The (183,14,1) 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 183 × 183 matrix was filled with ones in such a way that each row of the matrix contains exactly 14 ones and any two rows have exactly 1 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 = 183, k = 14, 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- (183,14,1) block plan is called the projective plane or Desarguean plane of order 13.
properties
This symmetrical block diagram has the parameters v = 183, k = 14, λ = 1 and thus the following properties:
It consists of 183 blocks and 183 points.
Each block contains exactly 14 points.
Every 2 blocks intersect in exactly 1 point.
Each point lies on exactly 14 blocks.
Each 2 points are connected by exactly 1 block.
Existence and characterization
There is at least one 2- (183,14,1) block plan (apart from isomorphism ). 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 13 14
1 15 16 17 18 19 20 21 22 23 24 25 26 27
1 28 29 30 31 32 33 34 35 36 37 38 39 40
1 41 42 43 44 45 46 47 48 49 50 51 52 53
1 54 55 56 57 58 59 60 61 62 63 64 65 66
1 67 68 69 70 71 72 73 74 75 76 77 78 79
1 80 81 82 83 84 85 86 87 88 89 90 91 92
1 93 94 95 96 97 98 99 100 101 102 103 104 105
1 106 107 108 109 110 111 112 113 114 115 116 117 118
1 119 120 121 122 123 124 125 126 127 128 129 130 131
1 132 133 134 135 136 137 138 139 140 141 142 143 144
1 145 146 147 148 149 150 151 152 153 154 155 156 157
1 158 159 160 161 162 163 164 165 166 167 168 169 170
1 171 172 173 174 175 176 177 178 179 180 181 182 183
2 15 28 41 54 67 80 93 106 119 132 145 158 171
2 16 29 42 55 68 81 94 107 120 133 146 159 172
2 17 30 43 56 69 82 95 108 121 134 147 160 173
2 18 31 44 57 70 83 96 109 122 135 148 161 174
2 19 32 45 58 71 84 97 110 123 136 149 162 175
2 20 33 46 59 72 85 98 111 124 137 150 163 176
2 21 34 47 60 73 86 99 112 125 138 151 164 177
2 22 35 48 61 74 87 100 113 126 139 152 165 178
2 23 36 49 62 75 88 101 114 127 140 153 166 179
2 24 37 50 63 76 89 102 115 128 141 154 167 180
2 25 38 51 64 77 90 103 116 129 142 155 168 181
2 26 39 52 65 78 91 104 117 130 143 156 169 182
2 27 40 53 66 79 92 105 118 131 144 157 170 183
3 15 29 43 57 71 85 99 113 127 141 155 169 183
3 16 30 46 64 76 83 105 106 126 143 151 166 175
3 17 33 44 60 78 90 97 107 119 140 157 165 180
3 18 38 47 58 74 92 104 111 121 132 154 159 179
3 19 37 52 61 72 88 94 118 125 135 145 168 173
3 20 31 51 66 75 86 102 108 120 139 149 158 182
3 21 40 45 65 68 89 100 116 122 134 153 163 171
3 22 28 42 59 79 82 103 114 130 136 148 167 177
3 23 35 41 56 73 81 96 117 128 144 150 162 181
3 24 39 49 54 70 87 95 110 131 142 146 164 176
3 25 34 53 63 67 84 101 109 124 133 156 160 178
3 26 36 48 55 77 80 98 115 123 138 147 170 174
3 27 32 50 62 69 91 93 112 129 137 152 161 172
4 15 30 44 58 72 86 100 114 128 142 156 170 172
4 16 33 51 63 70 92 93 113 130 138 153 162 173
4 17 31 47 65 77 84 94 106 127 144 152 167 176
4 18 34 45 61 79 91 98 108 119 141 146 166 181
4 19 39 48 59 75 81 105 112 122 132 155 160 180
4 20 38 53 62 73 89 95 107 126 136 145 169 174
4 21 32 52 55 76 87 103 109 121 140 150 158 183
4 22 29 46 66 69 90 101 117 123 135 154 164 171
