(109,28,7) block plan
The (109,28,7) block diagram is a special symmetrical block diagram . In order to be able to construct it, this combinatorial problem had to be solved: an empty 109 × 109 matrix was filled with ones in such a way that each row of the matrix contains exactly 28 ones and any two rows have exactly 7 ones in the same column (not more and not less). That sounds relatively simple, but it is not trivial to solve. There are only certain combinations of parameters (like here v = 109, k = 28, λ = 7) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .
properties
This symmetrical block diagram has the parameters v = 109, k = 28, λ = 7 and thus the following properties:
- It consists of 109 blocks and 109 points.
- Each block contains exactly 28 points.
- Every 2 blocks intersect at exactly 7 points.
- Each point lies on exactly 28 blocks.
- Each 2 points are connected by exactly 7 blocks.
Existence and characterization
There is (apart from isomorphism ) at least one 2- (109,28,7) block plan. This solution is:
List of blocks
All the blocks of this block plan are listed here; See this illustration to understand this list
- Solution 1
1 2 4 6 8 10 16 17 22 23 26 27 28 36 39 46 49 50 64 67 74 76 79 81 82 90 98 106 2 3 5 7 9 11 17 18 23 24 27 28 29 37 40 47 50 51 65 68 75 77 80 82 83 91 99 107 3 4 6 8 10 12 18 19 24 25 28 29 30 38 41 48 51 52 66 69 76 78 81 83 84 92 100 108 4 5 7 9 11 13 19 20 25 26 29 30 31 39 42 49 52 53 67 70 77 79 82 84 85 93 101 109 1 5 6 8 10 12 14 20 21 26 27 30 31 32 40 43 50 53 54 68 71 78 80 83 85 86 94 102 2 6 7 9 11 13 15 21 22 27 28 31 32 33 41 44 51 54 55 69 72 79 81 84 86 87 95 103 3 7 8 10 12 14 16 22 23 28 29 32 33 34 42 45 52 55 56 70 73 80 82 85 87 88 96 104 4 8 9 11 13 15 17 23 24 29 30 33 34 35 43 46 53 56 57 71 74 81 83 86 88 89 97 105 5 9 10 12 14 16 18 24 25 30 31 34 35 36 44 47 54 57 58 72 75 82 84 87 89 90 98 106 6 10 11 13 15 17 19 25 26 31 32 35 36 37 45 48 55 58 59 73 76 83 85 88 90 91 99 107 7 11 12 14 16 18 20 26 27 32 33 36 37 38 46 49 56 59 60 74 77 84 86 89 91 92 100 108 8 12 13 15 17 19 21 27 28 33 34 37 38 39 47 50 57 60 61 75 78 85 87 90 92 93 101 109 1 9 13 14 16 18 20 22 28 29 34 35 38 39 40 48 51 58 61 62 76 79 86 88 91 93 94 102 2 10 14 15 17 19 21 23 29 30 35 36 39 40 41 49 52 59 62 63 77 80 87 89 92 94 95 103 3 11 15 16 18 20 22 24 30 31 36 37 40 41 42 50 53 60 63 64 78 81 88 90 93 95 96 104 4 12 16 17 19 21 23 25 31 32 37 38 41 42 43 51 54 61 64 65 79 82 89 91 94 96 97 105 5 13 17 18 20 22 24 26 32 33 38 39 42 43 44 52 55 62 65 66 80 83 90 92 95 97 98 106 6 14 18 19 21 23 25 27 33 34 39 40 43 44 45 53 56 63 66 67 81 84 91 93 96 98 99 107 7 15 19 20 22 24 26 28 34 35 40 41 44 45 46 54 57 64 67 68 82 85 92 94 97 99 100 108 8 16 20 21 23 25 27 29 35 36 41 42 45 46 47 55 58 65 68 69 83 86 93 95 98 100 101 109 1 9 17 21 22 24 26 28 30 36 37 42 43 46 47 48 56 59 66 69 