• A library of over 200 orthogonal arrays. Maintained by N. J. A. Sloane (email address: njasloane@gmail.com; home page: NeilSloane.com/). Comments or contributions of new OA's welcomed.
• John Stufken, Dean De Cock, Warren Kuhfeld, Hongquan Xu and Yingshan Zhang have kindly contributed a number of orthogonal arrays to this library. Thanks also to Chung-yi Suen for some corrections.
• New constructions from Yingshan Zhang (2007)
  (Not yet added to the main tables, Jan 22 2007)

  OA(72, 2^7 3^7 6^5 12^1)
  OA(72, 2^15 3^7 4^1 6^5)
  OA(72, 2^8 3^8 4^1 6^5)
  OA(72, 2^6 3^7 4^1 6^6)
  OA(72, 2^6 3^3 6^6 12^1)
  OA(72, 2^14 3^3 4^1 6^6)
  OA(72, 2^7 3^4 4^1 6^6)
  OA(72, 2^5 3^3 4^1 6^7)
  

  OA(96, 2^12 4^20 24^1)
  OA(96, 2^18 4^22 12^1)
  OA(96, 2^26 4^23)
  OA(96, 2^19 3^1 4^23)
  OA(96, 2^17 4^23 6^1)
  

• Related files:
  • Parameters of arrays of strength 2 with up to 100 runs
  • Library of Hadamard matrices
  • Web site for the OA book
• Related sites
  • Pieter Eendebak and Eric Schoen, Complete series of non-isomorphic orthogonal arrays
• Contents:
  • Fixed-level or unmixed OAs with 2 levels and strength 2: - essentially equivalent to Hadamard matrices, which are in a separate library, although a handful can be found below.
  • Fixed-level or unmixed OAs with 2 levels and strength 3.
  • Fixed-level or unmixed OAs with 3 levels and strength 2.
  • Fixed-level or unmixed OAs with 3 levels and strength greater than 2.
  • Fixed-level or unmixed OAs with more than 3 levels.
  • Fixed-level OR mixed-level OAs with up to 100 runs.
  • Notes on certain constructions.
• File names: A file oa.N.k.s.t.name indicates an orthogonal array with N runs, k factors, s levels, and strength t.

  This is an array of size N by k, with entries from 0 to s-1, with the property that in any t columns you see each of the s^t possibilities equally often.

  We normally write OA(N, s^k, 2) to specify such an orthogonal array, or simply OA(N, s^k) if it has strength 2.

• Mixed arrays: The name MA.N.s1.k1.s2.k2.... indicates a mixed-level (or asymmetrical) orthogonal array with N runs, k1 factors at s1 levels, k2 factors at s2 levels, ..., and with strength 2.

  We normally write OA(N, s1^k1 s1^k1 ... ) to specify such an orthogonal array.

• Note that some people prefer to work with the transposed array. The notation used here follows Hedayat-Sloane-Stufken's book Orthogonal Arrays .
• REFERENCES. For references to the literature before 1999 see our book; for more recent references see the new references section of the web page for the book.
• Warning: The different levels in the name of a mixed orthogonal array are normally written in increasing order.
  For example MA.36.3.7.6.3 would indicate a 36-run array with seven 3-level factors and three 6-level factors.
  However, in the file containing this array the factors may appear in a different order, which can be seen from the file name at the top of the page (in this case MA.36.6.3.3.7.finney.txt).
  The reason for this is that these arrays have been obtained by many different constructions and many different sources.
• Format: If s is at most 10 the rows (or runs) are often written without spaces. If s exceeds 10 (and in some cases when s is less than 10) the symbols are separated by spaces.

• This library contains some OAs that are dominated by others (e.g. with more factors for the same number of runs). For example there is a 36-run OA( 36, 2⁴ 3⁵ 6¹) and a 36-run OA( 36, 2¹⁰ 3⁸ 6¹) even though the former is dominated by the latter. This is because it is often useful to have more than one construction for an OA with a given set of parameters.
• For OAs with two levels and strength 2, see also the accompanying Table of Hadamard Matrices (sometimes also called Plackett-Burman designs).