4 23 28 43 60 68 83 104 115 131 137 149 168 178
4 24 36 41 57 74 82 97 118 129 133 151 163 182
4 25 40 50 54 71 88 96 111 120 143 147 165 177
4 26 35 42 64 67 85 102 110 125 134 157 161 179
4 27 37 49 56 78 80 99 116 124 139 148 159 175
5 15 31 45 59 73 87 101 115 129 143 157 159 173
5 16 38 50 57 79 80 100 117 125 140 149 160 176
5 17 34 52 64 71 81 93 114 131 139 154 163 174
5 18 32 48 66 78 85 95 106 128 133 153 168 177
5 19 35 46 62 68 92 99 109 119 142 147 167 182
5 20 40 49 60 76 82 94 113 123 132 156 161 181
5 21 39 42 63 74 90 96 108 127 137 145 170 175
5 22 33 53 56 77 88 104 110 122 141 151 158 172
5 23 30 47 55 70 91 102 118 124 136 155 165 171
5 24 28 44 61 69 84 105 116 120 138 150 169 179
5 25 37 41 58 75 83 98 107 130 134 152 164 183
5 26 29 51 54 72 89 97 112 121 144 148 166 178
5 27 36 43 65 67 86 103 111 126 135 146 162 180
6 15 32 46 60 74 88 102 116 130 144 146 160 174
6 16 37 44 66 67 87 104 112 127 136 147 163 181
6 17 39 51 58 68 80 101 118 126 141 150 161 177
6 18 35 53 65 72 82 93 115 120 140 155 164 175
6 19 33 49 55 79 86 96 106 129 134 154 169 178
6 20 36 47 63 69 81 100 110 119 143 148 168 183
6 21 29 50 61 77 83 95 114 124 132 157 162 182
6 22 40 43 64 75 91 97 109 128 138 145 159 176
6 23 34 42 57 78 89 105 111 123 142 152 158 173
6 24 31 48 56 71 92 103 107 125 137 156 166 171
6 25 28 45 62 70 85 94 117 121 139 151 170 180
6 26 38 41 59 76 84 99 108 131 135 153 165 172
6 27 30 52 54 73 90 98 113 122 133 149 167 179
7 15 33 47 61 75 89 103 117 131 133 147 161 175
7 16 31 53 54 74 91 99 114 123 134 150 168 180
7 17 38 45 55 67 88 105 113 128 137 148 164 182
7 18 40 52 59 69 80 102 107 127 142 151 162 178
7 19 36 42 66 73 83 93 116 121 141 156 165 176
7 20 34 50 56 68 87 97 106 130 135 155 170 179
7 21 37 48 64 70 82 101 111 119 144 149 169 172
7 22 30 51 62 78 84 96 115 125 132 146 163 183
7 23 29 44 65 76 92 98 110 129 139 145 160 177
7 24 35 43 58 79 90 94 112 124 143 153 158 174
7 25 32 49 57 72 81 104 108 126 138 157 167 171
7 26 28 46 63 71 86 95 118 122 140 152 159 181
7 27 39 41 60 77 85 100 109 120 136 154 166 173
8 15 34 48 62 76 90 104 118 120 134 148 162 176
8 16 40 41 61 78 86 101 110 121 137 155 167 174
8 17 32 42 54 75 92 100 115 124 135 151 169 181
8 18 39 46 56 67 89 94 114 129 138 149 165 183
8 19 29 53 60 70 80 103 108 128 143 152 163 179
8 20 37 43 55 74 84 93 117 122 142 157 166 177
8 21 35 51 57 69 88 98 106 131 136 156 159 180
8 22 38 49 65 71 83 102 112 119 133 150 170 173
8 23 31 52 63 79 85 97 116 126 132 147 164 172
8 24 30 45 66 77 81 99 111 130 140 145 161 178
8 25 36 44 59 68 91 95 113 125 144 154 158 175
8 26 33 50 58 73 82 105 109 127 139 146 168 171
8 27 28 47 64 72 87 96 107 123 141 153 160 182
9 15 35 49 63 77 91 105 107 121 135 149 163 177
9 16 28 48 65 73 88 97 108 124 142 154 161 183
9 17 29 41 62 79 87 102 111 122 138 156 168 175
9 18 33 43 54 76 81 101 116 125 136 152 170 182
9 19 40 47 57 67 90 95 115 130 139 150 166 172
9 20 30 42 61 71 80 104 109 129 144 153 164 180
9 21 38 44 56 75 85 93 118 123 143 146 167 178
9 22 36 52 58 70 89 99 106 120 137 157 160 181
9 23 39 50 66 72 84 103 113 119 134 151 159 174
9 24 32 53 64 68 86 98 117 127 132 148 165 173
