: Q5492614
cites work: A characterization of some \(\{ 3v_ 2+v_ 3,3v_ 1+v_ 2; 3,3\}\)-minihypers and some \([15,4,9; 3]\)-codes with \(B_ 2=0\)
cites work: Q4370894
cites work: Q3694152
cites work: In \(\mathrm{AG}(3, q)\) any \(q^2\)-set of class \([1, m, n]_2\) is a cap
cites work: In \(\mathrm{AG}(3,q)\) any \(q^2\)-set of class \([0,m,n]_2\) containing a line is a cylinder
cites work: A note on line-Baer subspace partitions of \(\text{PG}(3,4)\)
cites work: Sets of type \((3,h)\) in \(\mathrm{PG}(3,q)\)
cites work: On sets of type \((q + 1, n)_{2}\) in finite three-dimensional projective spaces
cites work: On quasi-Hermitian varieties in \(\operatorname{PG}(3, q^2)\)
cites work: Q3810446
cites work: Q4153268
cites work: A combinatorial problem
cites work: Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields
DOI: 10.1007/S00022-017-0401-3
description: scientific article; zbMATH DE number 6818637