• That there is also a table showing the parameters of all known orthogonal arrays of strength 2 and up to 100 runs (both mixed and unmixed). That table lists many OA's that are not explicitly given here (yet) - however their parameters are listed here.
• At present the table contains the following orthogonal arrays:

• Two levels
  
  • Two levels and strength 2
    oa.4.3.2.2
    oa.8.5.2.2
    oa.8.7.2.2
    oa.12.11.2.2
    oa.16.15.2.2.0
    oa.16.15.2.2.1
    oa.16.15.2.2.2
    oa.16.15.2.2.3
    oa.16.15.2.2.4
    oa.20.19.2.2.pal
    oa.20.19.2.2.toniii
    oa.20.19.2.2.will
    

  • Two levels and strength 3
    oa.8.4.2.3
    oa.16.8.2.3
    oa.16.8.2.3.d
    oa.24.12.2.3
    oa.24.12.2.3.rev
    oa.32.16.2.3.0
    oa.32.16.2.3.1
    oa.32.16.2.3.2
    oa.32.16.2.3.3
    oa.32.16.2.3.4
    oa.40.20.2.3.halln
    oa.40.20.2.3.julin
    oa.40.20.2.3.pal
    oa.40.20.2.3.toniii
    oa.40.20.2.3.toniv
    oa.40.20.2.3.will
    oa.48.24.2.3.pal
    oa.48.24.2.3.1
    oa.48.24.2.3.2
    oa.48.24.2.3.3
    oa.48.24.2.3.4
    oa.48.24.2.3.5
    oa.48.24.2.3.6
    oa.48.24.2.3.7
    oa.48.24.2.3.8
    oa.48.24.2.3.9
    oa.48.24.2.3.10
    oa.48.24.2.3.11
    oa.48.24.2.3.12
    oa.48.24.2.3.13
    oa.48.24.2.3.14
    oa.48.24.2.3.15
    oa.48.24.2.3.16
    oa.48.24.2.3.17
    oa.48.24.2.3.18
    oa.48.24.2.3.19
    oa.48.24.2.3.20
    oa.48.24.2.3.21
    oa.48.24.2.3.22
    oa.48.24.2.3.23
    oa.48.24.2.3.24
    oa.48.24.2.3.25
    oa.48.24.2.3.26
    oa.48.24.2.3.27
    oa.48.24.2.3.28
    oa.48.24.2.3.29
    oa.48.24.2.3.30
    oa.48.24.2.3.31
    oa.48.24.2.3.32
    oa.48.24.2.3.33
    oa.48.24.2.3.34
    oa.48.24.2.3.35
    oa.48.24.2.3.36
    oa.48.24.2.3.37
    oa.48.24.2.3.38
    oa.48.24.2.3.39
    oa.48.24.2.3.40
    oa.48.24.2.3.41
    oa.48.24.2.3.42
    oa.48.24.2.3.43
    oa.48.24.2.3.44
    oa.48.24.2.3.45
    oa.48.24.2.3.46
    oa.48.24.2.3.47
    oa.48.24.2.3.48
    oa.48.24.2.3.49
    oa.48.24.2.3.50
    oa.48.24.2.3.51
    oa.48.24.2.3.52
    oa.48.24.2.3.53
    oa.48.24.2.3.54
    oa.48.24.2.3.55
    oa.48.24.2.3.56
    oa.48.24.2.3.57
    oa.48.24.2.3.58
    oa.48.24.2.3.59
    oa.48.24.2.3.60
    oa.56.28.2.3.pal2
    oa.56.28.2.3.will
    oa.64.32.2.3.pal
    oa.64.32.2.3.syl
    oa.64.32.2.3.t1
    oa.64.32.2.3.t2
    oa.64.32.2.3.t3
    oa.64.32.2.3.t4
    oa.72.36.2.3.pal2
    oa.72.36.2.3.will
    oa.80.40.2.3.tpal
    oa.80.40.2.3.ttoncheviv
    oa.80.40.2.3.twill
    oa.88.44.2.3.pal
    oa.88.44.2.3.will
    oa.96.48.2.3.pal
    oa.104.52.2.3.will
    oa.112.56.2.3.tpal2
    oa.112.56.2.3.twill
    oa.120.60.2.3.pal
    oa.120.60.2.3.pal2
    oa.120.60.2.3.will
    oa.128.64.2.3.syl
    oa.136.68.2.3.pal
    oa.136.68.2.3.will
    oa.144.72.2.3.pal
    oa.152.76.2.3.pal2
    oa.152.76.2.3.will
    oa.160.80.2.3.pal
    oa.168.84.2.3.pal
    oa.168.84.2.3.pal2
    oa.168.84.2.3.will
    oa.176.88.2.3.tpal
    oa.176.88.2.3.twill
    oa.184.92.2.3.will
    oa.192.96.2.3.tpal
    oa.200.100.2.3.will
    oa.208.104.2.3.pal
    oa.216.108.2.3.pal
    oa.216.108.2.3.pal2
    oa.216.108.2.3.will
    oa.224.112.2.3.ttpal2
    oa.224.112.2.3.ttwill
    oa.232.116.2.3.will
    oa.240.120.2.3.tpal
    oa.240.120.2.3.tpal2
    oa.240.120.2.3.twill
    oa.248.124.2.3.pal2
    oa.248.124.2.3.will
    oa.256.128.2.3.syl
    oa.264.132.2.3.pal
    oa.264.132.2.3.will
    oa.272.136.2.3.tpal
    oa.272.136.2.3.twill
    oa.280.140.2.3.pal
    oa.288.144.2.3.tpal
    oa.296.148.2.3.pal2
    oa.296.148.2.3.will
    oa.304.152.2.3.pal
    oa.312.156.2.3.will
    oa.320.160.2.3.tpal
    oa.328.164.2.3.pal
    oa.336.168.2.3.pal
    oa.344.172.2.3.will
    oa.352.176.2.3.ttpal
    oa.352.176.2.3.ttwill
    oa.360.180.2.3.pal
    oa.368.184.2.3.twill
    oa.376.188.2.3
    oa.384.192.2.3.pal
    oa.392.196.2.3.pal2
    oa.400.200.2.3.pal
    oa.512.256.2.3.syl
    