9 25 31 46 55 78 82 100 112 131 141 145 162 179
9 26 37 45 60 69 92 96 114 126 133 155 158 176
9 27 34 51 59 74 83 94 110 128 140 147 169 171
10 15 36 50 64 78 92 94 108 122 136 150 164 178
10 16 35 52 60 75 84 95 111 129 141 148 170 171
10 17 28 49 66 74 89 98 109 125 143 155 162 172
10 18 30 41 63 68 88 103 112 123 139 157 169 176
10 19 34 44 54 77 82 102 117 126 137 153 159 183
10 20 29 48 58 67 91 96 116 131 140 151 167 173
10 21 31 43 62 72 80 105 110 130 133 154 165 181
10 22 39 45 57 76 86 93 107 124 144 147 168 179
10 23 37 53 59 71 90 100 106 121 138 146 161 182
10 24 40 51 55 73 85 104 114 119 135 152 160 175
10 25 33 42 65 69 87 99 118 128 132 149 166 174
10 26 32 47 56 79 83 101 113 120 142 145 163 180
10 27 38 46 61 70 81 97 115 127 134 156 158 177
11 15 37 51 65 79 81 95 109 123 137 151 165 179
11 16 39 47 62 71 82 98 116 128 135 157 158 178
11 17 36 53 61 76 85 96 112 130 142 149 159 171
11 18 28 50 55 75 90 99 110 126 144 156 163 173
11 19 31 41 64 69 89 104 113 124 140 146 170 177
11 20 35 45 54 78 83 103 118 127 138 154 160 172
11 21 30 49 59 67 92 97 117 120 141 152 168 174
11 22 32 44 63 73 80 94 111 131 134 155 166 182
11 23 40 46 58 77 87 93 108 125 133 148 169 180
11 24 38 42 60 72 91 101 106 122 139 147 162 183
11 25 29 52 56 74 86 105 115 119 136 153 161 176
11 26 34 43 66 70 88 100 107 129 132 150 167 175
11 27 33 48 57 68 84 102 114 121 143 145 164 181
12 15 38 52 66 68 82 96 110 124 138 152 166 180
12 16 34 49 58 69 85 103 115 122 144 145 165 182
12 17 40 48 63 72 83 99 117 129 136 146 158 179
12 18 37 42 62 77 86 97 113 131 143 150 160 171
12 19 28 51 56 76 91 100 111 127 133 157 164 174
12 20 32 41 65 70 90 105 114 125 141 147 159 178
12 21 36 46 54 79 84 104 107 128 139 155 161 173
12 22 31 50 60 67 81 98 118 121 142 153 169 175
12 23 33 45 64 74 80 95 112 120 135 156 167 183
12 24 29 47 59 78 88 93 109 126 134 149 170 181
12 25 39 43 61 73 92 102 106 123 140 148 163 172
12 26 30 53 57 75 87 94 116 119 137 154 162 177
12 27 35 44 55 71 89 101 108 130 132 151 168 176
13 15 39 53 55 69 83 97 111 125 139 153 167 181
13 16 36 45 56 72 90 102 109 131 132 152 169 177
13 17 35 50 59 70 86 104 116 123 133 145 166 183
13 18 29 49 64 73 84 100 118 130 137 147 158 180
13 19 38 43 63 78 87 98 114 120 144 151 161 171
13 20 28 52 57 77 92 101 112 128 134 146 165 175
13 21 33 41 66 71 91 94 115 126 142 148 160 179
13 22 37 47 54 68 85 105 108 129 140 156 162 174
13 23 32 51 61 67 82 99 107 122 143 154 170 176
13 24 34 46 65 75 80 96 113 121 136 157 168 172
13 25 30 48 60 79 89 93 110 127 135 150 159 182
13 26 40 44 62 74 81 103 106 124 141 149 164 173
13 27 31 42 58 76 88 95 117 119 138 155 163 178
14 15 40 42 56 70 84 98 112 126 140 154 168 182
14 16 32 43 59 77 89 96 118 119 139 156 164 179
14 17 37 46 57 73 91 103 110 120 132 153 170 178
14 18 36 51 60 71 87 105 117 124 134 145 167 172
14 19 30 50 65 74 85 101 107 131 138 148 158 181
14 20 39 44 64 79 88 99 115 121 133 152 162 171
14 21 28 53 58 78 81 102 113 129 135 147 166 176
14 22 34 41 55 72 92 95 116 127 143 149 161 180
14 23 38 48 54 69 86 94 109 130 141 157 163 175
14 24 33 52 62 67 83 100 108 123 144 155 159 177
