Non-isomorphic conference designs

pieter.eendebak@gmail.com

On this page we present numbers of isomorphism classes for conference designs with a specified number of runs (or rows) and factors (or columns). For all the cases a set of representatives for the isomorphism classes is available. The algorithm used to generate the results is described in A Classification Criterion for Definitive Screening Designs and Enumeration and Classification of Definitive Screening Designs, in preparation.

Single conference designs

Conference designs are matrixes of size 2m x k with values 0, +1 or -1, where m ≥ 2 and k ≤ m. Each column contains exactly one zero and each row conains at most one zero. The columns are orthogonal to each other. Square single conference designs are conference matrices.

A definitive screening design can be constructed by appending a conference design with both its fold-over and a row of zeroes.

Number of rows
k 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4 1 1 2 3 3 5 4 7 5 9 6 11 7 13 8 15 9 17 10
5 1 1 2 2 5 7 13 15 28 30 59 55 101 97 176 158 277 252
6 1 1 2 5 12 30 92 219 637 1588 4135 ≥ 209 ≥ 217 ≥ 2628 ≥ 256 ? ? ?
7 1 1 2 7 48 201 1781 10962 87929 ≥ 660 ≥ 897 ≥ 8682 ? ≥ 31726 ? ? ?
8 1 1 2 7 77 251 5292 70859 1839474 ≥ 1585 ≥ 6769 ≥ 340714 ? ? ? ? ?
9 1 1 3 42 47 3640 78966 8259167 ≥ 6364 ≥ 48980 ≥ 29524 ? ? ? ? ?
10 1 1 3 37 26 2342 16865 8667156 ≥ 5790 ≥ 107503 ≥ 13573 ? ? ? ? ?
11 1 1 17 10 1589 101 4124471 ≥ 1982 ≥ 65517 ≥ 4827 ? ? ? ? ?
12 1 1 13 10 1172 21 2397144 ≥ 1307 ≥ 16184 ≥ 906 ? ? ? ? ?
13 1 3 4 689 0 1806230 ≥ 571 ≥ 2633 ≥ 114 ? ? ? ? ?
14 1 3 3 366 0 1353790 ≥ 203 ≥ 1088 ≥ 69 ? ? ? ? ?
15 1 1 142 0 888475 ≥ 78 ≥ 675 ≥ 30 ? ? ? ? ?
16 1 1 57 0 499614 ≥ 64 ≥ 455 ≥ 28 ? ? ? ? ?
17 1 13 0 234006 ≥ 63 ≥ 295 ? ? ? ? ? ?
18 1 5 0 91773 ≥ 61 ≥ 221 ? ? ? ? ? ?
19 2 0 28730 ≥ 61 ≥ 168 ? ? ? ? ? ?
20 2 0 7417 ≥ 55 ≥ 132 ? ? ? ? ? ?
21 0 1377 ≥ 55 ≥ 92 ? ? ? ? ? ?
22 0 232 ≥ 54 ≥ 76 ? ? ? ? ? ?
23 19 ≥ 52 ≥ 63 ? ? ? ? ? ?
24 9 ≥ 31 ≥ 57 ? ? ? ? ? ?
25 ≥ 8 ≥ 50 ? ? ? ? ? ?
26 4 ≥ 48 ? ? ? ? ? ?
27 ≥ 44 ? ? ? ? ? ?
28 41 ? ? ? ? ? ?
29 ? ? ? ? ? ?
30 ? ? ? ? ? ?
31 ? ? ? ? ?

Double conference designs (DCDs)

Double conference designs are matrices of size 4m x k with values 0, +1 or -1, where m ≥2 and k ≤ 2m. Each column contains exactly two zeros and each row contains at most one zero. The columns are orthogonal to each other.

DCDs with level balance and orthogonal interaction columns.

In double conference designs with level balance and orthogonal interaction columns, the elements of each column sums to zero. In addition, for any set of three columns, the vector formed by the element-wise product of the columns has elements that sum to zero too. A definitive screening design can be constructed by appending a double conference design with a row of zeroes.