  • Two levels and strength greater than 3
    oa.80.6.2.4
    oa.128.9.2.5
    oa.256.16.2.5
    oa.1024.24.2.5
    oa.2048.32.2.5
    oa.64.7.2.6
    oa.1024.12.2.7
    oa.2048.16.2.7
    oa.4096.12.2.7
    

• Three levels
  
  • Three levels and strength 2
    oa.9.4.3.2
    oa.18.7.3.2
    oa.18.7.3.2.ugly
    oa.27.13.3.2
    oa.36.13.3.2
    difference scheme D(24,24,3) - can be used to construct an OA(72,25,3,2) and a MA(72, 3²⁴24¹, 2)
    oa.81.40.3.2
    

  • Three levels and strength greater than 2
    oa.54.5.3.3.a
    oa.243.20.3.3
    oa.729.14.3.4
    oa.729.12.3.5
    

• Fixed-level arrays with more than three levels
  oa.16.5.4.2
  oa.16.5.4.2.a
  oa.32.9.4.2
  oa.32.9.4.2.a
  oa.64.21.4.2
  oa.64.6.4.3
  oa.256.17.4.3
  oa.4096.12.4.5
  oa.25.6.5.2
  oa.50.11.5.2
  oa.49.8.7.2
  oa.98.15.7.2
  oa.64.9.8.2
  oa.128.17.8.2
  oa.81.10.9.2
  oa.162.19.9.2
  oa.100.4.10.2
  oa.121.12.11.2
  oa.242.23.11.2
  oa.144.7.12.2
  oa.169.14.13.2
  oa.256.17.16.2
  oa.512.33.16.2
  oa.289.18.17.2
  

• Fixed-level OR mixed-level (i.e. symmetrical OR asymmetrical) orthogonal arrays of strength 2 with up to 100 runs
  

  Note: Every good OA with up to 100 runs known to me is listed here, although many are not yet given explicitly. (Further contributions are welcomed.)

  On the other hand, as mentioned above, some of the OAs given here are dominated by others.

  There are errors in the 72-run OAs in the printed version of the paper in Discrete Math. (2001) by Yingshan Zhang et al., but with Zhang's help they have been corrected here.