14 25 35 47 66 76 80 97 114 122 137 146 169 173
14 26 31 49 61 68 90 93 111 128 136 151 160 183
14 27 29 45 63 75 82 104 106 125 142 150 165 174
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 13 20 21 23 44 61 72 77 86 90 116 122 169
Orthogonal Latin Squares (MOLS)
This order 13 projective plane is equivalent to these 12 MOLS of order 13:
[
1
2
3
4th
5
6th
7th
8th
9
10
11
12
13
2
3
6th
11
10
4th
13
1
8th
12
7th
9
5
3
6th
4th
7th
12
11
5
2
1
9
13
8th
10
4th
11
7th
5
8th
13
12
6th
3
1
10
2
9
5
10
12
8th
6th
9
2
13
7th
4th
1
11
3
6th
4th
11
13
9
7th
10
3
2
8th
5
1
12
7th
13
5
12
2
10
8th
11
4th
3
9
6th
1
8th
1
2
6th
13
3
11
9
12
5
4th
10
7th
9
8th
1
3
7th
2
4th
12
10
13
6th
5
11
10
12
9
1
4th
8th
3
5
13
11
2
7th
6th
11
7th
13
10
1
5
9
4th
6th
2
12
3
8th
12
9
8th
2
11
1
6th
10
5
7th
3
13
4th
13
5
10
9
3
12
1
7th
11
6th
8th
4th
2
]
[
1
3
4th
5
6th
7th
8th
9
10
11
12
13
2
2
6th
11
10
4th
13
1
8th
12
7th
9
5
3
3
4th
7th
12
11
5
2
1
9
13
8th
10
6th
4th
7th
5
8th
13
12
6th
3
1
10
2
9
11
5
12
8th
6th
9
2
13
7th
4th
1
11
3
10
6th
11
13
9
7th
10
3
2
8th
5
1
12
4th
7th
5
12
2
10
8th
11
4th
3
9
6th
1
13
8th
2
6th
13
3
11
9
12
5
4th
10
7th
1
9
1
3
7th
2
4th
12
10
13
6th
5
11
8th
10
9
1
4th
8th
3
5
13
11
2
7th
6th
12
11
13
10
1
5
9
4th
6th
2
12
3
8th
7th
12
8th
2
11
1
6th
10
5
7th
3
13
4th
9
13
10
9
3
12
1
7th
11
6th
8th
4th
2
5
]
{\ Display style {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 2 & 3 & 4 5 & 6 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ 2 & 3 & 6 & 11 & 10 & 4 & 13 & 1 & 8 & 12 7 & 9 & 5 \\ 3 & 6 & 4 & 7 & 12 & 11 & 5 & 2 & 1 & 9 & 13 & 8 & 10 \\ 4 & 11 & 7 & 5 & 8 & 13 & 12 & 6 & 3 & 1 10 2 9 \\ 5 & 10 & 12 & 8 & 6 & 9 & 2 & 13 7 & 4 & 1 & 11 & 3 \\ 6 & 4 & 11 & 13 & 9 7 & 10 3 2 8 & 5 & 1 & 12 \\ 7 & 13 & 5 & 12 & 2 & 10 8 & 11 & 4 3 & 9 & 6 & 1 \\ 8 & 1 & 2 & 6 & 13 & 3 & 11 & 9 & 12 & 5 4 10 7 \\ 9 & 8 & 1 & 3 & 7 & 2 & 4 & 12 & 10 & 13 & 6 & 5 & 11 \ \ 10 12 & 9 & 1 & 4 & 8 & 3 & 5 & 13 & 11 & 2 & 7 & 6 \\ 11 & 7 & 13 & 10 & 1 & 5 & 9 & 4 & 6 & 2 & 12 & 3 & 8 \\ 12 & 9 & 8 & 2 & 11 & 1 & 6 & 10 5 & 7 & 3 & 13 & 4 \\ 13 & 5 & 10 9 & 3 & 12 & 1 & 7 & 11 & 6 & 8 & 4 & 2 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 3 & 4 5 & 6 7 & 8 & 9 & 10 & 11 & 12 & 13 & 2 \\ 2 & 6 & 11 & 10 & 4 & 13 & 1 & 8 & 12 7 & 9 & 5 & 3 \\ 3 4 7 & 12 & 11 & 5 & 2 & 1 & 9 & 13 & 8 & 10 6 \\ 4 & 7 & 5 & 8 & 13 & 12 & 6 & 3 & 1 10 2 9 & 11 \\ 5 & 12 & 8 & 6 & 9 & 2 & 13 7 & 4 & 1 & 11 & 3 & 10 \\ 6 & 11 & 13 & 9 7 & 10 3 2 8 & 5 & 1 & 12 & 4 \\ 7 & 5 & 12 & 2 & 10 8 & 11 & 4 3 & 9 & 6 & 1 & 13 \\ 8 & 2 & 6 & 13 & 3 & 11 & 9 & 12 & 5 4 10 7 & 1 \\ 9 & 1 & 3 & 7 & 2 & 4 & 12 & 10 & 13 & 6 & 5 & 11 & 8 \\ 10 9 & 1 & 4 & 8 & 3 & 5 & 13 & 11 & 2 & 7 & 6 & 12 \\ 11 & 13 & 10 & 1 & 5 & 9 & 4 & 6 & 2 & 12 & 3 & 8 & 7 \\ 12 & 8 & 2 & 11 & 1 & 6 & 10 5 & 7 & 3 & 13 & 4 & 9 \\ 13 & 10 9 & 3 & 12 & 1 & 7 & 11 & 6 & 8 & 4 2 5 \ end {array}} \ end {bmatrix}}}
[
1
4th
5
6th
7th
8th
9
10
11
12
13
2
3
2
11
10
4th
13
1
8th
12
7th
9
5
3
6th
3
7th
12
11
5
2
1
9
13
8th
10
6th
4th
4th
5
8th
13
12
6th
3
1
10
2
9
11
7th
5
8th
6th
9
2
13
7th
4th
1
11
3
10
12
6th
13
9
7th
10
3
2
8th
5
1
12
4th
11
7th
12
2
10
8th
11
4th
3
9
6th
