The (133,12,1) 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 133 × 133 matrix was filled with ones in such a way that each row of the matrix contains exactly 12 ones and every two rows have exactly one one 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 = 133, k = 12, λ = 1) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .
designation
This symmetrical 2- (133,12,1) block plan is called the projective plane or Desarguean plane of order 11.
properties
This symmetrical block diagram has the parameters v = 133, k = 12, λ = 1 and thus the following properties:
It consists of 133 blocks and 133 points.
Each block contains exactly 12 points.
Every 2 blocks intersect in exactly 1 point.
Each point lies on exactly 12 blocks.
Each 2 points are connected by exactly 1 block.
Existence and characterization
There is (apart from isomorphism ) at least one 2- (133,12,1) 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
1 2 3 4 5 6 7 8 9 10 11 12
1 13 14 15 16 17 18 19 20 21 22 23
1 24 25 26 27 28 29 30 31 32 33 34
1 35 36 37 38 39 40 41 42 43 44 45
1 46 47 48 49 50 51 52 53 54 55 56
1 57 58 59 60 61 62 63 64 65 66 67
1 68 69 70 71 72 73 74 75 76 77 78
1 79 80 81 82 83 84 85 86 87 88 89
1 90 91 92 93 94 95 96 97 98 99 100
1 101 102 103 104 105 106 107 108 109 110 111
1 112 113 114 115 116 117 118 119 120 121 122
1 123 124 125 126 127 128 129 130 131 132 133
2 13 24 35 46 57 68 79 90 101 112 123
2 14 25 36 47 58 69 80 91 102 113 124
2 15 26 37 48 59 70 81 92 103 114 125
2 16 27 38 49 60 71 82 93 104 115 126
2 17 28 39 50 61 72 83 94 105 116 127
2 18 29 40 51 62 73 84 95 106 117 128
2 19 30 41 52 63 74 85 96 107 118 129
2 20 31 42 53 64 75 86 97 108 119 130
2 21 32 43 54 65 76 87 98 109 120 131
2 22 33 44 55 66 77 88 99 110 121 132
2 23 34 45 56 67 78 89 100 111 122 133
3 13 25 37 49 61 73 85 97 109 121 133
3 14 26 44 51 64 78 79 96 105 115 131
3 15 33 38 56 63 76 80 90 108 117 127
3 16 29 45 50 58 75 88 92 101 120 129
3 17 31 41 47 62 70 87 100 104 112 132
3 18 34 43 53 59 74 82 99 102 116 123
3 19 24 36 55 65 71 86 94 111 114 128
3 20 30 35 48 67 77 83 98 106 113 126
3 21 28 42 46 60 69 89 95 110 118 125
3 22 27 40 54 57 72 81 91 107 122 130
3 23 32 39 52 66 68 84 93 103 119 124
4 13 26 38 50 62 74 86 98 110 122 124
4 14 33 40 53 67 68 85 94 104 120 125
4 15 27 45 52 65 69 79 97 106 116 132
4 16 34 39 47 64 77 81 90 109 118 128
4 17 30 36 51 59 76 89 93 101 121 130
4 18 32 42 48 63 71 88 91 105 112 133
4 19 25 44 54 60 75 83 100 103 117 123
4 20 24 37 56 66 72 87 95 102 115 129
4 21 31 35 49 58 78 84 99 107 114 127
4 22 29 43 46 61 70 80 96 111 119 126
4 23 28 41 55 57 73 82 92 108 113 131
5 13 27 39 51 63 75 87 99 111 113 125
5 14 29 42 56 57 74 83 93 109 114 132
5 15 34 41 54 58 68 86 95 105 121 126
5 16 28 36 53 66 70 79 98 107 117 133
5 17 25 40 48 65 78 82 90 110 119 129
5 18 31 37 52 60 77 80 94 101 122 131
5 19 33 43 49 64 72 89 92 106 112 124
5 20 26 45 55 61 76 84 91 104 118 123
5 21 24 38 47 67 73 88 96 103 116 130
5 22 32 35 50 59 69 85 100 108 115 128
5 23 30 44 46 62 71 81 97 102 120 127
