---

---

	Extracts from reviews.
	About the book.
	Table of orthogonal arrays of strength 2 with up to 100 runs (in a separate file).
	Tables of explicit examples of orthogonal arrays (in a separate file).
	Tables of explicit examples of Hadamard matrices (in a separate file).
	Updates and corrections to the book
	Further references dealing with orthogonal arrays
	How to order the book
	Table of Contents of Book
	The authors

Extracts from Reviews.

• The literature on orthogonal arrays and related structures is vast and this book attempts to put most of the available material in one place. The authors have done a splendid job of presenting the vast and scattered literature in the form of a book and thus deserve the congratulations of the academic community. ... I have enjoyed reading this book and find it very useful. I would strongly recommend this book to anyone even remotely interested in the combinatorics of orthogonal arrays.
  - Aloke Dey, Journal of Statistical Planning and Inference, 87 (2000), 371-373.

• ... it is a fascinating book for mathematicians. There is a feast of information and detail. ... it is a good book for the mathematically inclined statistician, who is interested in experimental design, to have on the bookshelf.
  - P.W.M. John, Short Book Reviews of the International Statistical Review, 20 (2000), 9-10.

• ... I like the statement of research problems throughout (...) many were admirably focused.
  - Deborah J. Street, Journal of the American Statistical Association, 95 (2000), 677-678.

• From the review by Dieter Jungnickel in Math Reviews, #2000h:05042:

  ... The book is well written and nice to read. It contains a wealth of concrete examples, many exercises (collected in problem sections at the end of each chapter), some research problems and a generally quite thorough discussion of the available literature. I can recommend it to anybody interested in discrete mathematics, in particular designs and codes, or in design of experiments.

About the book (an extract from the Preface).

Who should read this book? 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. Anyone
interested in one of the most fascinating areas of discrete mathematics,
connected to statistics and coding theory, with applications
to computer science and cryptography. This is the first book on the
subject since its introduction more than fifty years ago, and 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 exercises
and research problems. Most of the arrays that can be obtained by
the methods in this book are available electronically.

Your automobile lasts longer today because of orthogonal arrays ["The new mantra: MVT", Forbes, Mar. 11, 1996, pp. 114-118.]

The mathematical theory is extremely beautiful: orthogonal arrays are related to combinatorics, finite fields, geometry and error-correcting codes. The definition of an orthogonal array is simple and natural, and we know many elegant constructions - yet there are at least as many unsolved problems.

Here is an example of an orthogonal array of strength 2:

Pick any two columns, say the first and the last:

Each of the four possible rows we might see there,

does appear, and they all appear the same number of times (three times, in fact). That's the property that makes it an orthogonal array.

Only 0's and 1's appear in that array, but for use in statistics

in the first column might be replaced by

and in the second column by

and so on. Or

etc., depending on the application.

Since only 0's and 1's appear, this is called a 2-level array. There are 11 columns, which means we can vary the levels of up to 11 different variables, and 12 rows, which means we are going to bake 12 different cakes, or produce 12 different samples of the alloy. In short, we call this array an OA(12,11,2,2) The first "2" indicates the number of levels, and the second "2" the strength, which is the number of columns where we are guaranteed to see all the possibilities an equal number of times. In an orthogonal array of strength 3 (with two levels), in any three columns we would see each of the eight possibilities

As already mentioned, the main applications of orthogonal arrays are in planning experiments. The rows of the array represent the experiments or tests to be performed - cakes to be baked, samples of alloy to be produced, integrated circuits to be etched, test plots of crops to be grown, and so on.

The columns of the orthogonal array correspond to the different variables whose effects are being analyzed. The entries in the array specify the levels at which the variables are to be applied. If a row of the orthogonal array reads

this could mean that in that test the first, second, fourth variables (where the 1's occur) are to be set at their "high" levels, and the third, fifth, sixth variables (where the 0's occur) at their "low" levels.

By basing the experiment on an orthogonal array of strength t we ensure that all possible combinations of up to t of the variables occur together equally often.

