Super-simple balanced incomplete block designs with block size 4 and index 5
type: Item
instance of: scholarly article
title: Super-simple balanced incomplete block designs with block size 4 and index 5
zbMATH Open document ID: 1250.05027
author: H. Cao
author: Kejun Chen
author: Ruizhong Wei
published in: Discrete Mathematics
publication date: 2009-12-09T00:00:00Z
review text: A group divisible design (GDD) is a triple \((X, G, B)\), which satisfies the following conditions: {\parindent=6mm \begin{itemize}\item[1.] \(G\) is a partition of a set \(X\) (of points) into subsets called groups. \item[2.] \(B\) is a set of subsets of \(X\) (called blocks) such that a group and a block contain at most one common point. \item[3.] Every pair of points from distinct groups occurs in exactly \(\lambda\) blocks. \end{itemize}} The group type of a GDD is the multi-set \(\{|G|: G \in G\}\). A GDD with block sizes from a set of positive integers \(K\) is called a \((K, \lambda)\)-GDD, when \(K = \{k\}\), we use \(k\) for \(K\) ad if \(\lambda=1\), it is a \(k\)-GDD. A \((k, \lambda)\)-GDD with group type 1 is called a pairwise balanced design (PBD). A \((k, \lambda)\)-GDD with group type \(1v\) is called balanced incomplete block design (BIBD) is denoted by \((v, k, \lambda)\)-BIBD or \((v, b, r, k, \lambda)\) with \(v\) elements, \(b\) blocks, each element repeating in \(r\) blocks, each block is of size \(k\) and each pair of elements occurring in \(\lambda\) blocks. The necessary conditions for the existence of a BIBD are \(b \geq v\), \(bk = rv\), \(\lambda (v-1) = r (k-1)\). A transversal design \(TD (k, \lambda; n)\) is a \((k, \lambda)\)-GDD of a group type \(nk\) and block size \(k\), when \(\lambda=1\), we write \(TD (k,n)\), which is equivalent to \(k-2\) mutually orthogonal Latin squares (MOLS) of order \(n\). A design is called simple if it contains no repeated blocks. A design is said to be super-simple if the intersection of any two blocks has at most two elements. When \(k=3\), a super simple design is just a simple design. When \(\lambda=1\) the designs are necessarily super-simple. In this paper the authors addressed the existence problem of the super-simple BIBDs with \(k \geq 4\) and \(\lambda > 1\). Actually the term of super-simple designs was introduced by Gronau and Mullin [12]. The existence of super-simple designs is an interest…
reviewed by: Ratnakaram Nava Mohan
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 5644070
zbMATH Keywords: balanced incomplete block designs
zbMATH Keywords: super-simple designs
MaRDI profile type: Publication
full work available at URL: j.disc.2008.07.003
OpenAlex ID: W2011992755
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DOI: 10.1016/J.DISC.2008.07.003
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description: scientific article; zbMATH DE number 5644070