Round brackets () are used for blocks, [] is used for resolution classes, [[]] is used for 2-resolution classes, etc. Vertical bars | separate sub-blocks. Some of the designs here are of a more general type (Type II as opposed to Type I), in which corresponding individual sub-blocks within the main blocks constitute sub-designs (see the definitions). The sub-designs are demarcated by double vertical bars, thus ||.
Most of the the designs (the numbered ones) may be found in a more detailed collection in Morgan, Preece & Rees; No 39 is due to Deng, Greig & Östergård. Various (un-numbered) designs have been added at the end for use elsewhere: these are for non-primes only. The designs for v=18 are due to Sreenath. The designs for v=24 are derived from an OBIBD published in Abel,Greig & Rees, those for v=27 from an OBIBD here, while those for 30 and 36 are GWhDs published by Abel, Finizio & Greig. Designs for 28 and 40 may be derived from PMDs published by Abel, Bennett & Zhang, but are too complex to be included here (in the presented form).
If a design is known to exist for a particular set of parameters, no multiple of that parameter set is included in the table.
| No | v | b1 | b2 | r | k1 | k2 | Blocks |
|---|---|---|---|---|---|---|---|
| 1 | 5 | 5 | 10 | 4 | 4 | 2 | (1 4 | 2 3) mod 5 |
| 2 | 7 | 7 | 21 | 6 | 6 | 2 | (1 3 | 2 6 | 4 5) mod 7 |
| 3 | 7 | 7 | 14 | 6 | 6 | 3 | (1 2 4 || 6 5 3) mod 7 |
| 4 | 8 | 14 | 28 | 7 | 4 | 2 | [(0 1 | 4 2) (3 6 | 5 ∞ )] mod 7 |
| 5 | 9 | 18 | 36 | 8 | 4 | 2 | (01 02 | 10 20) (11 22 | 12 21) mod(3,3) |
| 6 | 9 | 12 | 36 | 8 | 6 | 2 | [[(1 2 | 3 6 | 4 ∞ ) (5 6 | 7 2 | 0 ∞ )(0 4 | 1 7 | 3 5 )]] PC(4) mod 8 |
| 7 | 9 | 12 | 24 | 8 | 6 | 3 | [[(1 3 4 || 2 6 ∞ ) (2 6 ∞ || 5 7 0) (5 7 0 || 1 3 4)]] PC(4) mod 8 |
| 8 | 9 | 9 | 36 | 8 | 8 | 2 | (01 02 | 10 20 | 11 22 | 12 21) mod(3,3) |
| 9 | 9 | 9 | 18 | 8 | 8 | 4 | (01 02 10 20 | 11 22 12 21) mod(3,3) |
| 10 | 10 | 15 | 45 | 9 | 6 | 2 | (00 20 | 30 21 | 31 41) (20 30 | 00 31 | 40 01) (00 01 | 10 31 | 21 41) mod 5 (suffixes fixed) |
| 11 | 10 | 15 | 30 | 9 | 6 | 3 | No NBIBD exists, but a double (No 47) and a triple (No 58) exist. |
