(101,25,6) block plan
The (101,25,6) 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 101 × 101 matrix was filled with ones in such a way that each row of the matrix contains exactly 25 ones and any two rows have exactly 6 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 = 101, k = 25, λ = 6) 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 = 101, k = 25, λ = 6 and thus the following properties:
- It consists of 101 blocks and 101 points.
- Each block contains exactly 25 points.
- Every 2 blocks intersect in exactly 6 points.
- Each point lies on exactly 25 blocks.
- Each 2 points are connected by exactly 6 blocks.
Existence and characterization
There is at least one 2- (101,25,6) 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
- Solution 1
1 5 16 19 24 25 31 36 37 52 54 56 58 68 71 78 79 80 81 84 87 88 92 95 97 2 6 17 20 25 26 32 37 38 53 55 57 59 69 72 79 80 81 82 85 88 89 93 96 98 3 7 18 21 26 27 33 38 39 54 56 58 60 70 73 80 81 82 83 86 89 90 94 97 99 4 8 19 22 27 28 34 39 40 55 57 59 61 71 74 81 82 83 84 87 90 91 95 98 100 5 9 20 23 28 29 35 40 41 56 58 60 62 72 75 82 83 84 85 88 91 92 96 99 101 1 6 10 21 24 29 30 36 41 42 57 59 61 63 73 76 83 84 85 86 89 92 93 97 100 2 7 11 22 25 30 31 37 42 43 58 60 62 64 74 77 84 85 86 87 90 93 94 98 101 1 3 8 12 23 26 31 32 38 43 44 59 61 63 65 75 78 85 86 87 88 91 94 95 99 2 4 9 13 24 27 32 33 39 44 45 60 62 64 66 76 79 86 87 88 89 92 95 96 100 3 5 10 14 25 28 33 34 40 45 46 61 63 65 67 77 80 87 88 89 90 93 96 97 101 1 4 6 11 15 26 29 34 35 41 46 47 62 64 66 68 78 81 88 89 90 91 94 97 98 2 5 7 12 16 27 30 35 36 42 47 48 63 65 67 69 79 82 89 90 91 92 95 98 99 3 6 8 13 17 28 31 36 37 43 48 49 64 66 68 70 80 83 90 91 92 93 96 99 100 4 7 9 14 18 29 32 37 38 44 49 50 65 67 69 71 81 84 91 92 93 94 97 100 101 1 5 8 10 15 19 30 33 38 39 45 50 51 66 68 70 72 82 85 92 93 94 95 98 101 1 2 6 9 11 16 20 31 34 39 40 46 51 52 67 69 71 73 83 86 93 94 95 96 99 2 3 7 10 12 17 21 32 35 40 41 47 52 53 68 70 72 74 84 87 94 95 96 97 100 3 4 8 11 13 18 22 33 36 41 42 48 53 54 69 71 73 75 85 88 95 96 97 98 101 1 4 5 9 12 14 19 23 34 37 42 43 49 54 55 70 72 74 76 86 89 96 97 98 99 2 5 6 10 13 15 20 24 35 38 43 44 50 55 56 71 73 75 77 87 90 97 98 99 100 3 6 7 11 14 16 21 25 36 39 44 45 51 56 57 72 74 76 78 88 91 98 99 100 101 1 4 7 8 12 15 17 22 26 37 40 45 46 52 57 58 73 75 77 79 89 92 99 100 101 1 2 5 8 9 13 16 18 23 27 38 41 46 47 53 58 59 74 76 78 80 90 93 100 101 1 2 3 6 9 10 14 17 19 24 28 39 42 47 48 54 59 60 75 77 79 81 91 94 101 1 2 3 4 7 10 11 15 18 20 25 29 40 43 48 49 55 60 61 76 78 80 82 92 95 2 3 4 5 8 11 12 16 19 21 26 30 41 44 49 50 56 61 62 77 79 81 83 93 96 3 4 5 6 9 12 13 17 20 22 27 31 42 45 50 51 57 62 63 78 80 82 84 94 97 4 5 6 7 10 13 14 18 21 23 28 32 43 46 51 52 58 63 64 79 81 83 85 95 98 5 6 7 8 11 14 15 19 22 24 29 33 44 47 52 53 59 64 65 80 82 84 86 96 99 6 7 8 9 12 15 16 20 23 25 30 34 45 48 53 54 60 65 66 81 83 85 87 97 100 7 8 9 10 13 16 17 21 24 26 31 35 46 49 54 55 61 66 67 82 84 86 88 98 101 1 8 9 10 11 14 17 18 22 25 27 32 36 47 50 55 56 62 67 68 83 85 87 89 99 2 9 10 11 12 15 18 19 23 26 28 33 37 48 51 56 57 63 68 69 84 86 88 90 100 3 10 11 12 13 16 19 20 24 27 29 34 38 49 