Orthogonal Arrays

Frontmatter

Chapter 1. Introduction

Abstract

In the 1940’s, in a series of seminal papers (Rao, 1946a, 1947, 1949), C. R. Rao introduced certain combinatorial arrangements with applications to statistics. Although Rao (1946a) at first considered only a subclass of these arrangements, the entire class became quickly known by their current name of orthogonal arrays (or OA’s). Since their introduction, many prominent researchers have found a source of inspiration in this fascinating subject. Both statisticians and mathematicians can be credited with significant contributions to this field.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 2. Rao’s Inequalities and Improvements

Abstract

If the other parameters of an orthogonal array are specified, there is a limit on the number of possible factors, imposed by the defining conditions. We shall discuss these restrictions in this chapter and in Chapter 4. Section 2.2 presents the celebrated inequalities found by Rao (dy1947). Section 2.3 discusses improvements on Rao’s bounds for orthogonal arrays of strength two and three. Results on improvements for arrays of general strength are contained in Section 2.4, while Section 2.5 pays special attention to orthogonal arrays in which all the factors are at two levels.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 3. Orthogonal Arrays and Galois Fields

Abstract

A large number of techniques are known for constructing orthogonal arrays. This chapter, the first of several describing these techniques, discusses some constructions due to Bush (1952b), Addelman and Kempthorne (1961a), Rao (1946a, 1947, 1949) and Bose and Bush (1952), that have the common theme of using Galois fields and finite geometries. We also describe a number of basic properties of orthogonal arrays, including the important concept of linearity.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 4. Orthogonal Arrays and Error-Correcting Codes

Abstract

In this chapter we introduce error-correcting codes and discuss their connections with orthogonal arrays. The two subjects are very closely related, since we can use the codewords in an error-correcting code as the runs of an orthogonal array, or conversely we can regard the runs of an orthogonal array as forming a code.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 5. Construction of Orthogonal Arrays from Codes

Abstract

In this chapter we present some of the most important families of codes and the orthogonal arrays that are derived from them.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 6. Orthogonal Arrays and Difference Schemes

Abstract

In this chapter we introduce the concept of a difference scheme and some of its generalizations. Difference schemes were first defined by Bose and Bush (1952), and are a simple but powerful tool for the construction of orthogonal arrays of strength two.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 7. Orthogonal Arrays and Hadamard Matrices

Abstract

Hadamard matrices are square matrices of +1’s and – 1’s whose rows are orthogonal. The study of two-level orthogonal arrays of strengths 2 and 3 is essentially equivalent to the study of these matrices. They are also the most important examples of two-level difference schemes.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 8. Orthogonal Arrays and Latin Squares

Abstract

The subject of pairwise or mutually orthogonal Latin squares has fascinated researchers for many years. Although there are a number of intriguing results in this area, many open problems remain to which the answers seem as elusive as ever. The known results, however, are well documented, for example in the books by Dénes and Keedwell (1974, 1991) and Laywine and Mullen (1998), or the article by Jungnickel (1990).

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 9. Mixed Orthogonal Arrays

Abstract

In this chapter we investigate orthogonal arrays in which the various factors may have different numbers of levels — these are called mixed or asymmetrical orthogonal arrays.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 10. Further Constructions and Related Structures

Abstract

This chapter discusses a number of different topics that do not quite fit into any of the earlier chapters. These are

• Constructions for orthologonal arrays inspired by coding theory
• Bounds on the size of orthogonal arrays with many factors
• Compound orthogonal arrays
• Orthogonal multi-arrays
• Transversal designs, resilient functions and nets
• Othogonal arrays and association schemes

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 11. Statistical Application of Orthogonal Arrays

Abstract

Rao (1947) introduced orthogonal arrays because of their desirable statistical properties when used in “fractional factorial” experiments. Nowadays the main statistical application of orthogonal arrays, with mixed levels or otherwise, is still as fractional factorials, although other applications have been discovered. We will present the main application in considerable detail, while only giving key references for the other applications. Unless stated otherwise, throughout this chapter the term orthogonal array is to be interpreted as including mixed level arrays.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Chapter 12. Tables of Orthogonal Arrays

Abstract

This chapter contains several tables: (a) Tables showing the smallest possible index (and hence the smallest number of runs) in 2-, 3- and 4-level orthogonal arrays with at most 32 factors and strengths between 2 and 10. (b) Tables summarizing most of the arrays constructed in this book, including a table of both mixed-and fixed-level orthogonal arrays of strength 2 with up to 100 runs. (c) A table summarizing the connections between orthogonal arrays and other combinatorial structures. We also discuss what can be done if the orthogonal array you want does not exist, or is not presently known to exist, or exists but is too large for your application.

A. S. Hedayat, N. J. A. Sloane, John Stufken

Backmatter