An extension theorem for Steiner systems
type: Item
instance of: scholarly article
title: An extension theorem for Steiner systems
zbMATH Open document ID: 0826.05004
DOI: 10.1016/0012-365X(93)E0190-F
author: John L. Blanchard
published in: Discrete Mathematics
publication date: 1995-11-26T00:00:00Z
review text: In the first of the two articles under review (this item), the author proves the following extension theorem for Steiner systems with \(t= 2\): There are functions \(v_0(k)\) and \(\lambda_0(k, v)\) such that for any \(\text{S}(2, k- 1; v){\mathbf D}\) with \(v\geq v_0(k)\) and any \(\lambda\geq \lambda_0(k, v)\) satisfying the necessary arithmetic conditions there exists a 3-design \(\text{S}_\lambda(3, k; v+ 1)\) for which the derived design at a suitable point consists of \(\lambda\) copies of \({\mathbf D}\). The main interest of this theorem lies in the fact that it is a principal ingredient in the proof of the following major result contained in the author's second article (next item). Given a prime power \(q\) and any positive integer \(v\) satisfying the (necessary) conditions \(v- 1\equiv 0\text{ mod}(q- 1), v(v^2- 1)\equiv 0\text{ mod} (q^2- 1)\), there exists a Steiner system \(\text{S}(3, q+ 1, vq^n+ 1)\) for all sufficiently large \(n\) (i.e., \(n\geq n_1(q, v)\)).
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 51E10
zbMATH DE Number: 778521
zbMATH Keywords: 3-design
zbMATH Keywords: derived design
zbMATH Keywords: extension theorem
zbMATH Keywords: Steiner systems
reviewed by: Dieter Jungnickel
MaRDI profile type: Publication
cites work: Q3686719
cites work: A construction for Steiner 3-designs
cites work: Q4774057
full work available at URL: 0012-365x(93)e0190-f
OpenAlex ID: W2005433165
description: scientific article; zbMATH DE number 778521