The aim here is to investigate not only the effects of the individual variables (or factors) on the outcome, but also how the variables interact. Obviously, even with a moderate number of factors and a small number of levels for each factor, the number of possible level combinations for the factors increases rapidly. It may therefore not be feasible to make even one observation at each of the level combinations. In such cases observations are made at only some of the level combinations, and the purpose of the orthogonal array is to specify which level combinations are to be used. Such experiments are called "fractional factorial" experiments. While there are nowadays other applications of orthogonal arrays in statistics (for example in computer experiments and survey sampling), the principal application is in the selection of level combinations for fractional factorial experiments.

Since the rows of an orthogonal array represent runs (or tests or samples) - which require money, time, and other resources - there are always practical constraints on the number of rows that can be used in an experiment. Finding the smallest possible number of rows is a problem of eminent importance. On the other hand, for a given number of runs we may want to know the largest number of columns that can be used in an orthogonal array, since this will tell us how many variables can be studied. We also want the strength to be large, though in many real-life applications this is set at 2, 3 or 4.

Then the main questions we ask are:

• for which values of the numbers of rows, columns, strength and levels does an orthogonal array exist?
• how can we construct the array, if it exists?

How to order the book:

Please contact Springer-Verlag directly. The ISBN number is 0-387-98766-5, and the publication date was June 1999.

The Authors:

Professor A.S. Hedayat
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
851 South Morgan Street
Chicago, IL 60607-7045 USA
Home page
Email: hedayat(AT)uic.edu

Dr. N.J.A. Sloane
Information Sciences Research Center
AT&T Shannon Labs
180 Park Avenue
Florham Park, NJ 07932-0971 USA
Home page
Email: njasloane@gmail.com

Professor John Stufken
Department of Statistics
Snedecor Hall
Iowa State University
Ames, IA 50011 USA
Current address:
Professor and Head of Statistics,
University of Georgia,
204A Statistics Buikding,
Athens GA, USA
Home page
Email: jstufken(AT)uga.edu

Updates and Corrections:

• On page xiii, Professor Hedayat's home page is now www.uic.edu/~hedayat/.
• On page 27: concerning (2.17) of Theorem 2.23, J. Quistorff points out that W. Heise and P. Quattrocchi (Informations- und Codierungstheorie, 3rd edition, Springer, 1995), establsh a stronger condition ("Satz 10" on pages 325-327), namely:

  k<=s+t-3 if s is even, k>=3 and (s-2)*s*(s+1)*...*(s+t-4) is not congruent to 0 mod (t-1)!.

  For instance k<=39 if t=6 and s=36.

• On page 53, Table 3.25, in the 7-th row, labeled 1 0 2, the last 9 entries should be changed from 1 0 2 1 0 2 1 0 2 to 1 2 0 1 2 0 1 2 0

• On page 73 we should not constrain B₀ to be zero, but simply require that it be positive (compare Sloane and Stufken, 1996). So replace lines 7 (part of (4.21)) and -6 (part of (4.26)) by
  B_i   >=   0,   0 <= i <= k

• On page 81, line 9: MacWilliams and Sloane (1977) [not (1997)].

• On page 119, after Definition 6.14, we could have pointed out that an a-resolvable design is automatically a'-resolvable for all a' >= a with a' dividing N/s.

• Page 165, Table 7.37: We have worked out that there are 7570 nonisomorphic OA(28,27,2,2)'s. Thus the sequence in the first row of that table now reads
  1    1    1    5    3    130    7570

  (This is sequence A048885 of the On-Line Encyclopedia of Integer Sequences.)

• On page 193, lines 2 and 3, change `the next "Fermat's Last Theorem"' to `the "Next Fermat Problem"'.
• On page 215, in Example 9.28, the bottom right-hand entry of the third matrix should be 1 not 0.

• On page 227, first sentence: We probably should have defined "translate".

  If A is an orthogonal array over an additive group (e.g. a field) then if we add a fixed vector u to every run we get a new orthogonal array with the same parameters as A; this is called a translate of A.

• Page 243: We apologize for misspelling Malcolm Greig's name on line 1! The same error appears in the index on page 407.

• Page 300, middle of page, displayed equations for SN_1 and SN_2: please change "m" to "N_2" in four places.