1
13
5
8th
6th
13
3
11
9
12
5
4th
10
7th
1
2
9
3
7th
2
4th
12
10
13
6th
5
11
8th
1
10
1
4th
8th
3
5
13
11
2
7th
6th
12
9
11
10
1
5
9
4th
6th
2
12
3
8th
7th
13
12
2
11
1
6th
10
5
7th
3
13
4th
9
8th
13
9
3
12
1
7th
11
6th
8th
4th
2
5
10
]
[
1
5
6th
7th
8th
9
10
11
12
13
2
3
4th
2
10
4th
13
1
8th
12
7th
9
5
3
6th
11
3
12
11
5
2
1
9
13
8th
10
6th
4th
7th
4th
8th
13
12
6th
3
1
10
2
9
11
7th
5
5
6th
9
2
13
7th
4th
1
11
3
10
12
8th
6th
9
7th
10
3
2
8th
5
1
12
4th
11
13
7th
2
10
8th
11
4th
3
9
6th
1
13
5
12
8th
13
3
11
9
12
5
4th
10
7th
1
2
6th
9
7th
2
4th
12
10
13
6th
5
11
8th
1
3
10
4th
8th
3
5
13
11
2
7th
6th
12
9
1
11
1
5
9
4th
6th
2
12
3
8th
7th
13
10
12
11
1
6th
10
5
7th
3
13
4th
9
8th
2
13
3
12
1
7th
11
6th
8th
4th
2
5
10
9
]
{\ Display style {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 4 5 & 6 7 & 8 & 9 & 10 & 11 & 12 & 13 & 2 & 3 \\ 2 & 11 & 10 & 4 & 13 & 1 & 8 & 12 7 & 9 & 5 & 3 & 6 \\ 3 & 7 & 12 & 11 & 5 & 2 & 1 & 9 & 13 & 8 & 10 6 & 4 \\ 4 & 5 & 8 & 13 & 12 & 6 & 3 & 1 10 2 9 & 11 & 7 \\ 5 & 8 & 6 & 9 & 2 & 13 7 & 4 & 1 & 11 & 3 & 10 & 12 \\ 6 & 13 & 9 7 & 10 3 2 8 & 5 & 1 & 12 & 4 & 11 \\ 7 & 12 & 2 & 10 8 & 11 & 4 3 & 9 & 6 & 1 & 13 & 5 \\ 8 & 6 & 13 & 3 & 11 & 9 & 12 & 5 4 10 7 & 1 & 2 \\ 9 & 3 & 7 & 2 & 4 & 12 & 10 & 13 & 6 & 5 & 11 & 8 & 1 \ \ 10 1 4 8 & 3 & 5 & 13 & 11 & 2 & 7 & 6 & 12 & 9 \\ 11 & 10 1 5 9 & 4 & 6 & 2 & 12 & 3 & 8 7 & 13 \\ 12 & 2 & 11 & 1 & 6 & 10 5 & 7 & 3 & 13 & 4 & 9 & 8 \\ 13 & 9 & 3 & 12 & 1 & 7 & 11 & 6 & 8 & 4 2 5 10 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 5 & 6 7 & 8 & 9 & 10 & 11 & 12 & 13 & 2 & 3 & 4 \\ 2 10 4 13 & 1 & 8 & 12 7 & 9 & 5 & 3 & 6 & 11 \\ 3 & 12 & 11 & 5 & 2 & 1 & 9 & 13 & 8 & 10 6 & 4 & 7 \\ 4 & 8 & 13 & 12 & 6 & 3 & 1 10 2 9 & 11 & 7 & 5 \\ 5 & 6 & 9 & 2 & 13 7 & 4 & 1 & 11 & 3 & 10 & 12 & 8 \\ 6 & 9 7 & 10 3 2 8 & 5 & 1 & 12 & 4 & 11 & 13 \\ 7 & 2 & 10 8 & 11 & 4 3 & 9 & 6 & 1 & 13 & 5 & 12 \\ 8 & 13 & 3 & 11 & 9 & 12 & 5 4 10 7 & 1 & 2 & 6 \\ 9 7 & 2 & 4 & 12 & 10 & 13 & 6 & 5 & 11 & 8 & 1 & 3 \\ 10 4 8 & 3 & 5 & 13 & 11 & 2 & 7 & 6 & 12 & 9 & 1 \\ 11 & 1 & 5 & 9 & 4 & 6 & 2 & 12 & 3 & 8 7 & 13 & 10 \\ 12 & 11 & 1 & 6 & 10 5 & 7 & 3 & 13 & 4 & 9 & 8 & 2 \\ 13 & 3 & 12 & 1 & 7 & 11 & 6 & 8 & 4 2 5 10 9 \ end {array}} \ end {bmatrix}}}
[
1
6th
7th
8th
9
10
11
12
13
2
3
4th
5
2
4th
13
1
8th
12
7th
9
5
3
6th
11
10
3
11
5
2
1
9
13
8th
10
6th
4th
7th
12
4th
13
12
6th
3
1
10
2
9
11
7th
5
8th
5
9
2
13
7th
4th
1
11
3
10
12
8th
6th
6th
7th
10
3
2
8th
5
1
12
4th
11
13
9
7th
10
8th
11
4th
3
9
6th
1
13
5
12
2
8th
3
11
9
12
5
4th
10
7th
1
2
6th
13
9
2
4th
12
10
13
6th
5
11
8th
1
3
7th
10
8th
3
5
13
11
2
7th
6th
12
9
1
4th
11
5
9
4th
6th
2
12
3
8th
7th
13
10
1
12
1
6th
10
5
7th
3
13
4th
9
8th
2
11
13
12
1
7th
11
6th
8th
4th
2
5
10
9
3
]
[
1
7th
8th
9
10
11
12
13
2
3
4th
5
6th
2
13
1
8th
12
7th
9
5
3
6th
11
10
4th
3
5
2
1
9
13
8th
10
6th
4th
7th
12
11
4th
12
6th
3
1
10
2
9
11
7th
5
8th
13
5
2
13
7th
4th
1
11
3
10
12
8th
6th
9
6th
10
3
2
8th
5
1
12
4th
11
13
9
7th
7th
8th
11
4th
3
9
6th
1
13
5
12
2
10
8th
11
9
12
5
4th
10
7th
1
2
6th
13
3
9
4th
12
10
13
6th
5
11
8th
1
3
7th
2
10
3
5
13
11
2
7th
6th
12
9
1
4th
8th
11
9
4th
6th
2
12
3
8th
7th
