Some pairwise balanced designs
type: Item
instance of: scholarly article
title: Some pairwise balanced designs
zbMATH Open document ID: 0938.05007
author: Malcolm Greig
published in: The Electronic Journal of Combinatorics
publication date: 2000-03-12T00:00:00Z
full work available at URL: https://eudml.org/doc/120314
full work available at URL: v7i1r13.html
review text: Summary: A pairwise balanced design, \(B(K;v)\), is a block design on \(v\) points, with block sizes taken from \(K\), and with every pair of points occurring in a unique block; for a fixed \(K\), \(B(K)\) is the set of all \(v\) for which a \(B(K;v)\) exists. A set, \(S\), is a PBD-basis for the set, \(T\), if \(T=B(S)\). Let \(N_{a(m)}=\{n:n\equiv a\bmod m\}\), and \(N_{\geq m}=\{n:n\geq m\}\); with \(Q\) the corresponding restriction of \(N\) to prime powers. This paper addresses the existence of three PBD-basis sets. (1) It is shown that \(Q_{1(8)}\) is a basis for \(N_{1(8)}\setminus E\), where \(E\) is a set of 5 definite and 117 possible exceptions. (2) We construct a 78 element basis for \(N_{1(8)}\) with, at most, 64 inessential elements. (3) \textit{F. E. Bennett} and \textit{L. Zhu} [Contemporary design theory, Collect. Surv., 41-96 (1992; Zbl 0765.05022)] have shown that \(Q_{\geq 8}\) is a basis for \(N_{\geq 8}\setminus E'\), where \(E'\) is a set of 43 definite and 606 possible exceptions. Their result is improved to 48 definite and 470 possible exceptions. (Constructions for 35 of these possible exceptions are known.) Finally, we provide brief details of some improvements and corrections to the generating/exception sets published in The CRC Handbook of Combinatorial Designs.
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 1413177
zbMATH Keywords: BIBD
zbMATH Keywords: pairwise balanced design
zbMATH Keywords: block design
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1413177