  Arrays with 4 runs:
  OA(4, 2^3), (see the book, Chapter 1 and also Sections 2.3, 3.4, 5.3, 5.5, 6.3)

  Arrays with 6 runs:
  MA(6, 2^1 3^1),     Complete factorial: see Note (c)
  

  Arrays with 8 runs:
  MA.8.2.4.4.1,     Hadamard: see Note (h)
  

  Arrays with 9 runs:
  OA(9, 3^4), Sections 2.2, 3.2, 3.4, 5.3, 5.5, 5.12, 6.3
  

  Arrays with 10 runs:
  MA(10, 2^1 5^1),     Complete factorial: see Note (c)
  

  Arrays with 12 runs:
  OA(12, 2^{11}), Section 4.4 Table 7.15
  MA.12.2.4.3.1,     Example 9.2
  MA.12.2.2.6.1,     Trivial: see Note (t)
  MA(12, 3^1 4^1),     Complete factorial: see Note (c)
  

  Arrays with 14 runs:
  OA(14, 2^1 7^1),     Complete factorial: see Note (c)
  

  Arrays with 15 runs:
  OA(15, 3^1 5^1),     Complete factorial: see Note (c)
  

  Arrays with 16 runs:
  OA(16, 2^8 8^1),     Hadamard: see Note (h)
  MA.16.2.6.4.3
  OA(16, 4^5), Sections 3.2, 3.4, 5.3, 5.5, 6.2, 6.3; Taguchi 1987
  

  Arrays with 18 runs:
  MA.18.3.6.6.1,     Example 9.19
  

  Arrays with 20 runs:
  OA(20, 2^{19}), Section 7.5
  MA.20.2.8.5.1, Wang and Wu 1992, Dey and Midha 1996
  MA.20.2.2.10.1,     Trivial: see Note (t)
  

  Arrays with 24 runs:
  OA(24, 2^{20} 4^1),     Example 9.17 Wang and Wu, 1991
  MA.24.2.13.3.1.4.1, Wang and Wu, 1991
  OA(24, 2^{12} 12^1),     Hadamard: see Note (h)
  OA(24, 2^{11} 4^1 6^1), Wang and Wu, 1991
  

  Arrays with 25 runs:
  OA(25, 5^6), Sections 3.2, 3.4, 5.3, 5.5, 6.3; Taguchi, 1987
  

  Arrays with 27 runs:
  OA(27, 3^9 9^1),     Example 9.19
  

  Arrays with 28 runs:
  OA(28, 2^{27}), Table 7.19
  MA.28.2.12.7.1 (from Dey and Midha, 1996; Suen, Comm. Stat.-Theory Meth., 1989)
  MA.28.2.2.14.1,     Trivial: see Note (t)
  

  Arrays with 32 runs:
  OA(32, 2^{16} 16^1),     Hadamard: see Note (h)
  OA(32, 4^8 8^1),     Example 9.19
  

  Arrays with 36 runs:
  OA(36, 2^{35}), Table 7.33
  MA.36.2.27.3.1,     Example 9.30
  MA.36.2.20.3.2,     Example 9.30
  MA.36.2.18.3.1.6.1,     Example 9.30
  MA.36.2.13.6.2 (from H. Xu, 2002)
  MA.36.2.13.9.1 (from Dey and Midha, 1996; Suen, Comm. Stat.-Theory Meth., 1989)
  MA.36.2.11.3.12 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.11.3.2.6.1,     Example 9.30
  MA.36.2.10.3.8.6.1 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.10.3.1.6.2 (from H. Xu, 2002)
  MA.36.2.9.3.4.6.2 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.9.3.1.6.2,     Example 9.30
  MA.36.2.8.6.3 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.7.3.2.6.2 (from H. Xu, 2002)
  MA.36.2.5.3.3.6.2 (from H. Xu, 2002)
  MA.36.2.4.3.13 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.4.3.5.6.1 (from H. Xu, 2002)
  MA.36.2.4.3.3.6.1,     Example 9.30
  MA.36.2.4.3.1.6.3 (from H. Xu, 2002)
  MA.36.2.3.3.9.6.1 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.3.3.6.6.1 (from H. Xu, 2002)
  MA.36.2.3.3.2.6.3 (from H. Xu, 2002)
  MA.36.2.2.3.12.6.1 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.2.3.5.6.2 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.2.3.2.6.2,     Example 9.30
  OA(36, 2^2 18^1),     Trivial: see Note (t)
  MA.36.2.1.3.8.6.2 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.1.3.3.6.3 (from H. Xu, 2002)
  MA.36.2.1.3.1.6.3 (from Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.2.1.6.3 (from Warren Kuhfeld)
  MA.36.3.12.12.1 (Example 9.19; Wang and Wu, 1991; Y. Zhang et al., Discrete Math. 238 (2001)
  MA.36.3.7.6.3 (from Finney, 1982)
  

