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.