• Page 304, middle: Sloane (1973) should be Sloane (1993).
• Page 321: The following updates should be made to Table 12.3.
      In column t=5, for k=7 and 8 change 2-4 to 2^{Edel (1996)}.
      In column t=5, for k=28 through 32 change 18-256^EB to 18-128^{Edel (1996)}.
      In column t=7, for k=11 change 4-16 to 4-8^{Edel (1996)}.
  
• Page 330, in the section on OA's with 36 runs, interpolate the following entries:

       2¹³ 6²   found by H. Xu (2002)
       2¹⁰ 3⁸ 6¹   found by Y. Zhang et al. (2001)
       2¹⁰ 3¹ 6²   found by H. Xu (2002)
       2⁹ 3⁴ 6²   found by Y. Zhang et al. (2001)
       2⁸ 6³   found by Y. Zhang et al. (2001)
       2⁴ 3¹ 6³   found by H. Xu (2002)
       2³ 3⁹ 6¹   found by Y. Zhang et al. (2001)
       2³ 3² 6³   found by H. Xu (2002)
       2² 3⁵ 6²   found by Y. Zhang et al. (2001)
       2¹ 3³ 6³   found by H. Xu (2002)
       2¹ 3¹ 6³   found by Y. Zhang et al. (2001)
       2¹ 6³   found by Warren Kuhfeld (2002)

  - see the on-line version for the most up-to-date information.

• On page 332, we now know exactly which 64-run orthogonal arrays exist. For this and much more about the lattices of parameter sets of orthogonal arrays introduced on page 335 see E. M. Rains, N. J. A. Sloane and J. Stufken, The Lattice of N-Run Orthogonal Arrays [Abstract, ps, pdf]

  In particular, the following entries should be added to Table 12.7 on page 332, in the section dealing with arrays with 64 runs:

          2⁵ 4¹⁷ 8¹
          2⁵ 4¹⁰ 8⁴
          4¹⁴ 8³
          4⁷ 8⁶ .
  

  The entry 4¹⁵ 8² should be labeled with the "SS" (or section) symbol, since it is now dominated by one of the new arrays.

• Also on page 332, many new 72-run OAs have been discovered by Y. Zhang et al. (2001). See the on-line version of this table for the most up-to-date information.
• Pages 329-334: Numerous other improvements have been made to Table 12.7 (parameter sets of known orthogonal arrays with at most 100 runs): see the on-line version for the most up-to-date information.
• Page 336, line -4: 'know' should be 'known'.
• Page 338, line -11, change OA(rs, s^cr¹, s) to OA(rs, s^cr¹, 2).

• Page 365, line 7:
  Bierbrauer, J. (1993). Construction of orthogonal arrays. J. Statist. Plann. Infer. 56, 207-221.
  should be
  Bierbrauer, J. (1996). Construction of orthogonal arrays. J. Statist. Plann. Infer. 56, 39-47.
  [Thanks to Martin Roetteler for noticing this.]

• Page 371, insert the following just before the Cochran reference:

  A. T. Clayman, K. M. Lawrence, G. L. Mullen, H. Niederreiter and N. J. A. Sloane (1999). Updated Tables of Parameters of (T,M,S)-Nets, J. Combinatorial Designs, 7, pages 381-393. For the full tables see the J. Combinatorial Designs web site.

• Page 384, Laywine, Mullin and Suchower (1990) should be Laywine, C. F., Mullen, G. L., and Suchower, S. J. (1990).
• Page 394: Rosenbaum (1997) has now appeared:
  Rosenbaum, P. R. (1999). Blocking in compound dispersion experiments. Technometrics, Vol. 41, 125-134.
• Page 397: Sloane (1973) should be Sloane (1993).
• Page 405: Zhang, Lu and Pang (1999) has now appeared:
  Y. S. Zhang, Y. Lu and S. Pang (1999). Orthogonal arrays obtained by orthogonal decomposition of projection matrices. Statistica Sinica 9, 595-604.
• Page 407, add
  Djokovic, 147, 373
• Page 409, add
  Mullen, 193, 389
  Sawade, 147, 395
• Page 409, change
  Sloane, 193, 245
  to
  Sloane, 193, 245, 304, 370, 386, 397

