The bigraphical t-wise balanced designs of index two

✍ Scribed by Lisa M. Weiss; Donald L. Kreher

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 998 KB
• Volume: 3
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.3180030309

✦ Synopsis

In this article we study the group S, X S, acting on the mn ordered pairs and classify all t-wise balanced designs of index 2 that have such an automorphism group. o 1995 John Wiley & Sons, Inc.

1. Introduction

A t-wise balanced design (tBD) of type t -( v , K , A ) is a pair ( X , 3 ) where X is a v-element set of points and 3 is a family of subsets of X called blocks such that (1) 3( is a set of positive integers strictly between t and v ; ( 2 ) the size of every block is in K ; and (3) every t-element subset of X is contained in exactly A blocks. Note that X is not a block. If 3 contains all k-element subsets of X for some k, then ( X , 3) is said to be a trivial design. If we give the parameters of a specific tBD, then we will choose a minimal 3(. If 1 x 1 = 1, then the tBD is called a t -( v , k, A) design, where K = {k}. The index of a tBD is A. All of the designs in this article have index 2. For some interesting results and conjectures on t-wise balanced designs the reader is directed to 151.

A fundamental question in the theory of combinatorial designs is to ask what are the designs that can be obtained with a particular type of automorphism group? In this article we study the group S , X S , acting on the set of m . n ordered pairs and classify all tBDs of index 2 that have such an automorphism group. It is natural to think of the m . n ordered pairs as the edges of K,,n and this motivates the following definition: