On a particular class of minihypers and its applications. I: The result for general \(q\)
type: Item
instance of: scholarly article
title: On a particular class of minihypers and its applications. I: The result for general \(q\)
zbMATH Open document ID: 1022.51004
DOI: 10.1023/A:1021823703405
published in: Designs, Codes and Cryptography
publication date: 2003-04-03T00:00:00Z
review text: A partial \(s\)-spread \(S\) of \(\Sigma = \text{PG}(d,q)\) is a collection of mutually skew \(s\)-dimensional subspaces in \(\Sigma\). If all points of \(\Sigma\) are covered, \(S\) is called an \(s\)-spread; this is possible if and only if \((s + 1)|(d + 1)\). A partial spread which is not a spread is said to have positive deficiency. A blocking set \(B\) in \(\pi = \text{PG}(2,q)\) is a set of points in \(\pi\) such that every line of \(\pi\) meets \(B\) in at least one point. A blocking set is called nontrivial if it does not contain a line. Upper bounds on the size of maximal partial 1-spreads of \(\text{PG}(3,q)\) with positive deficiency have recently been improved by using an important link with blocking sets of \(\text{PG}(2,q)\). In the paper under review similar improvements are made for maximal partial \(s\)-spreads of positive deficiency whenever \((s+1)|(d+1)\), again by exploiting a link with nontrivial blocking sets of \(\text{PG}(2,q)\). The key ingredient is the use of (weighted) minihypers, first introduced in [\textit{N. Hamada} and \textit{T. Helleseth}, Math. Jap. 38, 925-939 (1993; Zbl 0786.05016)]. The results are then generalized to \(s\)-covers of \(\text{PG}(d,q)\); that is, subsets of s-dimensional subspaces of \(\text{PG}(d,q)\) which cover every point.
Mathematics Subject Classification ID: 51E14
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 1892218
zbMATH Keywords: partial spread
zbMATH Keywords: cover
zbMATH Keywords: minihyper
author: Patrick Govaerts
author: Leo Storme
reviewed by: Gary L. Ebert
MaRDI profile type: Publication
full work available at URL: a:1021823703405
OpenAlex ID: W1512937197
Recommended article: Results on maximal partial spreads in \(PG(3,p^3)\) and on related minihypers
Recommended article: On a particular class of minihypers and its applications. II: Improvements for \(q\) square
Recommended article: Covers of \(\mathrm{PG}(3,q)\) and of finite generalized quadrangles
Recommended article: A note on maximal partial spreads with deficiency \(q+1\), \(q\) even
Recommended article: Q3070368
Recommended article: Large minimal covers of PG\((3,q)\)
Recommended article: Q5883641
Recommended article: Partial covers of \(\mathrm{PG}(n, q)\)
Recommended article: Q3203626
Recommended article: A computer search of maximal partial spreads in PG(3,q)
description: scientific article; zbMATH DE number 1892218