13
10
1
5
12
6th
10
5
7th
3
13
4th
9
8th
2
11
1
13
1
7th
11
6th
8th
4th
2
5
10
9
3
12
]
{\ Display style {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 6 7 & 8 & 9 & 10 & 11 & 12 & 13 & 2 & 3 & 4 & 5 \\ 2 & 4 & 13 & 1 & 8 & 12 7 & 9 & 5 & 3 & 6 & 11 & 10 \\ 3 & 11 & 5 & 2 & 1 & 9 & 13 & 8 & 10 6 & 4 & 7 & 12 \\ 4 & 13 & 12 & 6 & 3 & 1 10 2 9 & 11 & 7 & 5 & 8 \\ 5 & 9 & 2 & 13 7 & 4 & 1 & 11 & 3 & 10 & 12 & 8 & 6 \\ 6 & 7 & 10 3 2 8 & 5 & 1 & 12 & 4 & 11 & 13 & 9 \\ 7 & 10 8 & 11 & 4 3 & 9 & 6 & 1 & 13 & 5 & 12 & 2 \\ 8 & 3 & 11 & 9 & 12 & 5 4 10 7 & 1 & 2 & 6 & 13 \\ 9 & 2 & 4 & 12 & 10 & 13 & 6 & 5 & 11 & 8 & 1 & 3 & 7 \ \ 10 & 8 & 3 & 5 & 13 & 11 & 2 & 7 & 6 & 12 & 9 & 1 & 4 \\ 11 & 5 & 9 & 4 & 6 & 2 & 12 & 3 & 8 7 & 13 & 10 & 1 \\ 12 & 1 & 6 & 10 5 & 7 & 3 & 13 & 4 & 9 & 8 & 2 & 11 \\ 13 & 12 & 1 & 7 & 11 & 6 & 8 & 4 2 5 10 9 & 3 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 7 & 8 & 9 & 10 & 11 & 12 & 13 & 2 & 3 & 4 5 & 6 \\ 2 & 13 & 1 & 8 & 12 7 & 9 & 5 & 3 & 6 & 11 & 10 & 4 \\ 3 & 5 & 2 & 1 & 9 & 13 & 8 & 10 6 & 4 & 7 & 12 & 11 \\ 4 & 12 & 6 & 3 & 1 10 2 9 & 11 & 7 & 5 & 8 & 13 \\ 5 & 2 & 13 7 & 4 & 1 & 11 & 3 & 10 & 12 & 8 & 6 & 9 \\ 6 & 10 3 2 8 & 5 & 1 & 12 & 4 & 11 & 13 & 9 & 7 \\ 7 & 8 & 11 & 4 3 & 9 & 6 & 1 & 13 & 5 & 12 & 2 & 10 \\ 8 & 11 & 9 & 12 & 5 4 10 7 & 1 & 2 & 6 & 13 & 3 \\ 9 & 4 & 12 & 10 & 13 & 6 & 5 & 11 & 8 & 1 & 3 & 7 & 2 \\ 10 & 3 & 5 & 13 & 11 & 2 & 7 & 6 & 12 & 9 & 1 & 4 & 8 \\ 11 & 9 & 4 & 6 & 2 & 12 & 3 & 8 7 & 13 & 10 & 1 & 5 \\ 12 & 6 & 10 5 & 7 & 3 & 13 & 4 & 9 & 8 & 2 & 11 & 1 \\ 13 & 1 & 7 & 11 & 6 & 8 & 4 2 5 10 9 & 3 & 12 \ end {array}} \ end {bmatrix}}}
[
1
8th
9
10
11
12
13
2
3
4th
5
6th
7th
2
1
8th
12
7th
9
5
3
6th
11
10
4th
13
3
2
1
9
13
8th
10
6th
4th
7th
12
11
5
4th
6th
3
1
10
2
9
11
7th
5
8th
13
12
5
13
7th
4th
1
11
3
10
12
8th
6th
9
2
6th
3
2
8th
5
1
12
4th
11
13
9
7th
10
7th
11
4th
3
9
6th
1
13
5
12
2
10
8th
8th
9
12
5
4th
10
7th
1
2
6th
13
3
11
9
12
10
13
6th
5
11
8th
1
3
7th
2
4th
10
5
13
11
2
7th
6th
12
9
1
4th
8th
3
11
4th
6th
2
12
3
8th
7th
13
10
1
5
9
12
10
5
7th
3
13
4th
9
8th
2
11
1
6th
13
7th
11
6th
8th
4th
2
5
10
9
3
12
1
]
[
1
9
10
11
12
13
2
3
4th
5
6th
7th
8th
2
8th
12
7th
9
5
3
6th
11
10
4th
13
1
3
1
9
13
8th
10
6th
4th
7th
12
11
5
2
4th
3
1
10
2
9
11
7th
5
8th
13
12
6th
5
7th
4th
1
11
3
10
12
8th
6th
9
2
13
6th
2
8th
5
1
12
4th
11
13
9
7th
10
3
7th
4th
3
9
6th
1
13
5
12
2
10
8th
11
8th
12
5
4th
10
7th
1
2
6th
13
3
11
9
9
10
13
6th
5
11
8th
1
3
7th
2
4th
12
10
13
11
2
7th
6th
12
9
1
4th
8th
3
5
11
6th
2
12
3
8th
7th
13
10
1
5
9
4th
12
5
7th
3
13
4th
9
8th
2
11
1
6th
10
13
11
6th
8th
4th
2
5
10
9
3
12
1
7th
]
{\ Display style {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 8 & 9 & 10 & 11 & 12 & 13 & 2 & 3 & 4 5 & 6 & 7 \\ 2 & 1 & 8 & 12 7 & 9 & 5 & 3 & 6 & 11 & 10 & 4 & 13 \\ 3 & 2 & 1 & 9 & 13 & 8 & 10 6 & 4 & 7 & 12 & 11 & 5 \\ 4 & 6 & 3 & 1 10 2 9 & 11 & 7 & 5 & 8 & 13 & 12 \\ 5 13 7 & 4 & 1 & 11 & 3 & 10 & 12 & 8 & 6 & 9 & 2 \\ 6 & 3 2 & 8 & 5 & 1 & 12 & 4 & 11 & 13 & 9 7 & 10 \\ 7 & 11 & 4 3 & 9 & 6 & 1 & 13 & 5 & 12 & 2 & 10 8 \\ 8 & 9 & 12 & 5 4 10 7 & 1 & 2 & 6 & 13 & 3 & 11 \\ 9 & 12 & 10 & 13 & 6 & 5 & 11 & 8 & 