6 13 28 40 52 64 76 88 100 102 114 126
6 14 31 45 46 63 72 82 98 103 121 128
6 15 30 43 47 57 75 84 94 110 115 133
6 16 25 42 55 59 68 87 96 106 122 127
6 17 29 37 54 67 71 79 99 108 118 124
6 18 26 41 49 66 69 83 90 111 120 130
6 19 32 38 53 61 78 81 95 101 113 132
6 20 34 44 50 65 73 80 93 107 112 125
6 21 27 36 56 62 77 85 92 105 119 123
6 22 24 39 48 58 74 89 97 104 117 131
6 23 33 35 51 60 70 86 91 109 116 129
7 13 29 41 53 65 77 89 91 103 115 127
7 14 34 35 52 61 71 87 92 110 117 130
7 15 32 36 46 64 73 83 99 104 122 129
7 16 31 44 48 57 76 85 95 111 116 124
7 17 26 43 56 60 68 88 97 107 113 128
7 18 30 38 55 58 72 79 100 109 119 125
7 19 27 42 50 67 70 84 90 102 121 131
7 20 33 39 54 62 69 82 96 101 114 133
7 21 25 45 51 66 74 81 94 108 112 126
7 22 28 37 47 63 78 86 93 106 120 123
7 23 24 40 49 59 75 80 98 105 118 132
8 13 30 42 54 66 78 80 92 104 116 128
8 14 24 41 50 60 76 81 99 106 119 133
8 15 25 35 53 62 72 88 93 111 118 131
8 16 33 37 46 65 74 84 100 105 113 130
8 17 32 45 49 57 77 86 96 102 117 125
8 18 27 44 47 61 68 89 98 108 114 129
8 19 31 39 56 59 73 79 91 110 120 126
8 20 28 43 51 58 71 85 90 103 122 132
8 21 34 40 55 63 70 83 97 101 115 124
8 22 26 36 52 67 75 82 95 109 112 127
8 23 29 38 48 64 69 87 94 107 121 123
9 13 31 43 55 67 69 81 93 105 117 129
9 14 30 39 49 65 70 88 95 108 122 123
9 15 24 42 51 61 77 82 100 107 120 124
9 16 26 35 54 63 73 89 94 102 119 132
9 17 34 38 46 66 75 85 91 106 114 131
9 18 33 36 50 57 78 87 97 103 118 126
9 19 28 45 48 62 68 80 99 109 115 130
9 20 32 40 47 60 74 79 92 111 121 127
9 21 29 44 52 59 72 86 90 104 113 133
9 22 25 41 56 64 71 84 98 101 116 125
9 23 27 37 53 58 76 83 96 110 112 128
10 13 32 44 56 58 70 82 94 106 118 130
10 14 28 38 54 59 77 84 97 111 112 129
10 15 31 40 50 66 71 89 96 109 113 123
10 16 24 43 52 62 78 83 91 108 121 125
10 17 27 35 55 64 74 80 95 103 120 133
10 18 25 39 46 67 76 86 92 107 115 132
10 19 34 37 51 57 69 88 98 104 119 127
10 20 29 36 49 63 68 81 100 110 116 131
10 21 33 41 48 61 75 79 93 102 122 128
10 22 30 45 53 60 73 87 90 105 114 124
10 23 26 42 47 65 72 85 99 101 117 126
11 13 33 45 47 59 71 83 95 107 119 131
11 14 27 43 48 66 73 86 100 101 118 127
11 15 29 39 55 60 78 85 98 102 112 130
11 16 32 41 51 67 72 80 97 110 114 123
11 17 24 44 53 63 69 84 92 109 122 126
11 18 28 35 56 65 75 81 96 104 121 124
11 19 26 40 46 58 77 87 93 108 116 133
11 20 25 38 52 57 70 89 99 105 120 128
11 21 30 37 50 64 68 82 91 111 117 132
11 22 34 42 49 62 76 79 94 103 113 129
11 23 31 36 54 61 74 88 90 106 115 125
12 13 34 36 48 60 72 84 96 108 120 132
12 14 32 37 55 62 75 89 90 107 116 126
12 15 28 44 49 67 74 87 91 101 119 128
12 16 30 40 56 61 69 86 99 103 112 131
12 17 33 42 52 58 73 81 98 111 115 123
12 18 24 45 54 64 70 85 93 110 113 127
12 19 29 35 47 66 76 82 97 105 122 125
12 20 27 41 46 59 78 88 94 109 117 124
12 21 26 39 53 57 71 80 100 106 121 129
12 22 31 38 51 65 68 83 92 102 118 133
12 23 25 43 50 63 77 79 95 104 114 130
Cyclical representation
There is a cyclical representation ( Singer cycle ) 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.
