Complete series of non-isomorphic orthogonal arrays

Pieter Eendebak <pieter.eendebak@tno.nl>, Eric Schoen <EDSite, eric.schoen@ua.ac.be, eric.schoen@tno.nl>

Image of OA

Example of an array in OA(16; 4423; 2)

On this page we present numbers of non-isomorphic orthogonal arrays for many series. Most of them are generated by our Orthogonal Arrays program, some series are generated by hand. The algorithm used is described in our paper Complete Enumeration of Pure-Level and Mixed-Level Orthogonal Arrays, Journal of Combinatorial Designs, Volume 18, Issue 2, pages 123-140, 2010.

Software for manipulating the arrays can be found here. To obtain the actual arrays (for the larger series several GB of data), please contact one of the authors above.

The research leading to these results was performed at the Computer Vision and Statistics group of TNO and the Department of Mathematics, Statistics, and Actuarial Sciences of University of Antwerp.

For an introduction to orthogonal arrays see the book Orthogonal Arrays: Theory and Applications by Hedayat, Sloane and Stufken.

Image of OA

The lexicographically first array in OA(18; 2; 2135)

Strength 2

OA(4; 2; 2a) : 1, 0
OA(8; 2; 2a) : 2, 2, 1, 1, 1, 0
OA(8; 2; 412a) : 1, 1, 1, 0
OA(9; 2; 3a) : 1, 1, 0
OA(12; 2; 2a) : 2, 1, 2, 2, 1, 1, 1, 1, 1, 0
OA(12; 2; 312a) : 2, 3, 1, 0
OA(12; 2; 612a) : 1, 0
OA(16; 2; 2a) : 3, 5, 11, 27, 55, 80, 87, 78, 58, 36, 18, 10, 5, 0
OA(16; 2; 412a) : 3, 10, 28, 65, 110, 123, 110, 72, 38, 15, 8, 0
OA(16; 2; 4a) : 2, 1, 1, 0
OA(16; 2; 442a) : 2, 1, 1, 1, 1, 0
OA(16; 2; 432a) : 2, 2, 4, 4, 4, 2, 2, 0
OA(16; 2; 422a) : 2, 7, 17, 27, 30, 25, 14, 6, 3, 0
OA(16; 2; 812a) : 1, 1, 2, 1, 1, 1, 1, 0
OA(18; 2; 213a) : 3, 15, 48, 19, 12, 3, 0
OA(18; 2; 3a) : 4, 12, 10, 8, 3, 0
OA(18; 2; 613a) : 2, 3, 1, 1, 1
OA(20; 2; 2a) : 3, 3, 11, 75, 474, 1603, 2477, 2389, 1914, 1300, 730, 328, 124, 40, 11, 6, 3, 0
OA(20; 2; 512a) : 3, 10, 15, 38, 30, 4, 1, 0
OA(20; 2; 512a) : 3, 10, 15, 38, 30, 4, 1, 0
OA(20; 2; 1012a) : 1, 0
OA(24; 2; 2a) : 4, 10, 63, 1350, 57389, 1470157, 12952435, 38592861, 52912678, 51154497, 43092737, 31833387, 19960039, 10351396, 4385567, 1502242, 409478, 86725, 13833, 1604, 130, 0
OA(24; 2; 312a) : 4, 29, 573, 28745, 1089168, 14576216, 57436095, 71157023, 33893515, 10266252, 3305030, 981180, 220993, 32567, 2282, 0
OA(24; 2; 412a) : 4, 25, 552, 21757, 457768, 3113669, 8168256, 12605571, 15119461, 14961206, 12096092, 7855020, 4066838, 1665918, 532484, 129122, 22880, 2758, 238, 0
OA(24; 2; 41312a) : 5, 131, 6412, 226330, 2360583, 6467858, 5404183, 2341404, 963413, 350559, 94228, 16238, 1282, 0
OA(24; 2; 612a) : 4, 25, 226, 2663, 19323, 58112, 65679, 26454, 12243, 4882, 1543, 277, 45, 0
OA(24; 2; 61412a) : 3, 38, 400, 2060, 3911, 2200, 1357, 689, 283, 64, 14, 0
OA(24; 2; 1212a) : 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 0
OA(25; 2; 5a) : 2, 1, 1, 1
OA(27; 2; 3a) : 9, 711, 187188, 922548, 157829, 21688, 9793, 3766, 1252, 341, 68, 0
OA(27; 2; 913a) : 2, 7, 10, 13, 12, 9, 5, 4, 0
OA(28; 2; 2a) : 4, 7, 127, 17826, 5882186, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(28; 2; 712a) : 4, 25, 371, 21502, 395598, 1422094, 1005490, 135569, 4296, 104, 21, 0
OA(30; 2; 513121) : 6
OA(32; 2; 2a) : 5, 19, 491, 266217, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(32; 2; 412a) : 7, 158, 57857, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(32; 2; 4a) : 44, 32983, 108339, 31779, 6564, 283, 20, 0
OA(32; 2; 462a) : 44, 32983, 108339, 31779, 582942, 3880455, 17714295, 52127662, 105510372, 152752321, 162022665, 127424325, 74302445, 31659282, 9492964, 1841598, 183833, 0
OA(32; 2; 472a) : 44, 32983, 108339, 31779, 6564, 70932, 342203, 969642, 1770830, 2208501, 1920008, 1172827, 491748, 132333, 19379, 0
OA(32; 2; 482a) : 44, 32983, 108339, 31779, 6564, 283, 1549, 4098, 6522, 6607, 4282, 1727, 362, 0
OA(32; 2; 492a) : 44, 32983, 108339, 31779, 6564, 283, 20, 86, 116, 86, 40, 0
OA(32; 2; 432a) : 44, 32804, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(32; 2; 422a) : 12, 2548, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(32; 2; 81462a) : 10, 89, 200, 70, 22, 67, 137, 177, 137, 67, 22, 0
OA(32; 2; 81472a) : 10, 89, 200, 70, 22, 1, 1, 1, 1, 0
OA(32; 2; 81452a) : 10, 89, 200, 70, 489, 1848, 4226, 6331, 6515, 4667, 2274, 718, 130, 0
OA(32; 2; 81442a) : 10, 89, 200, 1978, 12452, 47738, 123554, 220792, 281678, 259750, 176744, 87561, 31595, 7552, 1213, 0
OA(36; 2; 2a) : 5, 15, 1242, 3352529, ?, ?, ?, ?
OA(36; 2; 312a) : 6, 138, 130852, ?, ?, ?
OA(36; 2; 322a) : 9, 1774, ?, ?, ?, ?
OA(36; 2; 3a) : 24, 221510, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(36; 2; 6a) : 12, 0
OA(36; 2; 633a) : 12, 274, 579, 550, 288, 38, 23, 8, 0
OA(36; 2; 63312a) : 12, 274, 2212, 5272, 1208, 147, 0
OA(36; 2; 632a) : 12, 131, 3456, 13132, 11014, 3159, 761, 118, 8, 0
OA(42; 2; 713121) : 12
OA(70; 2; 715121) : 1897