Selected new references dealing with orthogonal arrays (or earlier references that were omitted)

1. J. Bierbrauer (2002). Direct construction of additive codes. J. Combin. Designs 10, 207-216.
2. Brouwer, Andrie E., Cohen, Arjeh M. and Nguyen, V. M. (2006). Orthogonal arrays of strength 3 and small run sizes. Journal of Statistical Planning and Inference 136, pp. 3268-3280
3. Chai, Feng Shun, Mukerjee, Rahul and Suen, Chung-yi, Further results on orthogonal arrays plus one run plans. J. Statist. Plann. Inference, 106 (2002), 287-301.
4. Chateauneuf, M. and Kreher, D.L. (2002). On the state of strength-three covering arrays. J. Combin. Designs 10, 217-238.
5. D. De Cock and J. Stufken (2000). On Finding Mixed Orthogonal Arrays of Strength 2 With Many 2-Level Factors. Statistics and Probability Letters, 50, 383-388.
6. A. Dey and R. Mukerjee (1998). Techniques for constructing asymmetric orthogonal arrays. J. Combin. Inform. System Sc. 23, 351-366.
7. Wiebke S. Diestelkamp, The decomposability of simple orthogonal arrays on 3 symbols having t+1 rows and strength t, Journal of Combinatorial Designs, Vol. 8 (2000), 442-458. [Author's home page].

8. Wiebke S. Diestelkamp, Parameter inequalities for orthogonal arrays with mixed levels. Designs, Codes and Cryptography, Vol. 33 (2004), 187-197. [Author's home page].

9. Wiebke S. Diestelkamp and Jay H. Beder, On the decomposition of orthogonal arrays, Utilitas Mathematica, Vol. 61 (2002), 65-86. [Diestelkamp's home page].

10. Y. Edel (1996), Eine Verallgemeinerung von BCH-Codes (Dissertation).

11. Longcheen Huwang, C. F. J. Wu, and C. H. Chen (2002). The idle column method: design construction, properties and comparisons. Technometrics, 44, 347-355.

12. S. Kageyama and K. Urata, (2000). Bounds on orthogonal arrays, Bulletin Fac. Educ. Hiroshima Univ. , Part II, 49, 25-32.

13. Kamali, F., Kharaghani, H., and Khosrovshahi, G.B. (2003). Some Bush-type Hadamard matrices. J. Statistical Planning and Inference, Vol. 113, pp. 375-384.
14. A. V. Khalyavin, Estimates of the capacity of orthogonal arrays of large strength, Moscow University Mathematics Bulletin, Volume 65, 2010, Number 3, 130-131.
15. V. C. Mavron: A Construction Method for Complete Sets of Mutually Orthogonal Frequency Squares, Electronic J. Combin., #N5, 2000.

16. Klaus Metsch, Improvement of Bruck's completion theorem. Des. Codes Cryptogr. 1 (1991), no. 2, 99--116.

17. Mishima, Miwako, Jimbo, Masakazu, Shirakura, Teruhiro (2000). On the optimality of orthogonal arrays in case of correlated errors. J. Statist. Plann. Inference 88, 319-338.

18. Max D. Morris, Leslie M. Moore and Michael D. McKay. Using Orthogoanl Arrays in the Sensitivity Analysis of Computer Models, Technomtrics, Vo. 50, No. 2, 2008, pp. 205-215.
19. E. M. Rains, N. J. A. Sloane and J. Stufken, The Lattice of N-Run Orthogonal Arrays, J. Statist. Plann. Inference, 102 (2002), 477-500 [Abstract, ps, pdf].

20. Eric D. Schoen, Pieter T. Eendebak, Man V.M. Nguyen (2009). Complete Enumeration of Pure-level and Mixed-level Orthogonal Arrays. Journal of Combinatorial Designs, to appear.
21. K. Sinha, V. Dhar, G. M. Saha and S. Kageyama (2002). Balanced arrays of strength two from block designs. J. Combinatorial Designs, 10, 303-312.
22. Street, Deborah J. (1998). Orthogonal arrays as designed experiments. Bull. Inst. Combin. Appl. 24, 81--101.