1 10 11 13 27 31 68 75 83 110 115 121
Orthogonal Latin Squares (MOLS)
This projective plane of order 11 is equivalent to these 10 MOLS of order 11:
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{\ Display style {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 & 2 & 3 & 4 5 & 6 7 & 8 & 9 & 10 & 11 \\ 2 & 3 & 10 6 & 8 & 11 & 1 & 7 & 5 & 4 & 9 \\ 3 10 4 11 7 & 9 & 2 & 1 & 8 & 6 & 5 \\ 4 & 6 & 11 & 5 & 2 & 8 & 10 3 1 9 & 7 \\ 5 & 8 7 & 2 & 6 & 3 & 9 & 11 & 4 & 1 & 10 \\ 6 & 11 & 9 & 8 & 3 & 7 & 4 10 2 5 1 \\ 7 & 1 & 2 & 10 9 & 4 & 8 & 5 & 11 & 3 & 6 \\ 8 & 7 & 1 & 3 & 11 & 10 & 5 & 9 & 6 & 2 & 4 \\ 9 & 5 & 8 & 1 & 4 & 2 & 11 & 6 & 10 7 & 3 \ \ 10 4 & 6 & 9 & 1 & 5 3 2 7 & 11 & 8 \\ 11 & 9 & 5 & 7 & 10 1 6 & 4 3 & 8 & 2 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 & 3 & 4 5 & 6 7 & 8 & 9 & 10 & 11 & 2 \\ 2 10 6 & 8 & 11 & 1 & 7 & 5 & 4 & 9 & 3 \\ 3 4 11 7 & 9 & 2 & 1 & 8 & 6 & 5 & 10 \\ 4 & 11 & 5 & 2 & 8 & 10 3 1 9 7 & 6 \\ 5 & 7 & 2 & 6 & 3 & 9 & 11 & 4 1 10 8 \\ 6 & 9 & 8 & 3 & 7 & 4 10 2 5 1 11 \\ 7 & 2 & 10 9 & 4 & 8 & 5 & 11 & 3 & 6 & 1 \\ 8 & 1 & 3 & 11 & 10 & 5 & 9 & 6 & 2 & 4 & 7 \\ 9 & 8 & 1 & 4 & 2 & 11 & 6 & 10 7 & 3 & 5 \\ 10 6 & 9 & 1 & 5 3 2 7 & 11 & 8 & 4 \\ 11 & 5 & 7 & 10 1 6 & 4 3 & 8 & 2 & 9 \ end {array}} \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 & 4 5 & 6 7 & 8 & 9 & 10 & 11 & 2 & 3 \\ 2 & 6 & 8 & 11 & 1 & 7 & 5 & 4 & 9 & 3 & 10 \\ 3 11 7 & 9 & 2 & 1 & 8 & 6 & 5 & 10 4 \\ 4 & 5 & 2 & 8 & 10 3 1 9 7 & 6 & 11 \\ 5 & 2 & 6 & 3 & 9 & 11 & 4 1 10 8 & 7 \\ 6 & 8 & 3 & 7 & 4 10 2 5 1 11 & 9 \\ 7 & 10 9 & 4 & 8 & 5 & 11 & 3 & 6 & 1 & 2 \\ 8 & 3 & 11 & 10 & 5 & 9 & 6 & 2 & 4 & 7 & 1 \\ 9 & 1 & 4 & 2 & 11 & 6 & 10 7 & 3 & 5 & 8 \ \ 10 & 9 & 1 & 5 3 2 7 & 11 & 8 & 4 & 6 \\ 11 & 7 & 10 1 6 & 4 3 & 8 & 2 & 9 & 5 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 & 5 & 6 7 & 8 & 9 & 10 & 11 & 2 & 3 & 4 \\ 2 & 8 & 11 & 1 & 7 & 5 & 4 & 9 & 3 & 10 6 \\ 3 & 7 & 9 & 2 & 1 & 8 & 6 & 5 10 4 11 \\ 4 & 2 & 8 & 10 3 1 