70 84 87 94 96 99 101 102 2 10 18 22 23 25 27 29 31 37 38 43 44 47 48 49 57 60 67 70 71 85 88 95 97 100 102 103 3 11 19 23 24 26 28 30 32 38 39 44 45 48 49 50 58 61 68 71 72 86 89 96 98 101 103 104 4 12 20 24 25 27 29 31 33 39 40 45 46 49 50 51 59 62 69 72 73 87 90 97 99 102 104 105 5 13 21 25 26 28 30 32 34 40 41 46 47 50 51 52 60 63 70 73 74 88 91 98 100 103 105 106 6 14 22 26 27 29 31 33 35 41 42 47 48 51 52 53 61 64 71 74 75 89 92 99 101 104 106 107 7 15 23 27 28 30 32 34 36 42 43 48 49 52 53 54 62 65 72 75 76 90 93 100 102 105 107 108 8 16 24 28 29 31 33 35 37 43 44 49 50 53 54 55 63 66 73 76 77 91 94 101 103 106 108 109 1 9 17 25 29 30 32 34 36 38 44 45 50 51 54 55 56 64 67 74 77 78 92 95 102 104 107 109 1 2 10 18 26 30 31 33 35 37 39 45 46 51 52 55 56 57 65 68 75 78 79 93 96 103 105 108 2 3 11 19 27 31 32 34 36 38 40 46 47 52 53 56 57 58 66 69 76 79 80 94 97 104 106 109 1 3 4 12 20 28 32 33 35 37 39 41 47 48 53 54 57 58 59 67 70 77 80 81 95 98 105 107 2 4 5 13 21 29 33 34 36 38 40 42 48 49 54 55 58 59 60 68 71 78 81 82 96 99 106 108 3 5 6 14 22 30 34 35 37 39 41 43 49 50 55 56 59 60 61 69 72 79 82 83 97 100 107 109 1 4 6 7 15 23 31 35 36 38 40 42 44 50 51 56 57 60 61 62 70 73 80 83 84 98 101 108 2 5 7 8 16 24 32 36 37 39 41 43 45 51 52 57 58 61 62 63 71 74 81 84 85 99 102 109 1 3 6 8 9 17 25 33 37 38 40 42 44 46 52 53 58 59 62 63 64 72 75 82 85 86 100 103 2 4 7 9 10 18 26 34 38 39 41 43 45 47 53 54 59 60 63 64 65 73 76 83 86 87 101 104 3 5 8 10 11 19 27 35 39 40 42 44 46 48 54 55 60 61 64 65 66 74 77 84 87 88 102 105 4 6 9 11 12 20 28 36 40 41 43 45 47 49 55 56 61 62 65 66 67 75 78 85 88 89 103 106 5 7 10 12 13 21 29 37 41 42 44 46 48 50 56 57 62 63 66 67 68 76 79 86 89 90 104 107 6 8 11 13 14 22 30 38 42 43 45 47 49 51 57 58 63 64 67 68 69 77 80 87 90 91 105 108 7 9 12 14 15 23 31 39 43 44 46 48 50 52 58 59 64 65 68 69 70 78 81 88 91 92 106 109 1 8 10 13 15 16 24 32 40 44 45 47 49 51 53 59 60 65 66 69 70 71 79 82 89 92 93 107 2 9 11 14 16 17 25 33 41 45 46 48 50 52 54 60 61 66 67 70 71 72 80 83 90 93 94 108 3 10 12 15 17 18 26 34 42 46 47 49 51 53 55 61 62 67 68 71 72 73 81 84 91 94 95 109 1 4 11 13 16 18 19 27 35 43 47 48 50 52 54 56 62 63 68 69 72 73 74 82 85 92 95 96 2 5 12 14 17 19 20 28 36 44 48 49 51 53 55 57 63 64 69 70 73 74 75 83 86 93 96 97 3 6 13 15 18 20 21 29 37 45 49 50 52 54 56 58 64 65 70 71 74 75 76 84 87 94 97 98 4 7 14 16 19 21 22 30 38 46 50 51 53 55 57 59 65 66 71 72 75 76 77 85 88 95 98 99 5 8 15 17 20 22 23 31 39 47 51 52 54 56 58 60 66 67 72 73 76 77 78 86 89 96 99 100 6 9 16 18 21 23 24 32 40 48 52 53 55 57 59 61 67 68 73 74 77 78 79 87 90 97 100 101 7 10 17 19 22 24 25 33 41 49 53 54 56 58 60 62 68 69 74 75 78 79 80 88 91 98 101 102 8 11 18 20 23 25 26 34 42 50 54 55 57 59 61 63 69 70 75 76 79 80 81 89 92 99 102 103 9 12 19 21 24 26 27 35 43 51 55 56 58 60 62 64 70 71 76 77 80 81 82 90 93 100 103 