Orthogonal Arrays: Theory and Applications

by

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






This is the web site for our book, which was published by Springer-Verlag, New York in 
1999





  
  

     
    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.





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. 
  
Orthogonal arrays are beautiful and useful. They are 
essential in statistics and they are used in computer science and cryptography. 
In statistics they are primarily used in designing experiments, which simply 
means that they are immensely important in all areas of human investigation: for 
example in medicine, agriculture and manufacturing. 

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: 






  
  

    0 
    0 
    0 
    0 
    0 
    0 
    0 
    0 
    0 
    0 
    0 

  

    1 
    1 
    1 
    0 
    1 
    1 
    0 
    1 
    0 
    0 
    0 

  

    0 
    1 
    1 
    1 
    0 
    1 
    1 
    0 
    1 
    0 
    0 

  

    0 
    0 
    1 
    1 
    1 
    0 
    1 
    1 
    0 
    1 
    0 

  

    0 
    0 
    0 
    1 
    1 
    1 
    0 
    1 
    1 
    0 
    1 

  

    1 
    0 
    0 
    0 
    1 
    1 
    1 
    0 
    1 
    1 
    0 

  

    0 
    1 
    0 
    0 
    0 
    1 
    1 
    1 
    0 
    1 
    1 

  

    1 
    0 
    1 
    0 
    0 
    0 
    1 
    1 
    1 
    0 
    1 

  

    1 
    1 
    0 
    1 
    0 
    0 
    0 
    1 
    1 
    1 
    0 

  

    0 
    1 
    1 
    0 
    1 
    0 
    0 
    0 
    1 
    1 
    1 

  

    1 
    0 
    1 
    1 
    0 
    1 
    0 
    0 
    0 
    1 
    1 

  

    1 
    1 
    0 
    1 
    1 
    0 
    1 
    0 
    0 
    0 
    1 


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






  
  

    0 
    0

  

    1 
    0 

  

    0 
    0 

  

    0 
    0 

  

    0 
    1 

  

    1 
    0 

  

    0 
    1 

  

    1 
    1 

  

    1 
    0 

  

    0 
    1 

  

    1 
    1 

  

    1 
    1


Each of the four possible rows we might see there, 

0 0,       0 1,       1 0,     
  1 1, 


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 



0       or       1 


in the first column might be replaced by 



"butter"       or       "margarine" , 



and in the second column by 



"sugar"       or       "no sugar" , 



and so on. Or 






  
  

    "slow cooling" 
    or 
    "fast cooling", 

  

    "catalyst" 
    or 
    "no catalyst", 


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 



000, 001, 010, 011, 100, 101, 110, 111 


equally often. (The formal definition is given in the first chapter.) 

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 



110100 ... 


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: 








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: 
njas(AT)research.att.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:












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. V. C. Mavron: A Construction Method for Complete Sets of Mutually 
  Orthogonal Frequency Squares, Electronic J. Combin., #N5, 2000. 
  
    
  
  15. Klaus Metsch, Improvement of Bruck's completion theorem. Des. Codes 
  Cryptogr. 1 (1991), no. 2, 99--116. 
  
    
  
  16. 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. 
  
    
  
  17. 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. 
  
    
  
  18. 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]. 
  
    
  
  19. 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. 
  
    
  
  20. 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. 
  
    
  
  21. Street, Deborah J. (1998). Orthogonal arrays as designed experiments. 
  Bull. Inst. Combin. Appl. 24, 81--101. 
  
    
  
  22. Suen, Chung-yi (2005). Orthogonal arrays of strength three and size 2^r. 
  Statist. Sinica 15, 731-749 
  
    
  
  23. Suen, C.-Y, Das, A., and Dey, A. (2001). On the construction of asymmetric 
  orthogonal arrys. Statist. Sinica, 11, 241-260. 
  
    
  
  24. Suen, C.-Y. and Dey, A. (2003). Construction of asymmetric orthogonal 
  arrays through finite geometries. J. Statist. Plann. Inference, 115, 623-635. 
  
    
  
  25. Suen, Chung-yi, Das, Ashish, and Dey, Aloke (2001). On the construction of 
  asymmetrical orthogonal arrays. Statistica Sinica 11, 241-260.
  
    
  
  26. 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.] 
  
    
  
  27. V. D. Tonchev (2003), Affine designs and linear orthogonal arrays, 
  preprint. 
  
    
  
  28. H. Xu (2002), An Algorithm for Constructing Orthogonal and 
  Nearly-Orthogonal Arrays with Mixed Levels and Small Runs, Technometrics, 44, 
  356-368. 
  
    
  
  29. 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. 
  
    
  
  30. Y. Zhang (2007), Orthogonal arrays obtained by repeating-column difference 
  matrices, Discrete Math., 307, 246-261. 
  
    
  
  31. J.-Z. Zhang, Z.-S. You and Z.-L. Li (2000), Enumeration of binary 
  orthogonal arrays of strength 1, Discrete Math., 239 191-198. 
  
    
  
  32. 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 
  
    
  
  33. 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 





2 Rao's Inequalities and Improvements         11 





3 Orthogonal Arrays and Galois Fields         37 





4 Orthogonal Arrays and Error-Correcting Codes         
61 





5 Construction of Orthogonal Arrays from Codes         
87 





6 Orthogonal Arrays and Difference Schemes         113 






7 Orthogonal Arrays and Hadamard Matrices         145 






8 Orthogonal Arrays and Latin Squares         167 





9 Mixed Orthogonal Arrays         199 





10 Further Constructions and Related Structures         
223 





11 Statistical Application of Orthogonal Arrays         
247 





12 Tables of Orthogonal Arrays         317 





Appendix: Galois Fields         341 




Bibliography         363 







Up [ Return to top of page 
]