  Arrays with 40 runs:
  OA(40, 2^{36} 4^1),     Example 9.17; Dey and Ramakrishna 1977; Wang and Wu, 1991
  OA(40, 2^{25} 4^1 5^1), Wang and Wu, 1991
  OA(40, 2^{20} 20^1),     Hadamard: see Note (h)
  OA(40, 2^{19} 4^1 10^1), Agrawal and Dey 1982; Wang and Wu, 1991
  

  Arrays with 44 runs:
  OA(44, 2^{43}), Table 7.33
  OA(44, 2^{12} 11^1),     Juxtaposition: see Note (j)
  MA.44.2.2.22.1,     Trivial: see Note (t)
  

  Arrays with 45 runs:
  OA(45, 3^9 15^1),     Example 9.19
  

  Arrays with 48 runs:
  OA(48, 2^{40} 8^1), Dey, 1985, p. 72 Wang and Wu, 1991
  OA(48, 2^{33} 3^1 8^1), Wang and Wu, 1991
  OA(48, 2^{31} 6^1 8^1), Wang and Wu, 1991
  OA(48, 2^{24} 24^1),     Hadamard: see Note (h)
  OA(48, 4^{12} 12^1),     Example 9.19 Suen 1989b; Wang and Wu, 1991
  

  Arrays with 49 runs:
  OA(49, 7^8), Sections 3.2, 3.4, 5.3, 5.5
  

  Arrays with 50 runs:
  OA(50, 5^{10} 10^1),     Example 9.19
  

  Arrays with 52 runs:
  OA(52, 2^{51}), Table 7.33
  OA(52, 2^{12} 13^1),     Juxtaposition: see Note (j)
  MA.52.2.2.26.1,     Trivial: see Note (t)
  

  Arrays with 54 runs:
  OA(54, 3^{20} 6^1 9^1), Wang and Wu, 1991
  OA(54, 3^{18} 18^1),     Example 9.19
  

  Arrays with 56 runs:
  OA(56, 2^{52} 4^1),     Example 9.17 (lambda = 7, mu = 1/2, f = 1)
  OA(56, 2^{28} 28^1),     Hadamard: see Note (h)
  

  Arrays with 60 runs:
  OA(60, 2^{59}), Table 7.33
  MA.60.2.30.3.1 (from De Cock and Stufken 2000)
  OA(60, 2^{22} 5^1),     Juxtaposition: see Note (j)
  OA(60, 2^{18} 6^1),     Juxtaposition: see Note (j)
  OA(60, 2^{18} 10^1),     Juxtaposition: see Note (j)
  MA.60.2.15.3.1.5.1 (from De Cock and Stufken 2000)
  OA(60, 2^{13} 15^1),     Juxtaposition: see Note (j)
  MA.60.2.10.6.1.10.1 (from De Cock and Stufken 2000
  OA(60, 2^2 30^1),     Trivial: see Note (t)
  

  Arrays with 63 runs:
  OA(63, 3^{11} 21^1),     Example 9.19
  

  Arrays with 64 runs:
  OA(64, 2^{32} 32^1),     Hadamard: see Note (h)
  OA(64, 2^5 4^{17} 8^1), E. M. Rains, N. J. A. Sloane and J. Stufken, The Lattice of N-Run Orthogonal Arrays [postscript, pdf]
  OA(64, 2^5 4^{10} 8^4), E. M. Rains, N. J. A. Sloane and J. Stufken, The Lattice of N-Run Orthogonal Arrays [postscript, pdf]
  OA(64, 4^{16} 16^1),     Example 9.19; Hedayat, Pu and Stufken, 1992
  OA(64, 4^{14} 8^3), E. M. Rains, N. J. A. Sloane and J. Stufken, The Lattice of N-Run Orthogonal Arrays [postscript, pdf]
  OA(64, 4^7 8^6), E. M. Rains, N. J. A. Sloane and J. Stufken, The Lattice of N-Run Orthogonal Arrays [postscript, pdf]
  OA(64, 8^9), Sections 3.2, 3.4, 5.3, 5.5
  