23. Suen, Chung-yi (2005). Orthogonal arrays of strength three and size 2^r. Statist. Sinica 15, 731-749

24. Suen, C.-Y, Das, A., and Dey, A. (2001). On the construction of asymmetric orthogonal arrys. Statist. Sinica, 11, 241-260.

25. Suen, C.-Y. and Dey, A. (2003). Construction of asymmetric orthogonal arrays through finite geometries. J. Statist. Plann. Inference, 115, 623-635.

26. Suen, Chung-yi, Das, Ashish, and Dey, Aloke (2001). On the construction of asymmetrical orthogonal arrays. Statistica Sinica 11, 241-260.
27. Yu. Tarannikov, P. Korolev, A. Botev. Autocorrelation coefficients and correlation immunity of Boolean functions, Proceedings of Asiacrypt 2001, Gold Coast, Australia, December 9--13, 2001, Lecture Notes in Computer Science, V. 2248, pp. 460--479, Springer-Verlag, 2001. [Bound on OA's of large strength.]

28. V. D. Tonchev (2003), Affine designs and linear orthogonal arrays, preprint.

29. H. Xu (2002), An Algorithm for Constructing Orthogonal and Nearly-Orthogonal Arrays with Mixed Levels and Small Runs, Technometrics, 44, 356-368.

30. S. Yamammoto, Y. Hyodo, M. Mitsuoka, H. Yumiba and T. Takahashi (1998). Algorithm for the construction and classification of orthogonal arrays and its feasibility. J. Combin. Inform. System Sc. 23, 71-84.

31. Y. Zhang (2007), Orthogonal arrays obtained by repeating-column difference matrices, Discrete Math., 307, 246-261.

32. J.-Z. Zhang, Z.-S. You and Z.-L. Li (2000), Enumeration of binary orthogonal arrays of strength 1, Discrete Math., 239 191-198.

33. Y. Zhang, L. Duan, Y. Lu and Z. Zheng (2002), Construction of generalized Hadamard matrices D(r^m(r+1), r^m(r+1); p), Journal of Statistical Planning and Inference, 104, 239-258

34. Y. Zhang, S. Pang and Y. Wang, (2001), Orthogonal Arrays Obtained by Generalized Hadamard Product, Discrete Math., 238 151-170. Note however that there are errors in the 72-run OAs as printed in the journal. See the web page Library of orthogonal arrays for corrected versions of these OAs.

Table of Contents of Book:

Preface         vii

Foreword by C. R. Rao         xv

List of symbols         xxiii

1 Introduction         1

• 1.1 Problems         8

2 Rao's Inequalities and Improvements         11

• 2.1 Introduction         11
• 2.2 Rao's Inequalities         12
• 2.3 Improvements on Rao's Bounds for Strength 2 and 3         17
• 2.4 Improvements on Rao's Bounds for Arrays of Index Unity         22
• 2.5 Orthogonal Arrays with Two Levels         27
• 2.6 Concluding Remarks         32
• 2.7 Notes on Chapter 2         33
• 2.8 Problems         33

3 Orthogonal Arrays and Galois Fields         37

• 3.1 Introduction         37
• 3.2 Bush's Construction         38
• 3.3 Addelman and Kempthorne's Construction         44
• 3.4 The Rao-Hamming Construction         49
• 3.5 Conditions for a Matrix to be an Orthogonal Array         54
• 3.6 Concluding Remarks         56
• 3.7 Problems         56

4 Orthogonal Arrays and Error-Correcting Codes         61

• 4.1 An Introduction to Error-Correcting Codes         61
• 4.2 Linear Codes         63
• 4.3 Linear Codes and Linear Orthogonal Arrays         65
• 4.4 Weight Enumerators and Delsarte's Theorem         67
• 4.5 The Linear Programming Bound         72
• 4.6 Concluding Remarks         82
• 4.7 Notes on Chapter 4         82
• 4.8 Problems         85

