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# 17: Designs

• • Joy Morris
• Professor (Mathematics) at University of Lethbridge
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• 17.1: Balanced Incomplete Block Designs (BIBD)
A regular design is a design in which every point appears in the same number of blocks, r. A uniform design is a design in which every block contains the same number of points, k. A balanced design is a design in which every pair of points appear together in the same number of blocks, λ. A balanced incomplete block design (BIBD) is a regular, uniform, balanced design that is not complete. So it is a (b,v,r,k,λ)-design with k < v.
• 17.2: Constructing Designs and Existence of Designs
There are a number of nice methods for constructing designs. We will discuss some of these methods in this section. For some of them, you must start with one design, and use it to create a different design.
• 17.3: Fisher’s Inequality
There is one more important inequality that is not at all obvious, but is necessary for the existence of a BIBD (v,k,λ). This is known as Fisher’s Inequality since it was proven by Fisher. The proof we will give is somewhat longer than the standard proof. This is because the standard proof uses linear algebra, which is not required background for this course.
• 17.4: Summary
This page contains the summary of the topics covered in Chapter 17.