| 12 | 10 | 10 | 30 | 9 | 9 | 3 | (10 20 41 | 30 40 31 | 01 11 21) (20 31 00 | 10 21 30 | 11 40 41) mod 5 (suffixes fixed) |
| 13 | 6 | 15 | 30 | 10 | 4 | 2 | [(1 5 || 1 3) (4 0 || 5 2) (3 1 || 4 0)]] PC(3) [[(1 2 || 3 4) (3 4 || 5 0) (5 0 || 1 2)]] PC(2) both mod 6 |
| 14 | 11 | 11 | 55 | 10 | 10 | 2 | (1 10 | 2 9 | 3 8 | 4 7 | 5 6) mod 11 |
| 15 | 11 | 11 | 22 | 10 | 10 | 5 | (1 3 4 5 9 || 2 6 8 10 7) mod 11 |
| 16 | 12 | 33 | 66 | 11 | 4 | 2 | [(0 1 | 3 7) (10 2 | 9 4) (8 6 | 5 ∞ )] mod 11 |
| 17 | 12 | 22 | 66 | 11 | 6 | 2 | [(0 3 | 1 5 | 4 9) (8 10 | 7 6 | 2 ∞ )] mod 11 |
| 18 | 12 | 22 | 44 | 11 | 6 | 3 | [(0 1 3 | 4 5 9) (10 7 ∞ | 6 8 2)] mod 11 |
| 19 | 7 | 21 | 42 | 12 | 4 | 2 | (0 1 || 4 2) (0 2 || 1 4) (0 4 || 2 1) mod 7 |
| 20 | 13 | 39 | 78 | 12 | 4 | 2 | (1 12 | 5 8) (2 11 | 3 10) (4 9 | 6 7) mod 13 |
| 21 | 13 | 26 | 78 | 12 | 6 | 2 | (3 10 | 4 9 | 1 12) (5 8 | 11 2 | 6 7) mod 13 |
| 22 | 13 | 26 | 52 | 12 | 6 | 3 | (1 3 9 || 4 12 10) (2 6 5 || 7 8 11) mod 13 |
| 23 | 13 | 13 | 78 | 12 | 12 | 2 | (1 12 | 2 11 | 3 10 | 4 9 | 5 8 | 6 7) mod 13 |
| 24 | 13 | 13 | 52 | 12 | 12 | 3 | (1 3 9 | 2 6 5 || 4 12 10 | 7 8 11) mod 13 |
| 25 | 13 | 13 | 39 | 12 | 12 | 4 | (1 12 5 8 | 2 11 3 10 | 4 9 6 7) mod 13 |
| 26 | 13 | 13 | 26 | 12 | 12 | 6 | (1 3 9 4 12 10 | 2 6 5 7 8 11) mod 13 |
| 27 | 15 | 35 | 105 | 14 | 6 | 2 | (11 00 | 21 01 | 41 ∞ )
(00 30 | 01 50 | ∞ 60) (20 10 | 40 31 | 11 01) (20 01 | 50 11 | 31 30) (40 11 | 50 00 | 01 31) mod 7 (suffixes fixed) |
| 28 | 15 | 35 | 70 | 14 | 6 | 3 | (11 21 41 || 00 01 ∞ )
(00 01 ∞ || 30 50 60) (20 40 11 || 10 31 01) (20 50 31 || 01 11 30) (40 50 01 || 11 00 31) mod 7 (suffixes fixed) |
| 29 | 15 | 21 | 105 | 14 | 10 | 2 | No D1 exists, so no NBIBD exists, but a double exists (No 60). |
| 30 | 15 | 21 | 42 | 14 | 10 | 5 | No D1 exists, so no NBIBD exists, but a double exists (No 61). |
| 31 | 15 | 15 | 105 | 14 | 14 | 2 | (1 14 | 2 13 | 3 12 | 4 11 | 5 10 | 6 9 | 7 8) mod 15 |
| 32 | 15 | 15 | 30 | 14 | 14 | 7 | (I 40 10 11 20 41 21 | 01 61 51 50 31 60 30) (00 20 40 51 10 61 31 | ∞ 60 50 41 30 21 11) mod 7 (suffixes fixed) with (01 11 21 31 41 51 61 | 00 10 20 30 40 50 60) |
