Suppose D is a t-(v, k, λ) design with blocks B1, B2,…, Bb. The cardinality | Bj∩Bj |, i ≠ j, is called an intersection number of D. Assume that x1, x2,…, xs are the distinct intersection numbers of the design D. Specifying some of the xi's or the number s can sometimes provide very useful information about the design. For instance, any 2-design with exactly one intersection number must necessarily be symmetric. Any 2-design with exactly the two intersection numbers 0 and 1 must be a non-symmetric 2-(v, k, 1) design.
In this chapter, we discuss designs which are in a sense “close” to symmetric designs. These are t-(v, k, λ) designs with exactly two intersection numbers. Such designs are called quasi-symmetric. We believe this concept goes back to S.S. Shrikhande who considered duals of designs with λ = 1. We let x, y stand for the intersection numbers of a quasi-symmetric design with the standard convention that x < y.
Before proceeding further, we list below some well known examples of quasi-symmetric designs.
Example 3.1. Let D be a multiple of a symmetric 2-(v, k, λ) design. Then D is a quasi-symmetric 2-design with x = λ and y = k.
Example 3.2. Let D be a 2-(v, k, 1) design with b > v. Then obviously D is quasi-symmetric with x = 0 and y = 1.