  Arrays with 68 runs:
  OA(68, 2^{67}), Table 7.33
  OA(68, 2^{13} 17^1),     Juxtaposition: see Note (j)
  MA.68.2.2.34.1,     Trivial: see Note (t)
  

  Arrays with 72 runs:
  OA(72, 2^{68} 4^1),     Example 9.17 (lambda = 9, mu = 1/2, f = 1)
  OA(72, 2^{44} 3^{12} 4^1), Wang and Wu, 1991
  OA(72, 2^{37} 3^{13} 4^1), Wang and Wu, 1991
  OA(72, 2^{36} 36^1),     Hadamard: see Note (h)
  OA(72, 2^{35} 3^{12} 4^1 6^1), Wang and Wu, 1991
  MA.72.2.20.3.24.4.1 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.19.3.20.4.1.6.1 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.18.3.16.4.1.6.2 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.17.3.12.4.1.6.3 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.16.3.8.4.1.6.4 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.13.3.25.4.1 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.12.3.24.12.1 (from Zhang et al. Discrete Math. 238 (2001))
  MA.72.2.12.3.21.4.1.6.1 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.11.3.24.4.1.6.1 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.11.3.20.6.1.12.1 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.11.3.17.4.1.6.2 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.10.3.20.4.1.6.2 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.10.3.16.6.2.12.1 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.10.3.13.4.1.6.3 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.9.3.16.4.1.6.3 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.9.3.12.6.3.12.1 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.9.3.9.4.1.6.4 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.8.3.12.4.1.6.4 (From Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.8.3.8.6.4.12.1 (First discovered by Mandeli, JSPI 47 (1995); this version taken from Zhang et al., Discrete Math. 238 (2001))
  MA.72.2.7.3.8.4.1.6.5 (From Zhang et al., Discrete Math. 238 (2001))
  OA(72, 2^7 3^7 6^5 12^1), Wang, 1996a
  OA(72, 2^6 3^3 6^6 12^1), Wang, 1996a
  MA.72.3.24.24.1 (Example 9.19; Zhang et al. Discrete Math. 238 (2001))
  

  Arrays with 75 runs:
  OA(75, 5^7 15^1),     Example 9.19
  

  Arrays with 76 runs:
  OA(76, 2^{75}), Table 7.33
  OA(76, 2^{13} 19^1),     Juxtaposition: see Note (j)
  MA.76.2.2.38.1,     Trivial: see Note (t)
  

  Arrays with 80 runs:
  OA(80, 2^{61} 5^1 8^1), Wang, 1996b
  OA(80, 2^{55} 8^1 10^1), Wang and Wu, 1989
  OA(80, 2^{51} 4^3 20^1), Dey, 1985 Wang and Wu, 1989
  OA(80, 2^{40} 40^1),     Hadamard: see Note (h)
  OA(80, 4^8 20^1),     Example 9.19
  

  Arrays with 81 runs:
  OA(81, 3^{27} 27^1),     Example 9.19
  OA(81, 9^{10}), Sections 3.2, 3.4, 5.3, 5.5
  

  Arrays with 84 runs:
  OA(84, 2^{83}), Table 7.32
  MA.84.2.28.3.1 (from De Cock and Stufken 2000)
  OA(84, 2^{27} 7^1),     Juxtaposition: see Note (j)
  OA(84, 2^{26} 6^1),     Juxtaposition: see Note (j)
  OA(84, 2^{18} 14^1),     Juxtaposition: see Note (j)
  MA.84.2.15.3.1.7.1 (from De Cock and Stufken 2000)
  OA(84, 2^{13} 21^1),     Juxtaposition: see Note (j)
  MA.84.2.12.3.1.14.1 (from De Cock and Stufken 2000)
  MA.84.2.11.6.1.7.1 (from De Cock and Stufken 2000)
  MA.84.2.8.6.1.14.1 (from De Cock and Stufken 2000)
  OA(84, 2^2 42^1),     Trivial: see Note (t)
  

  Arrays with 88 runs:
  OA(88, 2^{84} 4^1),     Example 9.17
  OA(88, 2^{44} 44^1),     Hadamard: see Note (h)
  