| 33 | 16 | 60 | 120 | 15 | 4 | 2 | [(I 0 | 5 10)(1 2 | 4 8)(6 9 | 7 13)(11 3 | 12 14)] mod 15 |
| 34 | 16 | 40 | 120 | 15 | 6 | 2 | (0 1 | 9 3 | 5 12) (0 3 | 1 12 | 6 2) (11 9 | 1 3 | 0 8) PC(8) mod 16 |
| 35 | 16 | 40 | 80 | 15 | 6 | 3 | (01 02 03 | 12 22 32)
(I 31 41 | 01 32 43) (11 31 03 | 13 02 22) (I 22 43 | 01 32 13) (11 31 02 | 01 33 42) (I 32 13 | 01 12 33) (21 31 42 | 41 03 13) (01 02 03 | 13 23 43) mod 5 with indices fixed |
| 36 | 16 | 30 | 120 | 15 | 8 | 2 | [(I 0 | 3 14 | 1 4 | 9 7)(2 8 | 6 13 | 5 10 | 11 12)] mod 15 |
| 37 | 16 | 30 | 60 | 15 | 8 | 4 | [(0 1 3 7 | 4 9 14 ∞ ) (2 10 11 13 | 5 6 8 12)] mod 15 |
| 38 | 16 | 24 | 120 | 15 | 10 | 2 | (I1 01 | ∞ 2 02 | 12 13 | 22 24 | 23 14) (I2 03 | ∞ 3 02 | 11 24 | 21 23 | 13 14) (I1 03 | ∞ 3 01 | 11 22 | 21 14 | 12 24) (I2 24 | ∞ 3 23 | ∞ 4 02 | 01 22 | 11 12) (I1 02 | ∞ 3 24 | ∞ 4 03 | 21 13 | 12 23) (I1 24 | ∞ 2 11 | ∞ 4 01 | 21 03 | 22 13) (I4 04 | 11 21 | 12 22 | 13 23 | 14 24) mod 3 with v (I1 ∞ 2 | ∞ 3 ∞ 4 | 21 24 | 22 14 | 23 04) (I1 ∞ 3 | ∞ 2 ∞ 4 | 01 04 | 02 24 | 03 14) (I1 ∞ 4 | ∞ 2 ∞ 3 | 11 14 | 12 04 | 13 24) |
| 39 | 16 | 24 | 48 | 15 | 10 | 5 | (0,1,2,3,4 | 5,6,7,8,9)(0,1,2,3,5 | 4,6,10,11,12) (0,1,2,3,6 | 4,5,13,14,15)(0,1,10,11,12 | 2,3,7,8,9) (0,2,13,14,15 | 1,3,7,8,9)(0,3,13,14,15 | 1,2,10,11,12) (0,4,5,7,11 | 1,8,10,13,14)(0,4,5,9,10 | 1,7,12,13,15) (0,4,5,8,12 | 1,9,11,14,15)(0,6,7,10,13 | 2,4,8,11,14) (0,6,9,12,15 | 2,4,7,10,13) (0,6,8,11,14 | 2,4,9,12,15) (0,7,8,10,15 | 3,5,6,12,14) (0,7,9,12,14 | 3,5,6,11,13) (0,8,9,11,13 | 3,5,6,10,15) (1,5,7,12,14 | 2,6,8,10,15) (1,5,9,11,13 | 2,6,7,12,14) (1,5,8,9,15 | 2,6,9,11,13) (1,4,6,7,13 | 3,8,11,12,15) (1,4,6,9,15 | 3,7,10,11,14) (1,4,6,8,14 | 3,4,8,12,13) (2,5,7,11,15 | 3,4,8,12,13) (2,5,9,10,14 | 3,4,7,11,15) (2,5,8,12,13 | 3,4,9,10,14) |
| 40 | 16 | 20 | 120 | 15 | 12 | 2 |
[[[(1 2 | 4 8 | 6 13 | 7 9 | 0 5 | 10 ∞ ) (6 7 | 9 13 | 11 3 | 12 14 | 5 10 | 0 ∞ ) (11 12 | 14 3 | 1 8 | 2 4 | 10 0 | 5 ∞ ) (1 4 | 6 9 | 11 14 | 2 8 | 7 13 | 12 3)]]] PC(5) mod 15 |
| 41 | 16 | 20 | 80 | 15 | 12 | 3 |
[[[(0 1 5 | 2 8 10 | 6 7 9 | 13 4 ∞ ) (5 6 10 | 7 13 0 | 11 12 14 | 3 9 ∞ ) (10 11 0 | 12 3 5 | 1 2 4 | 8 14 ∞ ) (2 7 12 | 1 4 8 | 6 9 13 | 3 11 14) ]]] PC(5) mod 15 |
