Repeated Measurements and Cross-Over Designs

Chapter 9Some Constructions of Cross-Over Designs

9.1 INTRODUCTION

The construction and looking into the existence or nonexistence of designs is as important as providing the analysis for a given class of designs. Finite geometries, methods of differences, patchwork methods, and recursive methods are commonly used procedures for constructing designs. In Section 9.2, a brief review of Galois fields (GF) is provided, and in subsequent sections, some construction methods of the previously introduced designs are considered.

9.2 GALOIS FIELDS

Given a set of elements, F, a system is said to be a field, if the following conditions are satisfied:

1. a, b ∈ F ⇒ a + b ∈ F,  a · b ∈ F.
2. a + (b + c) = (a + b) + c,  a · (b · c) = (a · b) · c.
3. a + b = b + a,  a · b = b · a.
4. ∃0, such that a + 0 = a,   ∃1, such that a · 1 = a.
5. ∃ (-a), such that a + (-a) = 0.
6. ∀ a ≠ 0,   ∃ (a^– 1), such that a · (a^- 1) = 1.
7. a · (b + c) = a · b + a · c.

The number of elements, s, in a field is finite if s is a prime or prime power. These elements are called Galois fields and are written as GF(s). The quantity a is said to be congruent to b modulus s and is written a ≡ b(mod  s), iff, a – b is divisible by s. If s is a prime p, the set of integers {0, 1, …, p - 1} is a field if these are multiplied or added in the usual manner and are reduced by mod p. For example, with s = 5, a field of 5 elements is {0, 1, 2, ...