Strength 3

OA(8; 3; 2a) : 1, 0
OA(16; 3; 2a) : 2, 2, 1, 1, 1, 0
OA(16; 3; 412a) : 1, 0
OA(24; 3; 2a) : 2, 1, 2, 1, 1, 1, 1, 1, 1, 0
OA(24; 3; 312a) : 2, 3, 0
OA(24; 3; 612a) : 1, 0
OA(27; 3; 3a) : 1, 0
OA(32; 3; 2a) : 3, 5, 10, 17, 33, 34, 32, 22, 23, 12, 10, 5, 5, 0
OA(32; 3; 412a) : 3, 7, 7, 11, 8, 0
OA(32; 3; 422a) : 2, 2, 2, 0
OA(32; 3; 812a) : 1, 0
OA(36; 3; 322a) : 3
OA(40; 3; 2a) : 3, 3, 9, 25, 105, 213, 353, 260, 235, 132, 96, 36, 26, 7, 6, 3, 3, 0
OA(40; 3; 512a) : 3, 7, 1, 1, 0
OA(40; 3; 1012a) : 1, 0
OA(48; 3; 2a) : 4, 10, 45, 397, 8383, 166081, 1310006, 3528089, 4460865, 3980095, 3139653, 2165144, 1288460, 629705, 259346, 84495, 24012, 4919, 1129, 130, 60, 0
OA(48; 3; 312a) : 4, 21, 134, 938, 3056, 5018, 3, 0
OA(48; 3; 412a) : 4, 4, 24, 98, 387, 1362, 2595, 2217, 560, 0
OA(48; 3; 41312a) : 5, 35, 19, 0
OA(48; 3; 612a) : 4, 14, 30, 74, 45, 0
OA(48; 3; 61412a) : 3, 0
OA(48; 3; 1212a) : 1, 0
OA(54; 3; 213a) : 5, 33, 4, 0
OA(54; 3; 3a) : 7, 4, 0
OA(54; 3; 613a) : 2
OA(56; 3; 2a) : 4, 7, 86, 4049, 757190, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(56; 3; 712a) : 4, 14, 7, 10
OA(56; 3; 1412a) : 1, 0
OA(60; 3; 51312a) : 6
OA(64; 3; 2a) : 5, 19, 358, 91789, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(64; 3; 412a) : 7, 75, 4101, 251617, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?
OA(64; 3; 4a) : 5, 1, 1, 0
OA(64; 3; 462a) : 5, 1, 1, 0
OA(64; 3; 452a) : 5, 1, 1, 1, 0
OA(64; 3; 442a) : 5, 3, 5, 3, 3, 1, 1, 0
OA(64; 3; 432a) : 10, 107, 237, 255, 126, 35, 12, 2, 0
OA(64; 3; 422a) : 12, 267, 13903, 104949, 175297, 151708, 138825, 83409, 35807, 10030, 2159, 0
OA(64; 3; 812a) : 5, 26, 192, 1146, 924, 0
OA(64; 3; 81412a) : 4
OA(64; 3; 1612a) : 1, 0
OA(72; 3; 322a) : 9, 465, 380253, 32565625, 10620268, 953145, 653569, 207970, 22833, 69, 27, 0
OA(72; 3; 413221) : 18
OA(72; 3; 61312a) : 10, 265, 293, 0
OA(72; 3; 622a) : 6, 0
OA(72; 3; 912a) : 5, 26, 96, 498, 0
OA(80; 3; 2a) : 6, 33, 3494, ?, ?, ?, ?
