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A method of constructing pairwise balanced designs containing parallel classes

Douglas R. Stinson

Abstract

The obvious way to construct a GDD (group-divisible design) recursively is to use Wilson's Fundamental Construction for GDDs (WFC). Then a PBD (pairwise balanced design) is often obtained by adding a new point to each group of the GDD. However, after constructing such a PBD, it might be the case that we then want to identify a parallel class of blocks. In this short note, we explore some possible ways of doing this.

A method of constructing pairwise balanced designs containing parallel classes

Abstract

The obvious way to construct a GDD (group-divisible design) recursively is to use Wilson's Fundamental Construction for GDDs (WFC). Then a PBD (pairwise balanced design) is often obtained by adding a new point to each group of the GDD. However, after constructing such a PBD, it might be the case that we then want to identify a parallel class of blocks. In this short note, we explore some possible ways of doing this.
Paper Structure (2 sections, 5 theorems, 8 equations)

This paper contains 2 sections, 5 theorems, 8 equations.

Table of Contents

  1. Introduction and Definitions
  2. Constructions

Key Result

Theorem 2.1

Suppose there is a TD$(\ell+1,m)$ and a TD$(\ell,u)$, where $u \leq m$. Suppose $\ell \in K$ and suppose that there is a $K$-GDD of type $u^{\ell} v^1$. Finally, suppose there exists an $(mu+1,K)$-PBD. Then there exists a $K$-GDD of type $\ell^{mu}{(tv+1)}^1$ for all $t$ such that $0 \leq t \leq m -

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark
  • Example 2.3
  • Theorem 2.4
  • proof
  • Remark
  • Lemma 2.5
  • ...and 4 more