9 7 & 6 & 11 & 5 \\ 5 & 6 & 3 & 9 & 11 & 4 1 10 8 7 & 2 \\ 6 & 3 & 7 & 4 10 2 5 1 11 & 9 & 8 \\ 7 & 9 & 4 & 8 & 5 & 11 & 3 & 6 & 1 & 2 & 10 \\ 8 & 11 & 10 & 5 & 9 & 6 & 2 & 4 & 7 & 1 & 3 \\ 9 & 4 & 2 & 11 & 6 & 10 7 & 3 & 5 & 8 & 1 \\ 10 1 5 3 2 7 & 11 & 8 & 4 & 6 & 9 \\ 11 & 10 1 6 & 4 3 & 8 & 2 & 9 & 5 & 7 \ end {array}} \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 & 6 7 & 8 & 9 & 10 & 11 & 2 & 3 & 4 & 5 \\ 2 & 11 & 1 & 7 & 5 & 4 & 9 & 3 10 6 & 8 \\ 3 & 9 & 2 & 1 & 8 & 6 & 5 10 4 11 & 7 \\ 4 & 8 & 10 3 1 9 7 & 6 & 11 & 5 & 2 \\ 5 & 3 & 9 & 11 & 4 1 10 8 7 & 2 & 6 \\ 6 & 7 & 4 10 2 5 1 11 & 9 & 8 & 3 \\ 7 & 4 & 8 & 5 & 11 & 3 & 6 & 1 & 2 & 10 9 \\ 8 & 10 5 9 & 6 & 2 & 4 & 7 & 1 & 3 & 11 \\ 9 & 2 & 11 & 6 & 10 7 & 3 & 5 & 8 & 1 & 4 \ \ 10 5 3 2 7 & 11 & 8 & 4 & 6 & 9 & 1 \\ 11 & 1 & 6 & 4 3 & 8 & 2 & 9 & 5 & 7 & 10 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 7 & 8 & 9 & 10 & 11 & 2 & 3 & 4 5 & 6 \\ 2 & 1 & 7 & 5 & 4 & 9 & 3 10 6 & 8 & 11 \\ 3 2 1 8 & 6 & 5 10 4 11 7 & 9 \\ 4 10 3 1 9 7 & 6 & 11 & 5 & 2 & 8 \\ 5 & 9 & 11 & 4 1 10 8 7 & 2 & 6 & 3 \\ 6 & 4 10 2 5 1 11 & 9 & 8 & 3 & 7 \\ 7 & 8 & 5 & 11 & 3 & 6 & 1 & 2 & 10 9 & 4 \\ 8 & 5 & 9 & 6 & 2 & 4 & 7 & 1 & 3 & 11 & 10 \\ 9 & 11 & 6 & 10 7 & 3 & 5 & 8 & 1 & 4 2 \\ 10 3 2 7 & 11 & 8 & 4 & 6 & 9 & 1 & 5 \\ 11 & 6 & 4 3 & 8 & 2 & 9 & 5 & 7 & 10 1 \ end {array}} \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 & 8 & 9 & 10 & 11 & 2 & 3 & 4 5 & 6 & 7 \\ 2 & 7 & 5 & 4 & 9 & 3 10 6 & 8 & 11 & 1 \\ 3 & 1 & 8 & 6 & 5 10 4 11 7 & 9 & 2 \\ 4 & 3 & 1 & 9 7 & 6 & 11 & 5 & 2 & 8 & 10 \\ 5 11 & 4 1 10 8 7 & 2 & 6 & 3 & 9 \\ 6 & 10 2 5 1 11 & 9 & 8 & 3 & 7 & 4 \\ 7 & 5 & 11 & 3 & 6 & 1 & 2 & 10 9 & 4 & 8 \\ 8 & 9 & 6 & 2 & 4 & 7 & 1 & 3 & 11 & 10 & 5 \\ 9 & 6 & 10 7 & 3 & 5 & 8 & 1 & 4 & 2 & 11 \ \ 10 2 7 & 11 & 8 & 4 & 6 & 9 & 1 & 5 & 3 \\ 11 & 4 3 & 8 & 2 & 9 & 5 & 7 & 10 1 6 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 & 9 & 10 & 11 & 2 & 3 & 4 5 & 6 7 & 8 \\ 2 5 4 9 & 3 10 