OA(80; 3; 51412a) : 26, 3127, 409509, 217319, 4814, 0
OA(81; 3; 3a) : 32, 17056, 1099, 486, 235, 1, 1, 0
OA(81; 3; 913a) : 3, 2, 0
OA(84; 3; 71312a) : 12
OA(90; 3; 513221) : 39
OA(100; 3; 522a) : 61
OA(108; 3; 322a) : 20
OA(108; 3; 332a) : 39, 5384016
OA(108; 3; 613221) : 122
OA(112; 3; 71412a) : 129
OA(120; 3; 51413121) : 2395
OA(120; 3; 61512a) : 422
OA(150; 3; 523121) : 84422

Strength 4

OA(16; 4; 2a) : 1, 0
OA(32; 4; 2a) : 2, 2, 0
OA(32; 4; 412a) : 1, 0
OA(48; 4; 2a) : 2, 0
OA(48; 4; 312a) : 2, 3, 0
OA(48; 4; 612a) : 1
OA(64; 4; 2a) : 3, 5, 7, 3, 0
OA(64; 4; 412a) : 3, 7, 2, 0
OA(64; 4; 422a) : 2
OA(64; 4; 812a) : 1
OA(72; 4; 322a) : 3
OA(80; 4; 2a) : 3, 1
OA(80; 4; 512a) : 3, 7
OA(81; 4; 3a) : 1
OA(96; 4; 2a) : 4, 9, 4, 0
OA(96; 4; 312a) : 4, 21, 64, 5, 0
OA(96; 4; 412a) : 4, 1, 0
OA(96; 4; 41312a) : 5, 35
OA(96; 4; 612a) : 4, 14, 0
OA(96; 4; 61412a) : 3, 0
OA(108; 4; 332a) : 5
OA(112; 4; 2a) : 4, 3
OA(112; 4; 712a) : 4, 14
OA(120; 4; 51312a) : 6
OA(128; 4; 2a) : 5, 17, 123, 1288, 4132, 1683, 477, 2, 1, 1, 1, 0
OA(128; 4; 412a) : 7, 75, 870, 2649, 1014, 275, 0
OA(128; 4; 432a) : 10, 6, 0
OA(128; 4; 422a) : 12, 198, 202, 77, 0
OA(128; 4; 812a) : 5, 26, 19, 11, 10, 0
OA(128; 4; 81412a) : 4, 0
OA(128; 4; 1612a) : 1, 0
OA(144; 4; 2a) : 5, 7, 35, 20, 0
OA(144; 4; 312a) : 6, 78, 130, 0
OA(144; 4; 322a) : 9, 319, 11232, 0
OA(144; 4; 41322a) : 18
OA(144; 4; 612a) : 6, 0
OA(144; 4; 61312a) : 10, 265
OA(144; 4; 622a) : 6, 0
OA(144; 4; 912a) : 5, 26, 0
OA(160; 4; 2a) : 6, 29, 450, 99618, ?, ?, ?
OA(160; 4; 412a) : 9, 43, 1965, 534, 0
OA(160; 4; 512a) : 9, 230, 11016, 19153, 0
OA(160; 4; 51412a) : 26, 2277, 28, 0
OA(160; 4; 1012a) : 6, ?, ?
OA(162; 4; 3a) : 14, 0
OA(162; 4; 3521) : 14, 46
OA(162; 4; 3421) : 13
OA(168; 4; 71312a) : 12
OA(180; 4; 51322a) : 39
OA(200; 4; 522a) : 61
OA(216; 4; 332a) : 39, 464307
OA(216; 4; 413321) : 215
OA(216; 4; 61322a) : 122
OA(252; 4; 71322a) : 308
OA(256; 4; 4a) : 26, 0
OA(256; 4; 452a) : 26, 0