5 Construction of Orthogonal Arrays from Codes         87

• 5.1 Extending a Code by Adding More Coordinates         87
• 5.2 Cyclic Codes         88
• 5.3 The Rao-Hamming Construction Revisited         91
• 5.4 BCH Codes         93
• 5.5 Reed-Solomon Codes         95
• 5.6 MDS Codes and Orthogonal Arrays of Index Unity         96
• 5.7 Quadratic Residue and Golay Codes         99
• 5.8 Reed-Muller Codes         99
• 5.9 Codes from Finite Geometries         101
• 5.10 Nordstrom-Robinson and Related Codes         102
• 5.11 Examples of Binary Codes and Orthogonal Arrays         103
• 5.12 Examples of Ternary Codes and Orthogonal Arrays         105
• 5.13 Examples of Quaternary Codes and Orthogonal Arrays         106
• 5.14 Notes on Chapter 5         108
• 5.15 Problems         109

6 Orthogonal Arrays and Difference Schemes         113

• 6.1 Difference Schemes         113
• 6.2 Orthogonal Arrays Via Difference Schemes         118
• 6.3 Bose and Bush's Recursive Construction         123
• 6.4 Difference Schemes of Index 2         127
• 6.5 Generalizations and Variations         132
• 6.6 Concluding Remarks         138
• 6.7 Notes on Chapter 6         140
• 6.8 Problems         141

7 Orthogonal Arrays and Hadamard Matrices         145

• 7.1 Introduction         145
• 7.2 Basic Properties of Hadamard Matrices         146
• 7.3 The Connection Between Hadamard Matrices and Orthogonal Arrays         148
• 7.4 Constructions for Hadamard Matrices         148
• 7.5 Hadamard Matrices of Orders up to 200         155
• 7.6 Notes on Chapter 7         163
• 7.7 Problems         165

8 Orthogonal Arrays and Latin Squares         167

• 8.1 Latin Squares and Orthogonal Latin Squares         168
• 8.2 Frequency Squares and Orthogonal Frequency Squares         173
• 8.3 Orthogonal Arrays from Pairwise Orthogonal Latin Squares         183
• 8.4 Concluding Remarks         191
• 8.5 Problems         196

9 Mixed Orthogonal Arrays         199

• 9.1 Introduction         199
• 9.2 The Rao Inequalities for Mixed Orthogonal Arrays         201
• 9.3 Constructing Mixed Orthogonal Arrays         203
• 9.4 Further Constructions         211
• 9.5 Notes on Chapter 9         219
• 9.6 Problems         220

10 Further Constructions and Related Structures         223

• 10.1 Constructions Inspired by Coding Theory         223
• 10.2 The Juxtaposition Construction         224
• 10.3 The (u,u+v) Construction         225
• 10.4 Construction X4         226
• 10.5 Orthogonal Arrays from Union of Translates of a Linear Code         228
• 10.6 Bounds on Large Orthogonal Arrays         228
• 10.7 Compound Orthogonal Arrays         230
• 10.8 Orthogonal Multi-Arrays         236
• 10.9 Transversal Designs, Resilient Functions and Nets         242
• 10.10 Schematic Orthogonal Arrays         245
• 10.11 Problems         246

11 Statistical Application of Orthogonal Arrays         247

• 11.1 Factorial Experiments         247
• 11.2 Notation and Terminology         249
• 11.3 Factorial Effects         251
• 11.4 Analysis of Experiments Based on Orthogonal Arrays         258
• 11.5 Two-Level Fractional Factorials with a Defining Relation         272
• 11.6 Blocking for a 2^(k-n) Fractional Factorial         282
• 11.7 Orthogonal Main-Effects Plans and Orthogonal Arrays         288
• 11.8 Robust Design         298
• 11.9 Other Types of Designs         302
• 11.10 Notes on Chapter 11         305
• 11.11 Problems         308

12 Tables of Orthogonal Arrays         317

• 12.1 Tables of Orthogonal Arrays of Minimal Index         317
• 12.2 Description of Tables 12.1--12.3         318
• 12.3 Index Tables         324
• 12.4 If No Suitable Orthogonal Array Is Available         336
• 12.5 Connections with Other Structures         338
• 12.6 Other Tables         339

Appendix: Galois Fields         341

Bibliography         363