1 & 3 & 7 & 2 & 4 \ \ 10 & 5 & 13 & 11 & 2 & 7 & 6 & 12 & 9 & 1 & 4 & 8 & 3 \\ 11 & 4 & 6 & 2 & 12 & 3 & 8 7 & 13 & 10 & 1 & 5 & 9 \\ 12 & 10 5 & 7 & 3 & 13 & 4 & 9 & 8 & 2 & 11 & 1 & 6 \\ 13 & 7 & 11 & 6 & 8 & 4 2 5 10 9 & 3 & 12 & 1 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 9 & 10 & 11 & 12 & 13 & 2 & 3 & 4 5 & 6 7 & 8 \\ 2 & 8 & 12 7 & 9 & 5 & 3 & 6 & 11 & 10 & 4 & 13 & 1 \\ 3 & 1 & 9 & 13 & 8 & 10 6 & 4 & 7 & 12 & 11 & 5 & 2 \\ 4 & 3 & 1 & 10 & 2 & 9 & 11 & 7 & 5 & 8 & 13 & 12 & 6 \\ 5 & 7 & 4 & 1 & 11 & 3 & 10 & 12 & 8 & 6 & 9 & 2 & 13 \\ 6 & 2 & 8 & 5 & 1 & 12 & 4 & 11 & 13 & 9 7 & 10 3 \\ 7 & 4 3 & 9 & 6 & 1 & 13 & 5 & 12 & 2 & 10 8 & 11 \\ 8 & 12 & 5 4 10 7 & 1 & 2 & 6 & 13 & 3 & 11 & 9 \\ 9 & 10 & 13 & 6 & 5 & 11 & 8 & 1 & 3 & 7 & 2 & 4 & 12 \\ 10 & 13 & 11 & 2 & 7 & 6 & 12 & 9 & 1 & 4 & 8 & 3 & 5 \\ 11 & 6 & 2 & 12 & 3 & 8 7 & 13 & 10 & 1 & 5 & 9 & 4 \\ 12 & 5 & 7 & 3 & 13 & 4 & 9 & 8 & 2 & 11 & 1 & 6 & 10 \\ 13 & 11 & 6 & 8 & 4 2 5 10 9 & 3 & 12 & 1 & 7 \ end {array}} \ end {bmatrix}}}
[
1
10
11
12
13
2
3
4th
5
6th
7th
8th
9
2
12
7th
9
5
3
6th
11
10
4th
13
1
8th
3
9
13
8th
10
6th
4th
7th
12
11
5
2
1
4th
1
10
2
9
11
7th
5
8th
13
12
6th
3
5
4th
1
11
3
10
12
8th
6th
9
2
13
7th
6th
8th
5
1
12
4th
11
13
9
7th
10
3
2
7th
3
9
6th
1
13
5
12
2
10
8th
11
4th
8th
5
4th
10
7th
1
2
6th
13
3
11
9
12
9
13
6th
5
11
8th
1
3
7th
2
4th
12
10
10
11
2
7th
6th
12
9
1
4th
8th
3
5
13
11
2
12
3
8th
7th
13
10
1
5
9
4th
6th
12
7th
3
13
4th
9
8th
2
11
1
6th
10
5
13
6th
8th
4th
2
5
10
9
3
12
1
7th
11
]
[
1
11
12
13
2
3
4th
5
6th
7th
8th
9
10
2
7th
9
5
3
6th
11
10
4th
13
1
8th
12
3
13
8th
10
6th
4th
7th
12
11
5
2
1
9
4th
10
2
9
11
7th
5
8th
13
12
6th
3
1
5
1
11
3
10
12
8th
6th
9
2
13
7th
4th
6th
5
1
12
4th
11
13
9
7th
10
3
2
8th
7th
9
6th
1
13
5
12
2
10
8th
11
4th
3
8th
4th
10
7th
1
2
6th
13
3
11
9
12
5
9
6th
5
11
8th
1
3
7th
2
4th
12
10
13
10
2
7th
6th
12
9
1
4th
8th
3
5
13
11
11
12
3
8th
7th
13
10
1
5
9
4th
6th
2
12
3
13
4th
9
8th
2
11
1
6th
10
5
7th
13
8th
4th
2
5
10
9
3
12
1
7th
11
6th
]
{\ Display style {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 10 & 11 & 12 & 13 & 2 & 3 & 4 5 & 6 7 & 8 & 9 \\ 2 12 7 & 9 & 5 & 3 & 6 & 11 & 10 & 4 & 13 & 1 & 8 \\ 3 & 9 & 13 & 8 & 10 6 & 4 & 7 & 12 & 11 & 5 & 2 & 1 \\ 4 & 1 & 10 & 2 & 9 & 11 & 7 & 5 & 8 & 13 & 12 & 6 & 3 \\ 5 4 1 11 & 3 & 10 & 12 & 8 & 6 & 9 & 2 & 13 & 7 \\ 6 & 8 & 5 & 1 & 12 & 4 & 11 & 13 & 9 7 & 10 3 2 \\ 7 & 3 & 9 & 6 & 1 & 13 & 5 & 12 & 2 & 10 8 & 11 & 4 \\ 8 & 5 4 10 7 & 1 & 2 & 6 & 13 & 3 & 11 & 9 & 12 \\ 9 & 13 & 6 & 5 & 11 & 8 & 1 & 3 & 7 & 2 & 4 & 12 & 10 \ \ 10 & 11 & 2 & 7 & 6 & 12 & 9 & 1 & 4 & 8 & 3 & 5 & 13 \\ 11 & 2 & 12 & 3 & 8 7 & 13 & 10 & 1 & 5 & 9 & 4 & 6 \\ 12 & 7 & 3 & 13 & 4 & 9 & 8 & 2 & 11 & 1 & 6 & 10 5 \\ 13 & 6 & 8 & 4 2 5 10 9 & 3 & 12 & 1 & 7 & 11 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 11 & 12 & 13 & 2 & 3 & 4 5 & 6 7 & 8 & 9 & 10 \\ 2 & 7 & 9 & 5 & 3 & 6 & 11 & 10 & 4 & 13 & 1 & 8 & 12 \\ 3 & 13 & 8 & 10 6 & 4 & 7 & 12 & 11 & 5 & 2 & 1 & 9 \\ 4 10 2 9 & 11 & 7 & 5 & 8 & 13 & 12 & 6 & 3 & 1 \\ 5 & 1 & 11 & 3 & 10 & 12 & 8 & 6 & 9 & 2 & 13 7 & 4 \\ 6 & 5 & 1 & 12 & 4 & 11 & 13 & 9 7 & 10 3 2 8 \\ 7 & 9 & 6 & 1 & 13 & 5 & 12 & 2 & 10 8 & 11 & 4 & 3 \\ 8 & 4 10 7 & 1 & 2 & 6 & 13 & 3 & 11 & 9 & 12 & 5 \\ 9 & 6 & 5 & 11 & 8 & 1 & 3 & 7 & 2 & 4 & 12 & 10 & 13 \\ 10 2 7 & 6 & 12 & 9 & 1 & 4 & 8 & 3 & 5 & 13 & 11 \\ 11 & 12 & 3 & 8 7 & 13 & 10 & 1 & 5 & 9 & 4 & 6 & 2 \\ 12 & 3 & 13 & 4 & 9 & 8 & 2 & 11 & 1 & 6 & 10 5 & 7 \\ 13 & 8 & 4 2 5 10 9 & 3 & 12 & 1 & 7 & 11 & 6 \ end {array}} \ end {bmatrix}}}