104 10 13 20 22 25 27 28 36 44 52 56 57 59 61 63 65 71 72 77 78 81 82 83 91 94 101 104 105 11 14 21 23 26 28 29 37 45 53 57 58 60 62 64 66 72 73 78 79 82 83 84 92 95 102 105 106 12 15 22 24 27 29 30 38 46 54 58 59 61 63 65 67 73 74 79 80 83 84 85 93 96 103 106 107 13 16 23 25 28 30 31 39 47 55 59 60 62 64 66 68 74 75 80 81 84 85 86 94 97 104 107 108 14 17 24 26 29 31 32 40 48 56 60 61 63 65 67 69 75 76 81 82 85 86 87 95 98 105 108 109 1 15 18 25 27 30 32 33 41 49 57 61 62 64 66 68 70 76 77 82 83 86 87 88 96 99 106 109 1 2 16 19 26 28 31 33 34 42 50 58 62 63 65 67 69 71 77 78 83 84 87 88 89 97 100 107 2 3 17 20 27 29 32 34 35 43 51 59 63 64 66 68 70 72 78 79 84 85 88 89 90 98 101 108 3 4 18 21 28 30 33 35 36 44 52 60 64 65 67 69 71 73 79 80 85 86 89 90 91 99 102 109 1 4 5 19 22 29 31 34 36 37 45 53 61 65 66 68 70 72 74 80 81 86 87 90 91 92 100 103 2 5 6 20 23 30 32 35 37 38 46 54 62 66 67 69 71 73 75 81 82 87 88 91 92 93 101 104 3 6 7 21 24 31 33 36 38 39 47 55 63 67 68 70 72 74 76 82 83 88 89 92 93 94 102 105 4 7 8 22 25 32 34 37 39 40 48 56 64 68 69 71 73 75 77 83 84 89 90 93 94 95 103 106 5 8 9 23 26 33 35 38 40 41 49 57 65 69 70 72 74 76 78 84 85 90 91 94 95 96 104 107 6 9 10 24 27 34 36 39 41 42 50 58 66 70 71 73 75 77 79 85 86 91 92 95 96 97 105 108 7 10 11 25 28 35 37 40 42 43 51 59 67 71 72 74 76 78 80 86 87 92 93 96 97 98 106 109 1 8 11 12 26 29 36 38 41 43 44 52 60 68 72 73 75 77 79 81 87 88 93 94 97 98 99 107 2 9 12 13 27 30 37 39 42 44 45 53 61 69 73 74 76 78 80 82 88 89 94 95 98 99 100 108 3 10 13 14 28 31 38 40 43 45 46 54 62 70 74 75 77 79 81 83 89 90 95 96 99 100 101 109 1 4 11 14 15 29 32 39 41 44 46 47 55 63 71 75 76 78 80 82 84 90 91 96 97 100 101 102 2 5 12 15 16 30 33 40 42 45 47 48 56 64 72 76 77 79 81 83 85 91 92 97 98 101 102 103 3 6 13 16 17 31 34 41 43 46 48 49 57 65 73 77 78 80 82 84 86 92 93 98 99 102 103 104 4 7 14 17 18 32 35 42 44 47 49 50 58 66 74 78 79 81 83 85 87 93 94 99 100 103 104 105 5 8 15 18 19 33 36 43 45 48 50 51 59 67 75 79 80 82 84 86 88 94 95 100 101 104 105 106 6 9 16 19 20 34 37 44 46 49 51 52 60 68 76 80 81 83 85 87 89 95 96 101 102 105 106 107 7 10 17 20 21 35 38 45 47 50 52 53 61 69 77 81 82 84 86 88 90 96 97 102 103 106 107 108 8 11 18 21 22 36 39 46 48 51 53 54 62 70 78 82 83 85 87 89 91 97 98 103 104 107 108 109 1 9 12 19 22 23 37 40 47 49 52 54 55 63 71 79 83 84 86 88 90 92 98 99 104 105 108 109 1 2 10 13 20 23 24 38 41 48 50 53 55 56 64 72 80 84 85 87 89 91 93 99 100 105 106 109 1 2 3 11 14 21 24 25 39 42 49 51 54 56 57 65 73 81 85 86 88 90 92 94 100 101 106 107 2 3 4 12 15 22 25 26 40 43 50 52 55 57 58 66 74 82 86 87 89 91 93 95 101 102 107 108 3 4 5 13 16 23 26 27 41 44 51 53 56 58 59 67 75 83 87 88 90 92 94 96 102 103 108 109 1 4 5 6 14 17 24 27 28 42 45 52 54 57 59 60 68 76 84 88 89 91 93 95 97 103 104 109 1 2 5 6 7 15 18 25 28 29 43 46 53 55 58 60 61 69 77 85 89 90 92 94 