| 42 | 16 | 20 | 60 | 15 | 12 | 4 |
[[[(1 2 4 8 || 6 7 9 13 || 0 5 10 ∞ ) (6 7 9 13 || 5 10 0 ∞ || 11 12 14 3) ( 10 0 5 ∞ || 11 12 14 3 || 1 2 4 8) (11 12 14 3 || 1 2 4 8 || 6 7 9 13)]]] PC(5) mod 15 |
| 43 | 16 | 20 | 40 | 15 | 12 | 6 |
[[[(0 5 1 4 7 13 | 2 6 8 10 9 ∞ ) (5 10 6 9 12 3 | 7 11 13 0 14 ∞ ) (10 0 11 14 2 8 | 12 1 3 5 4 ∞ ) (1 9 6 14 11 4 | 8 7 13 12 3 2)]]] PC(5) mod 15 |
| 44 | 16 | 16 | 80 | 15 | 15 | 3 | (0001 0110 0111 | 0010 1100 1110 | 0100 1011 1111 |1000 0101 1101 | 0011 1010 1001) mod(2,2,2,2) |
| 45 | 16 | 16 | 48 | 15 | 15 | 5 |
(0001 1000 1100 1010 1111 |
0010 0011 1011 0111 1101 | 0100 0110 0101 1110 1001) mod(2,2,2,2) |
| 46 | 10 | 45 | 90 | 18 | 4 | 2 |
(1 2 | 3 5) (1 4 | 3 7) (0 2 | 4 5) (0 3 | 1 6) mod 10 with (1 7 | 2 6) PC(5) |
| 47 | 10 | 30 | 60 | 18 | 6 | 3 | (1 2 4 | 5 6 9) (1 2 7 | 3 5 8) (1 2 4 | 3 5 9) mod 10 |
| 48 | 13 | 26 | 78 | 18 | 9 | 3 | (2 6 5 || 4 12 10 || 8 11 7) (1 3 9 || 8 11 7 || 4 12 10) mod 13 |
| 49 | 11 | 55 | 110 | 20 | 4 | 2 | (10 9 || 1 2) (9 7 || 2 4) (7 3 || 4 8) (3 6 || 8 5) (6 1 || 5 10) mod 11 |
| 50 | 8 | 28 | 84 | 21 | 6 | 2 | [[[(1 3 || 2 6 || 4 5) (I 0 || 4 5 || 2 6) (4 5 || ∞ 0 || 1 3) (2 6 || 1 3 || ∞ 0)]]] mod 7 |
| 51 | 8 | 28 | 56 | 21 | 6 | 3 | [[[(1 2 4 | 3 6 5) (I 4 2 | 0 5 6) (4 ∞ 1 | 5 0 3) (2 1 ∞ | 6 3 0)]]] mod 7 |
| 52 | 15 | 35 | 105 | 21 | 9 | 3 | (01 41 52 | 11 31 42 | 21 51 22) (01 11 31 | 02 12 32 | 22 42 52 ) (01 22 32 | 21 31 12 | 41 62 ∞ ) (11 12 42 | 21 31 02 | 61 62 ∞ ) (01 21 12 | 11 32 52 | 51 42 ∞ ) mod 7 |
| 53 | 12 | 33 | 132 | 22 | 8 | 2 | [[(2 10 | 4 9 | 6 8 | 5 ∞ ) (0 1 | 3 7 | 6 8 | 5 ∞ ) (0 1 | 3 7 | 2 10 | 4 9)]] mod 11 |
| 54 | 12 | 33 | 66 | 22 | 8 | 4 | [[( 2 10 4 9 | 6 8 5 ∞ )( 0 1 3 7 | 6 8 5 ∞ ) ( 0 1 3 7 | 2 10 4 9) ]] mod 11 |
| 55 | 13 | 39 | 156 | 24 | 8 | 2 | (1 12 | 5 8 | 2 11 | 3 10) (4 9 | 6 7 | 1 12 | 5 8) (2 11 | 3 10 | 4 9 | 6 7) mod 13 |
| 56 | 13 | 39 | 78 | 24 | 8 | 4 | (1 12 5 8 || 2 11 3 10) (4 9 6 7 || 1 12 5 8) (2 11 3 10 || 4 9 6 7) mod 13 |
| 57 | 14 | 91 | 182 | 26 | 4 | 2 | (0 1 || 9 8) (0 2 || 5 3) (0 8 || 1 9) (0 3 || 2 5) (0 4 || 7 11) (I 0 || 2 9) (3 9 || ∞ 0) mod 13 |