6 & 8 & 11 & 1 & 7 \\ 3 & 8 & 6 & 5 10 4 11 7 & 9 & 2 & 1 \\ 4 & 1 & 9 7 & 6 & 11 & 5 & 2 & 8 & 10 & 3 \\ 5 4 1 10 8 7 & 2 & 6 & 3 & 9 & 11 \\ 6 & 2 & 5 & 1 & 11 & 9 & 8 & 3 & 7 & 4 & 10 \\ 7 & 11 & 3 & 6 & 1 & 2 & 10 9 & 4 & 8 & 5 \\ 8 & 6 & 2 & 4 & 7 & 1 & 3 & 11 & 10 & 5 & 9 \\ 9 & 10 7 & 3 & 5 & 8 & 1 & 4 & 2 & 11 & 6 \\ 10 & 7 & 11 & 8 & 4 & 6 & 9 & 1 & 5 3 2 \\ 11 & 3 & 8 & 2 & 9 & 5 & 7 & 10 1 6 & 4 \ end {array}} \ end {bmatrix}}}
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4th
10
2
5
7th
6th
1
2
10
9
4th
8th
5
11
3
8th
4th
7th
1
3
11
10
5
9
6th
2
9
3
5
8th
1
4th
2
11
6th
10
7th
10
8th
4th
6th
9
1
5
3
2
7th
11
11
2
9
5
7th
10
1
6th
4th
3
8th
]
{\ Display style {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 & 10 & 11 & 2 & 3 & 4 5 & 6 7 & 8 & 9 \\ 2 & 4 & 9 & 3 10 6 & 8 & 11 & 1 & 7 & 5 \\ 3 & 6 & 5 10 4 11 7 & 9 & 2 & 1 & 8 \\ 4 & 9 7 & 6 & 11 & 5 & 2 & 8 & 10 3 1 \\ 5 & 1 & 10 & 8 7 & 2 & 6 & 3 & 9 & 11 & 4 \\ 6 & 5 & 1 & 11 & 9 & 8 & 3 & 7 & 4 10 2 \\ 7 & 3 & 6 & 1 & 2 & 10 9 & 4 & 8 & 5 & 11 \\ 8 & 2 & 4 & 7 & 1 & 3 & 11 & 10 & 5 & 9 & 6 \\ 9 & 7 & 3 & 5 & 8 & 1 & 4 & 2 & 11 & 6 & 10 \ \ 10 & 11 & 8 & 4 & 6 & 9 & 1 & 5 3 2 7 \\ 11 & 8 & 2 & 9 & 5 & 7 & 10 1 6 & 4 & 3 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {11} {c}} 1 11 & 2 & 3 & 4 5 & 6 7 & 8 & 9 & 10 \\ 2 & 9 & 3 10 6 & 8 & 11 & 1 & 7 & 5 & 4 \\ 3 5 10 4 11 7 & 9 & 2 & 1 & 8 & 6 \\ 4 & 7 & 6 & 11 & 5 & 2 & 8 & 10 3 1 9 \\ 5 10 8 7 & 2 & 6 & 3 & 9 & 11 & 4 & 1 \\ 6 & 1 & 11 & 9 & 8 & 3 & 7 & 4 10 2 5 \\ 7 & 6 & 1 & 2 & 10 9 & 4 & 8 & 5 & 11 & 3 \\ 8 & 4 & 7 & 1 & 3 & 11 & 10 & 5 & 9 & 6 & 2 \\ 9 & 3 & 5 & 8 & 1 & 4 & 2 & 11 & 6 & 10 & 7 \\ 10 & 8 & 4 & 6 & 9 & 1 & 5 3 2 7 & 11 \\ 11 & 2 & 9 & 5 & 7 & 10 1 6 & 4 3 & 8 \ end {array}} \ end {bmatrix}}}
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:
1 2 13 25 38 55 63 78 84 94 103 131
literature
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.
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