96 98 104 105 2 3 6 7 8 16 19 26 29 30 44 47 54 56 59 61 62 70 78 86 90 91 93 95 97 99 105 106 3 4 7 8 9 17 20 27 30 31 45 48 55 57 60 62 63 71 79 87 91 92 94 96 98 100 106 107 4 5 8 9 10 18 21 28 31 32 46 49 56 58 61 63 64 72 80 88 92 93 95 97 99 101 107 108 5 6 9 10 11 19 22 29 32 33 47 50 57 59 62 64 65 73 81 89 93 94 96 98 100 102 108 109 1 6 7 10 11 12 20 23 30 33 34 48 51 58 60 63 65 66 74 82 90 94 95 97 99 101 103 109 1 2 7 8 11 12 13 21 24 31 34 35 49 52 59 61 64 66 67 75 83 91 95 96 98 100 102 104 2 3 8 9 12 13 14 22 25 32 35 36 50 53 60 62 65 67 68 76 84 92 96 97 99 101 103 105 3 4 9 10 13 14 15 23 26 33 36 37 51 54 61 63 66 68 69 77 85 93 97 98 100 102 104 106 4 5 10 11 14 15 16 24 27 34 37 38 52 55 62 64 67 69 70 78 86 94 98 99 101 103 105 107 5 6 11 12 15 16 17 25 28 35 38 39 53 56 63 65 68 70 71 79 87 95 99 100 102 104 106 108 6 7 12 13 16 17 18 26 29 36 39 40 54 57 64 66 69 71 72 80 88 96 100 101 103 105 107 109 1 7 8 13 14 17 18 19 27 30 37 40 41 55 58 65 67 70 72 73 81 89 97 101 102 104 106 108 2 8 9 14 15 18 19 20 28 31 38 41 42 56 59 66 68 71 73 74 82 90 98 102 103 105 107 109 1 3 9 10 15 16 19 20 21 29 32 39 42 43 57 60 67 69 72 74 75 83 91 99 103 104 106 108 2 4 10 11 16 17 20 21 22 30 33 40 43 44 58 61 68 70 73 75 76 84 92 100 104 105 107 109 1 3 5 11 12 17 18 21 22 23 31 34 41 44 45 59 62 69 71 74 76 77 85 93 101 105 106 108 2 4 6 12 13 18 19 22 23 24 32 35 42 45 46 60 63 70 72 75 77 78 86 94 102 106 107 109 1 3 5 7 13 14 19 20 23 24 25 33 36 43 46 47 61 64 71 73 76 78 79 87 95 103 107 108 2 4 6 8 14 15 20 21 24 25 26 34 37 44 47 48 62 65 72 74 77 79 80 88 96 104 108 109 1 3 5 7 9 15 16 21 22 25 26 27 35 38 45 48 49 63 66 73 75 78 80 81 89 97 105 109
Cyclical representation
There is a cyclical representation ( Singer cycle ) for solution 1 of this block diagram, it is isomorphic to the above list of blocks. Starting from the block shown, the remaining blocks of the block plan are obtained by cyclic permutation of the points it contains.
- Solution 1
1 2 4 6 8 10 16 17 22 23 26 27 28 36 39 46 49 50 64 67 74 76 79 81 82 90 98 106
oval
An oval of the block plan is a set of its points, no three of which are on a block. Here is an example of a maximum order oval from this block diagram:
- Solution 1
1 2 9
literature
- Thomas Beth , Dieter Jungnickel , Hanfried Lenz : Design Theory . 1st edition. BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1985, ISBN 3-411-01675-2 .
- Albrecht Beutelspacher : Introduction to Finite Geometry. Volume 1: Block Plans . BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1982, ISBN 3-411-01632-9 .
Individual evidence
- ^ Rudolf Mathon, Alexander Rosa : 2- (ν, κ, λ) Designs of Small Order. In: Charles J. Colbourn , Jeffrey H. Dinitz (Eds.): Handbook of Combinatorial Designs. 2nd edition. Chapman and Hall / CRC, Boca Raton FL et al. 2007, ISBN 978-1-4200-1054-1 , pp. 25-57.