| 58 | 10 | 45 | 90 | 27 | 6 | 3 | (2 3 4 | 1 6 ∞ ) (1 2 5 | 3 6 ∞ ) ( 1 3 5 | 2 8 ∞ ) (1 3 5 | 2 8 7) (1 2 7 | 3 4 6) mod 9 |
| 59 | 15 | 105 | 210 | 28 | 4 | 2 | (1 6 || 3 10) (1 13 || 2 8) (1 2 || 6 8) (7 14 || 1 2 ) (2 8 || 1 4 ) (2 4 || 1 12) (1 5 || 2 7) mod 15 |
| 60 | 15 | 42 | 210 | 28 | 10 | 2 | (0 2 | 3 11 | 4 13 | 5 12 | 6 9) (0 1 | 2 5 | 3 13 | 4 10 | 9 ∞ ) (0 1 | 3 5 | 7 11 | 8 13 | 10 ∞ ) mod 14 |
| 61 | 15 | 42 | 84 | 28 | 10 | 5 | (0 7 8 9 11 | 2 3 4 5 10) (0 5 6 9 10 | 1 4 7 8 ∞ ) (0 2 5 7 11 | 4 8 10 12 ∞ ) mod 14 |
| 62 | 15 | 35 | 210 | 28 | 12 | 2 | [[[[(12 02 | 62 52 | 32 31 | 42 ∞ | 01 51 | 21 41) (52 02 | 41 11 | 61 22 | 12 01 | 62 ∞ | 31 32) (11 ∞ | 61 42 | 21 52 | 02 32 | 62 51 | 22 01) (01 11 | 02 51 | 21 ∞ | 42 61 | 12 41 | 22 31) (52 32 | 31 61 | 12 41 | 42 51 | 22 62 | 11 21)]]]] mod 7 (suffixes fixed) |
| 63 | 15 | 35 | 140 | 28 | 12 | 3 | (0 3 14 || 7 8 11 || 9 10 13 || 4 6 12 ) (0 2 8 || 4 11 13 || 5 12 14 || 6 7 10 ) (1 6 11 || 2 7 12 || 3 8 13 || 4 9 14 ) mod 15, last block PC(5) |
| 64 | 15 | 35 | 105 | 28 | 12 | 4 | [[[[(12 02 62 52 | 32 31 42 ∞ | 01 51 21 41) (52 02 41 11 | 61 22 12 01 | 62 ∞ 31 32) (11 ∞ 61 42 | 21 52 02 32 | 62 51 22 01) (01 11 02 51 | 21 ∞ 42 61 | 12 41 22 31) (52 32 31 61 | 12 41 42 51 | 22 62 11 21)]]]] mod 7 (suffixes fixed) |
| 65 | 15 | 35 | 70 | 28 | 12 | 6 | (01 02 ∞ 21 51 42 | 32 52 62 41 31 12) (11 61 22 41 31 12 | 01 02 ∞ 32 52 62) (21 51 42 32 52 62 | 11 61 22 01 02 ∞ ) (41 31 12 01 02 ∞ | 21 51 42 11 61 22) (32 52 62 11 61 22 | 41 31 12 21 51 42) mod 7 (suffixes fixed) |
| 66 | 11 | 55 | 165 | 30 | 6 | 2 | (0 2 || 8 7 || 10 6) (0 8 || 10 6 || 7 2) (0 10 || 7 2 || 6 8) (0 7 || 6 8 || 2 10) (0 6 || 2 10 || 8 7) mod 11 |
| 67 | 11 | 55 | 110 | 30 | 6 | 3 |
(10 4 6 || 1 7 5) (4 6 9 || 7 5 2) (6 9 8 || 5 2 3) (9 8 1 || 2 3 10) (8 1 7 || 3 10 4) mod 11 |
| 68 | 16 | 80 | 160 | 30 | 6 | 3 | (1111 0100 1011 | 0011 1010 1001) (0011 1010 1001 | 1110 0010 1100) (1110 0010 1100 | 1000 0101 1101) (1000 0101 1101 | 0111 0001 0110) (0111 0001 0110 | 0100 1011 1111 mod (2,2,2,2) |
| 69 | 16 | 48 | 96 | 30 | 10 | 5 | (1101 0011 0111 0010 1011 | 0110 1110 0100 0101 1001) (0110 1110 0100 0101 1001 | 1111 1000 1010 0001 1100) (1111 1000 1010 0001 1100 | 0011 0111 0010 1011 1101) mod (2,2,2,2) |
