# 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:

*a*,*b*∈*F*⇒*a*+*b*∈*F*,*a*·*b*∈*F*.*a*+ (*b*+*c*) = (*a*+*b*) +*c*,*a*· (*b*·*c*) = (*a*·*b*) ·*c*.-
*a*+*b*=*b*+*a*,*a*·*b*=*b*·*a*. - ∃0, such that
*a*+ 0 =*a*, ∃1, such that*a*· 1 =*a*. - ∃ (-
*a*), such that*a*+ (-*a*) = 0. - ∀
*a*≠ 0, ∃ (*a*^{– 1}), such that*a*· (*a*^{- 1}) = 1. -
*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, ...

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