Number of rows
k 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4 0 1 1 2 3 3 5 4 7 5 9 6 11 7 13 8 15 ≥ 9 ≥ 17 10
5 0 1 1 2 2 6 8 15 17 34 38 74 72 135 135 ≥ 250 ≥ 230 ≥ 412 387
6 0 1 1 2 5 12 30 93 220 664 1691 4621 11345 ≥ 29165 ≥ 70480 ≥ 170345 ≥ 399563 ≥ 769435 ≥ 1220834
7 0 0 1 1 2 7 48 201 1781 11007 88691 ≥ 488248 ≥ 2763828 ≥ 2159796 ≥ 4503006 ≥ 1925678 ≥ 3143998 ≥ 1331877 ≥ 1697269
8 0 0 1 1 2 7 77 251 5292 70880 1840689 ≥ 1923908 ≥ 2030735 ≥ 1602270 ≥ 3238383 ≥ 1740801 ≥ 4073120 ≥ 2502673 ≥ 3971401
9 0 0 1 1 3 42 47 3640 78970 8259277 ≥ 939481 ≥ 809137 ≥ 682922 ≥ 1873207 ≥ 858485 ≥ 4668023 ≥ 792978 ≥ 2860616
10 0 0 1 1 3 37 26 2342 16866 8667156 ≥ 314593 ≥ 251597 ≥ 132276 ≥ 556721 ≥ 593656 ≥ 1356109 ≥ 208929 ≥ 1912806
11 0 0 0 1 1 17 10 1589 101 4124471 ≥ 73425 ≥ 45044 ≥ 44278 ≥ 111004 ≥ 331405 ≥ 283393 ≥ 37076 ≥ 1014231
12 0 0 0 1 1 13 10 1172 21 2397144 ≥ 43878 ≥ 4041 ≥ 5702 ≥ 20534 ≥ 89673 ≥ 74946 ≥ 9424 ≥ 301830
13 0 0 0 1 3 4 689 0 1806230 ≥ 9443 ≥ 1566 ≥ 784 ≥ 2785 ≥ 14544 ≥ 16202 ≥ 1038 ≥ 97028
14 0 0 0 1 3 3 366 0 1353790 ≥ 2723 ≥ 977 ≥ 100 ≥ 409 ≥ 2382 ≥ 4674 ≥ 69 ≥ 39204
15 0 0 0 0 1 1 142 0 888475 ≥ 829 ≥ 574 ≥ 4 ≥ 18 ≥ 180 ≥ 462 ≥ 2 ≥ 7661
16 0 0 0 0 1 1 57 0 499614 ≥ 253 ≥ 244 ? ? ? ≥ 34 ? ≥ 941
17 0 0 0 0 1 13 0 234006 ≥ 42 ≥ 62 ? ? ? ≥ 2 ? ≥ 334
18 0 0 0 0 1 5 0 91773 ≥ 6 ≥ 7 ? ? ? ? ? ≥ 83
19 0 0 0 0 0 2 0 28730 ? ? ? ? ? ? ? ≥ 5
20 0 0 0 0 0 2 0 7417 ? ? ? ? ? ? ? ?
21 0 0 0 0 0 0 1377 ? ? ? ? ? ? ? ?
22 0 0 0 0 0 0 232 ? ? ? ? ? ? ? ?
23 0 0 0 0 0 0 19 ? ? ? ? ? ? ? ?
24 0 0 0 0 0 0 9 ? ? ? ? ? ? ? ?
25 0 0 0 0 ? ? ? ? ? ? ? ? ?
26 0 0 0 0 ? ? ? ? ? ? ? ? ?

DCDs with level balance

In double conference designs with level balance, the elements of each column sum to zero.

Number of rows
k 4 6 8 10 12 14 16 18 20 22 24 26
2 1 1 1 2 1 2 1 2 1 2 1 2
3 0 0 1 2 1 0 1 2 1 0 1 2
4 0 0 1 2 2 0 6 8 14 0 25 72
5 0 0 1 2 0 13 60 159 0 1866 16732
6 0 1 2 0 21 403 640 0 77963 ?
7 0 0 0 12 1221 447 0 ? ?
8 0 0 6 1928 268 0 ? ?
9 0 0 1858 89 0 ? ?
10 0 1206 26 0 ? ?
11 577 0 0 ? ?
12 ? 0 0 ? ?
13 ? 0 ? ?

DCDs with orthogonality of interaction columns

In double conference designs with orthogonal interaction columns, for any set of three columns, the vector formed by the element-wise product of the columns has elements that sum to zero.

Number of rows
k 4 6 8 10 12 14 16 18 20 22 24
2 2 7 10 14 17 21 24 28 31 35 38
3 1 5 10 14 19 22 31 31 38 39 50
4 0 1 1 6 4 3 3 21 10 6 6
5 0 0 1 1 0 1 11 4 0 6
6 0 0 0 1 0 1 5 2 0 5
7 0 0 0 0 1 2 1 0 2
8 0 0 0 0 1 2 1 0 2
9 0 0 0 0 1 1 0 1
10 0 0 0 0 0 1 0 1
11 0 0 0 0 0 0 1
12 0 0 0 0 0 0 1
13 0 0 0 0 0 0

Plain DCDs

In double conference designs with orthogonal interaction columns, for any set of three columns, the vector formed by the element-wise product of the columns has elements that sum to zero.

Number of rows
k 4 6 8 10 12 14 16 18 20
2 2 8 17 26 41 54 75 92 119
3 1 15 61 207 500 1017 1893 3157 5085
4 0 12 106 930 6410 34333 ? ? ?
5 5 93 1222 ? ? ? ? ?
6 2 52 1146 ? ? ? ? ?
7 18 697 ? ? ? ? ?
8 4 267 ? ? ? ? ?
9 0 62 ? ? ? ? ?
10 9 ? ? ? ? ?
11 0 ? ? ? ? ?
12 ? ? ? ? ?

Weighing matrices

Plain DCDs (full isomorphism class)

For the isomorphism class level-permutatations of the rows are allowed. We also allow multiple zeros in a row. The square double conference designs of this type form a complete non-isomorphic set of weighing matrices of type W(N, N-2). The square double conference matrices are a complete non-isomorphic set of weighing matrices of type W(N, N-2).

Number of rows
k 4 6 8 10 12 14 16 18 20 22 24
2 2 2 2 2 2 2 2 2 2 2 469
3 1 3 3 4 4 4 4 4 4 4 ?
4 0 3 6 11 19 24 37 42 57 57 ?
5 1 3 11 38 102 352 ? ? ? ?
6 1 3 13 81 244 3244 ? ? ? ?
7 1 8 61 37 8165 ? ? ? ?
8 1 6 61 16 ? ? ? ? ?
9 1 27 5 ? ? ? ? ?
10 1 20 4 ? ? ? ? ?
11 5 2 ? ? ? ? ?
12 5 2 ? ? ? ? ?
13 0 ? ? ? ? ?
14 0 ? ? ? ? ?

If you make use of these results, please cite the paper A Classification Criterion for Definitive Screening Designs.

The square conference designs are conference matrices.