A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa

Satoru Kadowaki,Sanpei Kageyama

doi:10.11648/j.ijtam.20200602.12

Abstract

A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized toα-resolvability and affine α-resolvability in 1963. When α = 1, they are simply called a resolvable or an affine resolvable design, respectively. In literature these combinatorial arguments are mostly done for a class of block designs with property of balanced incomplete block (BIB) designs and α-resolvability. Due to these backgrounds, Kadowaki and Kageyama have tried to clarify the existence of affine α-resolvable partially balanced incomplete block (PBIB) designs having association schemes of two associate classes. The known 2-associate PBIB designs have been mainly classified into the following types depending on association schemes, i.e., group divisible (GD), triangular, Latin-square (L2), cyclic. First, it could be proved that an affine α-resolvable cyclic 2-associate PBIB design does not exist for any α ≥ 1. Also, Kageyama proved the non-existence of an affine α-resolvable triangular design for 1 ≤ α ≤ 10 in 2008. Furthermore, the existence of affine resolvable GD designs and affine resolvable L2 designs with parameters υ ≤ 100 and r, k ≤ 20 was mostly clarified by Kadowaki and Kageyama in 2009 and 2012. As a result, only three designs (i.e., two semi-regular GD designs, only one L2 design) are left unknown on existence within the practical range of parameters. In the present paper, a necessary condition for the existence of a certain resolvable pairwise balanced (PB) design (i.e., some block sizes are not equal) is newly provided. Existence problems on PB designs are far from the complete solution. By use of the necessary condition derived here, we can also show a non-existence result of the affine resolvable L2 design which is left as only one unknown among L2 designs.

Highlights

A pairwise balanced (PB) design with parameters v, b, r, K, λ is a collection of b subsets of a set of v treatments such that the size of each block is an element of a set K, each treatment occurs in exactly r different blocks and any two treatments occur in exactly λ blocks
The existence of affine resolvable group divisible (GD) designs and affine resolvable L2 designs with parameters v ≤ 100 and r, k ≤ 20 was mostly clarified by Kadowaki and Kageyama in 2009 and 2012
By use of the necessary condition derived here, we can show a non-existence result of the affine resolvable L2 design which is left as only one unknown among L2 designs

Summary

Introduction

A PB design with parameters v, b, r, K, λ is a collection of b subsets (blocks) of a set of v treatments such that the size of each block is an element of a set K, each treatment occurs in exactly r different blocks and any two treatments occur in exactly λ blocks. A block design is said to be resolvable if its b blocks can be grouped into r resolution sets of b/r blocks each such that in each resolution set any treatment occurs exactly once. By use of the necessary condition, a non-existence result of an affine resolvable L2 design is newly shown. This L2 design has been left as only one unknown case in Kadowaki and Kageyama [8]

Statements

Concluding Remark