Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- I Basic results from designs
- II Strongly regular graphs and partial geometries
- III Basic results on quasi-symmetric designs
- IV Some configurations related to strongly regular graphs and quasi-symmetric designs
- V Strongly regular graphs with strongly regular decompositions
- VI The Witt designs
- VII Extensions of symmetric designs
- VIII Quasi-symmetric 2-designs
- IX Towards a classification of quasi-symmetric 3-designs
- X Codes and quasi-symmetric designs
- References
- Index
III - Basic results on quasi-symmetric designs
Published online by Cambridge University Press: 05 May 2010
- Frontmatter
- Contents
- Preface
- Acknowledgments
- I Basic results from designs
- II Strongly regular graphs and partial geometries
- III Basic results on quasi-symmetric designs
- IV Some configurations related to strongly regular graphs and quasi-symmetric designs
- V Strongly regular graphs with strongly regular decompositions
- VI The Witt designs
- VII Extensions of symmetric designs
- VIII Quasi-symmetric 2-designs
- IX Towards a classification of quasi-symmetric 3-designs
- X Codes and quasi-symmetric designs
- References
- Index
Summary
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.
- Type
- Chapter
- Information
- Quasi-symmetric Designs , pp. 34 - 48Publisher: Cambridge University PressPrint publication year: 1991