[
1
12
13
2
3
4th
5
6th
7th
8th
9
10
11
2
9
5
3
6th
11
10
4th
13
1
8th
12
7th
3
8th
10
6th
4th
7th
12
11
5
2
1
9
13
4th
2
9
11
7th
5
8th
13
12
6th
3
1
10
5
11
3
10
12
8th
6th
9
2
13
7th
4th
1
6th
1
12
4th
11
13
9
7th
10
3
2
8th
5
7th
6th
1
13
5
12
2
10
8th
11
4th
3
9
8th
10
7th
1
2
6th
13
3
11
9
12
5
4th
9
5
11
8th
1
3
7th
2
4th
12
10
13
6th
10
7th
6th
12
9
1
4th
8th
3
5
13
11
2
11
3
8th
7th
13
10
1
5
9
4th
6th
2
12
12
13
4th
9
8th
2
11
1
6th
10
5
7th
3
13
4th
2
5
10
9
3
12
1
7th
11
6th
8th
]
[
1
13
2
3
4th
5
6th
7th
8th
9
10
11
12
2
5
3
6th
11
10
4th
13
1
8th
12
7th
9
3
10
6th
4th
7th
12
11
5
2
1
9
13
8th
4th
9
11
7th
5
8th
13
12
6th
3
1
10
2
5
3
10
12
8th
6th
9
2
13
7th
4th
1
11
6th
12
4th
11
13
9
7th
10
3
2
8th
5
1
7th
1
13
5
12
2
10
8th
11
4th
3
9
6th
8th
7th
1
2
6th
13
3
11
9
12
5
4th
10
9
11
8th
1
3
7th
2
4th
12
10
13
6th
5
10
6th
12
9
1
4th
8th
3
5
13
11
2
7th
11
8th
7th
13
10
1
5
9
4th
6th
2
12
3
12
4th
9
8th
2
11
1
6th
10
5
7th
3
13
13
2
5
10
9
3
12
1
7th
11
6th
8th
4th
]
{\ Display style {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 & 12 & 13 & 2 & 3 & 4 5 & 6 7 & 8 & 9 & 10 & 11 \\ 2 & 9 & 5 & 3 & 6 & 11 & 10 & 4 & 13 & 1 & 8 & 12 & 7 \\ 3 & 8 & 10 6 & 4 & 7 & 12 & 11 & 5 & 2 & 1 & 9 & 13 \\ 4 & 2 & 9 & 11 & 7 & 5 & 8 & 13 & 12 & 6 & 3 & 1 & 10 \\ 5 & 11 & 3 & 10 & 12 & 8 & 6 & 9 & 2 & 13 7 & 4 & 1 \\ 6 & 1 & 12 & 4 & 11 & 13 & 9 7 & 10 3 2 8 & 5 \\ 7 & 6 & 1 & 13 & 5 & 12 & 2 & 10 8 & 11 & 4 3 & 9 \\ 8 & 10 7 & 1 & 2 & 6 & 13 & 3 & 11 & 9 & 12 & 5 & 4 \\ 9 & 5 & 11 & 8 & 1 & 3 & 7 & 2 & 4 & 12 & 10 & 13 & 6 \ \ 10 & 7 & 6 & 12 & 9 & 1 & 4 & 8 & 3 & 5 & 13 & 11 & 2 \\ 11 & 3 & 8 7 & 13 & 10 & 1 & 5 & 9 & 4 & 6 & 2 & 12 \\ 12 & 13 & 4 & 9 & 8 & 2 & 11 & 1 & 6 & 10 5 & 7 & 3 \\ 13 & 4 2 5 10 9 & 3 & 12 & 1 & 7 & 11 & 6 & 8 \ end {array}} \ end {bmatrix}} \ quad {\ begin {bmatrix} {\ begin {array} {* {13} {c}} 1 13 & 2 & 3 & 4 5 & 6 7 & 8 & 9 & 10 & 11 & 12 \\ 2 5 3 6 & 11 & 10 & 4 & 13 & 1 & 8 & 12 7 & 9 \\ 3 10 6 & 4 & 7 & 12 & 11 & 5 & 2 & 1 & 9 & 13 & 8 \\ 4 & 9 & 11 & 7 & 5 & 8 & 13 & 12 & 6 & 3 & 1 10 2 \\ 5 & 3 & 10 & 12 & 8 & 6 & 9 & 2 & 13 7 & 4 & 1 & 11 \\ 6 & 12 & 4 & 11 & 13 & 9 7 & 10 3 2 8 & 5 & 1 \\ 7 & 1 & 13 & 5 & 12 & 2 & 10 8 & 11 & 4 3 & 9 & 6 \\ 8 & 7 & 1 & 2 & 6 & 13 & 3 & 11 & 9 & 12 & 5 4 10 \\ 9 & 11 & 8 & 1 & 3 & 7 & 2 & 4 & 12 & 10 & 13 & 6 & 5 \\ 10 6 & 12 & 9 & 1 & 4 & 8 & 3 & 5 & 13 & 11 & 2 & 7 \\ 11 & 8 7 & 13 & 10 & 1 & 5 & 9 & 4 & 6 & 2 & 12 & 3 \\ 12 & 4 & 9 & 8 & 2 & 11 & 1 & 6 & 10 5 & 7 & 3 & 13 \\ 13 & 2 & 5 & 10 9 & 3 & 12 & 1 & 7 & 11 & 6 & 8 & 4 \ 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 15 29 44 64 69 102 112 131 143 149 165 176
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.
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">