Existence results for near resolvable designs

✍ Scribed by Steven Furino

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 572 KB
• Volume: 3
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.3180030204

✦ Synopsis

A near resolvable design, NRB(v, k), is a balanced incomplete block design whose block set can be partitioned into v classes such that each class contains every point of the design but one, and each point is missing from exactly one class. The necessary conditions for the existence of near resolvable designs are v = 1 mod k and A = k -1. These necessary conditions have been shown to be sufficient for k E {2,3,4} and almost always sufficient for k E (56). We are able to show that there exists an integer n&) so that NRB(v,k) exist for all v > no@) and v = 1 mod k. Using some new direct constructions we show that there are many k for which it is easy to compute an explicit bound on no(&). These direct constructions also allow us to build previously unknown NRB(v, 5 ) and NRB(v, 6). 0 1995 John Wiley 81 Sons, he.

1. Introduction

This first section contains definitions and examples as well as a brief review of current results. The next two sections contain direct and recursive constructions. The main results, bounds on the existence of NRB(v, k ) for various values of k, are contained in the fourth section. The article closes with an examination of some of the applications of near resolvable designs to other problems in design theory.

We assume that the reader is familiar with the rudiments of design theory and that works such as [1,2] are available for reference. We begin with some definitions.

A group divisible design (GDD) with index A is a triple (V, G, 3) where 1. Ir is a finite set of points, 2. G is a set of subsets of Ir, called groups, which partition Ir,