Strength 5

OA(32; 5; 2a) : 1, 0
OA(64; 5; 2a) : 2, 2, 0
OA(64; 5; 412a) : 1, 0
OA(96; 5; 2a) : 2, 0
OA(96; 5; 312a) : 2, 3, 0
OA(128; 5; 2a) : 3, 5, 7, 2, 0
OA(128; 5; 412a) : 3, 7, 0
OA(128; 5; 422a) : 2, 0
OA(128; 5; 812a) : 1, 0
OA(144; 5; 322a) : 3
OA(160; 5; 2a) : 3, 1, 0
OA(160; 5; 512a) : 3, 7, 0
OA(192; 5; 2a) : 4, 9, 3, 0
OA(192; 5; 312a) : 4, 21, 64, 0
OA(192; 5; 412a) : 4, 0
OA(192; 5; 41312a) : 5, 35, 0
OA(192; 5; 612a) : 4, 14, 0
OA(192; 5; 61412a) : 3, 0
OA(216; 5; 332a) : 5
OA(224; 5; 2a) : 4, 3, 0
OA(224; 5; 712a) : 4, 14, 0
OA(240; 5; 51312a) : 6
OA(256; 5; 422a) : 12, 198, 10, 0
OA(256; 5; 432a) : 10, 0
OA(288; 5; 41322a) : 18
OA(360; 5; 51322a) : 39
OA(384; 5; 412a) : 14, 827, 2644, 0

Strength 6

OA(64; 6; 2a) : 1, 0
OA(128; 6; 2a) : 2, 2, 0
OA(128; 6; 412a) : 1, 0
OA(192; 6; 2a) : 2, 0
OA(192; 6; 312a) : 2, 3, 0
OA(256; 6; 2a) : 3, 5, 7, 0
OA(256; 6; 412a) : 3, 7, 0
OA(256; 6; 422a) : 2, 0
OA(288; 6; 322a) : 3
OA(320; 6; 2a) : 3, 0
OA(384; 6; 2a) : 4, 9, 1, 0
OA(384; 6; 312a) : 4, 21, 64, 0
OA(384; 6; 412a) : 4, 0
OA(384; 6; 41312a) : 5, 35, 0

Strength 7

OA(128; 7; 2a) : 1, 0
OA(256; 7; 2a) : 2, 2, 0
OA(256; 7; 412a) : 1, 0
OA(384; 7; 2a) : 2, 0
OA(384; 7; 312a) : 2, 3, 0
OA(384; 7; 612a) : 1, 0
OA(512; 7; 2a) : 3, 5, 7, 0
OA(512; 7; 412a) : 3, 7, 0
OA(512; 7; 422a) : 2, 0

Strength 8

OA(256; 8; 2a) : 1, 0
OA(512; 8; 2a) : 2, 2, 0
OA(512; 8; 412a) : 1, 0