  Arrays with 90 runs:
  OA(90, 3^{30} 30^1),     Example 9.19
  

  Arrays with 92 runs:
  OA(92, 2^{91}), Table 7.32
  OA(92, 2^{13} 23^1),     Juxtaposition: see Note (j)
  MA.92.2.2.46.1,     Trivial: see Note (t)
  

  Arrays with 96 runs:
  OA(96, 2^{80} 16^1), Dey and Midha, 1996 Wang and Wu, 1989
  OA(96, 2^{77} 8^1 12^1), Wang and Wu, 1989
  OA(96, 2^{73} 3^1 16^1), Wang and Wu, 1989
  OA(96, 2^{71} 6^1 16^1), Wang and Wu, 1989
  OA(96, 2^{48} 48^1),     Hadamard: see Note (h)
  OA(96, 2^{46} 3^1 4^{11} 6^1 8^1), Wang, 1996b (DOES NOT EXIST! 18 does not divide 96)
  OA(96, 2^{44} 4^{14} 6^1), Wang and Wu, 1991 Wang, 1996b
  OA(96, 2^{43} 4^{15} 8^1), Wang, 1996b
  OA(96, 2^{43} 4^{12} 6^1 8^1), Wang, 1996b
  OA(96, 2^{40} 3^1 4^{16}), Wang and Wu, 1991 Wang, 1996b
  OA(96, 2^{39} 3^1 4^{14} 8^1), Wang, 1996b
  OA(96, 2^{36} 4^{12} 24^1), Dey and Midha, 1996
  OA(96, 4^{16} 24^1),     Example 9.19
  

  Arrays with 98 runs:
  OA(98, 7^{14} 14^1),     Example 9.19
  

  Arrays with 99 runs:
  OA(99, 3^{11} 33^{1}),     Example 9.19
  

  Arrays with 100 runs:
  OA(100, 2^{99}), Table 7.32
  OA(100, 2^{18} 10^1),     Juxtaposition: see Note (j)
  MA.100.2.17.10.2 (from De Cock and Stufken 2000)
  OA(100, 2^{13} 25^1),     Juxtaposition: see Note (j)
  OA(100, 2^2 50^1),     Trivial: see Note (t)
  OA(100, 5^{20} 20^1),     Example 9.19
  OA(100, 5^8 10^3), Mandeli 1995 Wang, 1996a
  OA(100, 10^4), Theorem 8.28 Mandeli, 1995
  

Notes on Certain Constructions.

• Note (c): Complete factorial. Whenever N can be factorized as
  N     =     s₁ s₂ . . . s_a

  where the s_i are relatively prime, there is a complete factorial OA(N, s₁¹ s₂¹ . . . s_a¹) consisting of all possible runs in which the first factor takes any level between 0 and s₁-1, the second factor takes any level between 0 and s₂-1, etc. (Of course this is simply the direct product of the appropriate number of trivial orthogonal arrays.)

• Note (h): Hadamard construction. If A is an n X n Hadamard matrix with entries +1, -1, and u is a single column
  [0, 1, . . . , n - 1 ]^tr,

  then

  u	A
  u	-A

  is an OA( 2n, n¹ 2ⁿ). The Hadamard matrices used can be found in the accompanying table.

• Note (j): Juxtaposition. There are two versions of this trivial construction.
  1. Combining an OA(N', (s'₁)¹ s₂^k) and an OA(N'', (s''₁)¹ s₂^k) satisfying     N'/s'₁ = N''/s''₁     we obtain an OA(N = N'+N'', (s'₁+s''₁)¹ s₂^k).
     For example, from an OA(16, 4¹ 2¹²) and an OA(28, 7¹ 2¹²) we obtain an OA(44, 11¹ 2¹²).
  2. Combining an OA(N', s₁¹ s₂^k) and an OA(N'', s₁¹ s₂^k) we obtain an OA(N = N'+N'', s₁¹ s₂^k).
     For example, from an OA(24, 6¹ 2¹⁴) and an OA(36, 6¹ 2¹⁴) we obtain an OA(60, 6¹ 2¹⁴).

• Note (t): Trivial construction. Any OA with <= 3 factors is trivial! (Cf. Problem 9.12 in the OA book.