| 18 | 51 | 153 | 17 | 6 | 2 | (∞,2 | 0,8 | 12,9) (0,7 | 1,3 | 8,12) (0,1 | 2,7 | 15,4) mod 17 | |
| 18 | 51 | 102 | 17 | 6 | 3 | (∞, 0, 12 | 2, 8, 9) (0, 1, 8 | 7, 3, 12) (0, 2, 15 | 1, 7, 4) mod 17 | |
| 21 | 70 | 210 | 20 | 6 | 2 | (20, 1 | 0, 3 | 5, 7) (0, 7 | 1, 9 | 4, 15) (0, 5 | 1, 15 | 8, 12) (0, 1 | 2, 12 | 10, 11) mod 20, last block PC(10) | |
| 21 | 70 | 140 | 20 | 6 | 3 | (20, 5, 7 | 0, 1, 3) (0, 4, 7 | 1, 9, 15) (0, 5, 15 | 1, 8, 12) (0, 1, 12 | 2, 10, 11) mod 20, last block PC(10) | |
| 22 | 77 | 231 | 21 | 6 | 2 | (0, 1 | 2, 5 | 8, 17) (0, 16 | 1, 5 | 3, 8) (0, 14 | 2, 9 | 6, 18) (0, 2 | 1, 12 | 11, 13) mod 22, last block PC(11) | |
| 24 | 92 | 276 | 23 | 6 | 2 | (0,11 | 1,8 | 13,17) (0,10 | 2,22 | 17,3) (0,6 | 15,13 | 10,11) (0,5 | 22,14 | 3,∞) mod 23 | |
| 24 | 92 | 184 | 23 | 6 | 3 | (0,1,13 | 11,8,17 ) (0,2,17 | 10,22,3 ) (0,15,10 | 6,13,11 ) (0,22,3 | 5,14,∞ ) mod 23 | |
| 27 | 117 | 351 | 26 | 6 | 2 |
[[(∞,1 | 0,9 | 13,3 )(1,14 | 9,16 | 3,22) (14,∞ | 16,0 | 22,13) (2,10 | 21, 6 | 12,11 ) (10,4 | 6,18 | 11,7 ) (4,2 | 18,21 | 7,12) (15,17 | 15,20 | 8,5 ) (17,23 | 20,24 | 5,19 ) (23,15 | 24,25 | 19,8)]] mod 26, in steps of 2 |
|
| 27 | 117 | 234 | 26 | 6 | 3 |
[[(∞,0,13 || 1,9,3 )(1,9,3 || 14, 16, 22) (14,16,22 || ∞,0,13)(2,21,12 || 10, 6, 11) (10,6,11 || 4,18,7) (4,18,7 | 2,21,12) (15,25,8 || 17, 20, 5) (17,20,5 || 23,24,19) (23,24,19 || 15,25,8)]] mod 26, in steps of 2 |
|
| 30 | 145 | 435 | 29 | 6 | 2 | (∞,0 | 26,3 | 27,14) (28,25 | 7,16 | 9,24) (2,4 | 6,17 | 10,22) (11,21 | 1,23 | 15,20) (8,12 | 5,13 | 18,19) mod 29 | |
| 30 | 145 | 290 | 29 | 6 | 3 | [(∞,0,7 | 18,20,26) (11,15,17 | 5,16,3) (14,4,1 | 10,21,25) (28,8,9 | 6,22,23) (19,24,2 | 12,3,7)] mod 29 | |
| 36 | 210 | 630 | 35 | 6 | 2 | (∞,0 | 16,25 | 34,2) (23,6 | 22,12 | 8,29) (27,28 | 3,33 | 14,20) (7,26 | 30,10 | 15,17) (5,1 | 4,31 | 19,32) (9,21 | 13,24 | 11,18) mod 35 | |
| 36 | 210 | 420 | 35 | 6 | 3 | [(∞,0,33 | 14,23,32) (1,25,26 | 12,18,31) (6,20,21 | 4,10,27) (7,11,22 | 9,16,19) (2,29,34 | 3,17,30) (5,24,28 | 8,13,15)] mod 35 |
Last Updated: 23rd May 2005