Orthogonal Arrays

by Hedayat, A.; Stufken, John; Sloane, N. J. A.

• ISBN13: 9780387987668
• ISBN10: 0387987665
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: Springer Verlag

Summary

This is the first book on the subject since its introduction more than fifty years ago, and it can be used as a graduate text or as a reference work. It features all of the key results, many very useful tables, and a large number of research problems. The book will be of interest to those interested in one of the most fascinating areas of discrete mathematics, connected to statistics and coding theory, with applications to computer science and cryptography. It will be useful for anyone who is running experiments, whether in a chemistry lab or a manufacturing plant (trying to make those alloys stronger), or in agricultural or medical research. Sam Hedayat is Professor of Statistics and Senior Scholar in the Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago. Neil J.A. Sloane is with AT&T Bell Labs (now AT&T Labs). John Stufken is Professor Statistics at Iowa State University.

Table of Contents

Preface	vii
Foreword	xv
C. R. Rao
List of Symbols	xxiii
Introduction	1	(10)
Problems	8	(3)
Rao's Inequalities and Improvements	11	(26)
Introduction	11	(1)
Rao's Inequalities	12	(5)
Improvements on Rao's Bounds for Strength 2 and 3	17	(5)
Improvements on Rao's Bounds for Arrays of Index Unity	22	(5)
Orthogonal Arrays with Two Levels	27	(5)
Concluding Remarks	32	(1)
Notes on Chapter 2	33	(1)
Problems	33	(4)
Orthogonal Arrays and Galois Fields	37	(24)
Introduction	37	(1)
Bush's Construction	38	(6)
Addelman and Kempthorne's Construction	44	(5)
The Rao-Hamming Construction	49	(5)
Conditions for a Matrix	54	(2)
Concluding Remarks	56	(1)
Problems	56	(5)
Orthogonal Arrays and Error-Correcting Codes	61	(26)
An Introduction to Error-Correcting Codes	61	(2)
Linear Codes	63	(2)
Linear Codes and Linear Orthogonal Arrays	65	(2)
Weight Enumerators and Delsarte's Theorem	67	(5)
The Linear Programming Bound	72	(10)
Concluding Remarks	82	(1)
Notes on Chapter 4	82	(3)
Problems	85	(2)
Construction of Orthogonal Arrays from Codes	87	(26)
Extending a Code by Adding More Coordinates	87	(1)
Cyclic Codes	88	(3)
The Rao-Hamming Construction Revisited	91	(2)
BCH Codes	93	(2)
Reed-Solomon Codes	95	(1)
MDS Codes and Orthogonal Arrays of Index Unity	96	(3)
Quadratic Residue and Golay Codes	99	(1)
Reed-Muller Codes	99	(2)
Codes from Finite Geometries	101	(1)
Nordstrom-Robinson and Related Codes	102	(1)
Examples of Binary Codes and Orthogonal Arrays	103	(2)
Examples of Ternary Codes and Orthogonal Arrays	105	(1)
Examples of Quaternary Codes and Orthogonal Arrays	106	(2)
Notes on Chapter 5	108	(1)
Problems	109	(4)
Orthogonal Arrays and Difference Schemes	113	(32)
Difference Schemes	113	(5)
Orthogonal Arrays Via Difference Schemes	118	(5)
Bose and Bush's Recursive Construction	123	(4)
Difference Schemes of Index 2	127	(5)
Generalizations and Variations	132	(6)
Concluding Remarks	138	(2)
Notes on Chapter 6	140	(1)
Problems	141	(4)
Orthogonal Arrays and Hadamard Matrices	145	(22)
Introduction	145	(1)
Basic Properties of Hadamard Matrices	146	(2)
The Connection Between Hadamard Matrices and Orthogonal Arrays	148	(1)
Constructions for Hadamard Matrices	148	(7)
Hadamard Matrices of Orders up to 200	155	(8)
Notes on Chapter 7	163	(2)
Problems	165	(2)
Orthogonal Arrays and Latin Squares	167	(32)
Latin Squares and Orthogonal Latin Squares	168	(5)
Frequency Squares and Orthogonal Frequency Squares	173	(10)
Orthogonal Arrays from Pairwise Orthogonal Latin Squares	183	(8)
Concluding Remarks	191	(5)
Problems	196	(3)
Mixed Orthogonal Arrays	199	(24)
Introduction	199	(2)
The Rao Inequalities for Mixed Orthogonal Arrays	201	(2)
Constructing Mixed Orthogonal Arrays	203	(8)
Further Constructions	211	(8)
Notes on Chapter 9	219	(1)
Problems	220	(3)
Further Constructions and Related Structures	223	(24)
Constructions Inspired by Coding Theory	223	(1)
The Juxtaposition Construction	224	(1)
The (u,u + v) Construction	225	(1)
Construction X4	226	(2)
Orthogonal Arrays from Union of Translates of a Linear Code	228	(1)
Bounds on Large Orthogonal Arrays	228	(2)
Compound Orthogonal Arrays	230	(6)
Orthogonal Multi-Arrays	236	(6)
Transversal Designs, Resilient Functions and Nets	242	(3)
Schematic Orthogonal Arrays	245	(1)
Problems	246	(1)
Statistical Application of Orthogonal Arrays	247	(70)
Factorial Experiments	247	(2)
Notation and Terminology	249	(2)
Factorial Effects	251	(7)
Analysis of Experiments Based on Orthogonal Arrays	258	(14)
Two-Level Fractional Factorials with a Defining Relation	272	(10)
Blocking for a 2k-n Fractional Factorial	282	(6)
Orthogonal Main-Effects Plans and Orthogonal Arrays	288	(10)
Robust Design	298	(4)
Other Types of Designs	302	(3)
Notes on Chapter 11	305	(3)
Problems	308	(9)
Tables of Orthogonal Arrays	317	(24)
Tables of Orthogonal Arrays of Minimal Index	317	(1)
Description of Tables 12.1-12.3	318	(6)
Index Tables	324	(12)
If No Suitable Orthogonal Array Is Available	336	(2)
Connections with Other Structures	338	(1)
Other Tables	339	(2)
Appendix A: Galois Fields	341	(22)
A.1 Definition of a Field	341	(3)
A.2 The Construction of Galois Fields	344	(7)
A.3 The Existence of Galois Fields	351	(6)
A.4 Quadratic Residues in Galois Fields	357	(2)
A.5 Problems	359	(4)
Bibliography	363	(43)
Author Index	406	(5)
Subject Index	411