52 57 58 64 69 70 85 87 89 91 101 1 4 11 12 13 14 17 20 21 25 28 30 35 39 50 53 58 59 65 70 71 86 88 90 92 2 5 12 13 14 15 18 21 22 26 29 31 36 40 51 54 59 60 66 71 72 87 89 91 93 3 6 13 14 15 16 19 22 23 27 30 32 37 41 52 55 60 61 67 72 73 88 90 92 94 4 7 14 15 16 17 20 23 24 28 31 33 38 42 53 56 61 62 68 73 74 89 91 93 95 5 8 15 16 17 18 21 24 25 29 32 34 39 43 54 57 62 63 69 74 75 90 92 94 96 6 9 16 17 18 19 22 25 26 30 33 35 40 44 55 58 63 64 70 75 76 91 93 95 97 7 10 17 18 19 20 23 26 27 31 34 36 41 45 56 59 64 65 71 76 77 92 94 96 98 8 11 18 19 20 21 24 27 28 32 35 37 42 46 57 60 65 66 72 77 78 93 95 97 99 9 12 19 20 21 22 25 28 29 33 36 38 43 47 58 61 66 67 73 78 79 94 96 98 100 10 13 20 21 22 23 26 29 30 34 37 39 44 48 59 62 67 68 74 79 80 95 97 99 101 1 11 14 21 22 23 24 27 30 31 35 38 40 45 49 60 63 68 69 75 80 81 96 98 100 2 12 15 22 23 24 25 28 31 32 36 39 41 46 50 61 64 69 70 76 81 82 97 99 101 1 3 13 16 23 24 25 26 29 32 33 37 40 42 47 51 62 65 70 71 77 82 83 98 100 2 4 14 17 24 25 26 27 30 33 34 38 41 43 48 52 63 66 71 72 78 83 84 99 101 1 3 5 15 18 25 26 27 28 31 34 35 39 42 44 49 53 64 67 72 73 79 84 85 100 2 4 6 16 19 26 27 28 29 32 35 36 40 43 45 50 54 65 68 73 74 80 85 86 101 1 3 5 7 17 20 27 28 29 30 33 36 37 41 44 46 51 55 66 69 74 75 81 86 87 2 4 6 8 18 21 28 29 30 31 34 37 38 42 45 47 52 56 67 70 75 76 82 87 88 3 5 7 9 19 22 29 30 31 32 35 38 39 43 46 48 53 57 68 71 76 77 83 88 89 4 6 8 10 20 23 30 31 32 33 36 39 40 44 47 49 54 58 69 72 77 78 84 89 90 5 7 9 11 21 24 31 32 33 34 37 40 41 45 48 50 55 59 70 73 78 79 85 90 91 6 8 10 12 22 25 32 33 34 35 38 41 42 46 49 51 56 60 71 74 79 80 86 91 92 7 9 11 13 23 26 33 34 35 36 39 42 43 47 50 52 57 61 72 75 80 81 87 92 93 8 10 12 14 24 27 34 35 36 37 40 43 44 48 51 53 58 62 73 76 81 82 88 93 94 9 11 13 15 25 28 35 36 37 38 41 44 45 49 52 54 59 63 74 77 82 83 89 94 95 10 12 14 16 26 29 36 37 38 39 42 45 46 50 53 55 60 64 75 78 83 84 90 95 96 11 13 15 17 27 30 37 38 39 40 43 46 47 51 54 56 61 65 76 79 84 85 91 96 97 12 14 16 18 28 31 38 39 40 41 44 47 48 52 55 57 62 66 77 80 85 86 92 97 98 13 15 17 19 29 32 39 40 41 42 45 48 49 53 56 58 63 67 78 81 86 87 93 98 99 14 16 18 20 30 33 40 41 42 43 46 49 50 54 57 59 64 68 79 82 87 88 94 99 100 15 17 19 21 31 34 41 42 43 44 47 50 51 55 58 60 65 69 80 83 88 89 95 100 101 1 16 18 20 22 32 35 42 43 44 45 48 51 52 56 59 61 66 70 81 84 89 90 96 101 1 2 17 19 21 23 33 36 43 44 45 46 49 52 53 57 60 62 67 71 82 85 90 91 97 2 3 18 20 22 24 34 37 44 45 46 47 50 53 54 58 61 63 68 72 83 86 91 92 98 3 4 19 21 23 25 35 38 45 46 47 48 51 54 55 59 62 64 69 73 84 87 92 93 99 4 5 20 22 24 26 36 39 46 47 48 49 52 55 56 60 63 65 70 74 85 88 93 94 100 5 6 21 23 25 27 37 40 47 48 49 50 53 56 57 61 64 66 71 75 86 89 94 95 101 1 6 7 22 24 26 28 38 41 48 49 50 51 54 57 58 62 65 67 72 76 87 90 95 96 2 7 8 23 25 27 29 39 42 49 50 51 52 55 58 59 63 66 68 73 77 88 91 96 97 3 8 9 24 26 28 30 40 43 50 51 52 53 56 59 60 64 67 69 74 78 89 92 97 98 4 9 10 25 27 29 31 41 44 51 52 53 54 57 60 61 65 68 70 75 79 90 93 98 99 5 10 11 26 28 30 32 42 45 52 53 54 55 58 61 62 66 69 71 76 80 91 94 99 100 6 11 12 27 29 31 33 43 46 53 54 55 56 59 62 63 67 70 72 77 81 92 95 100 101 1 7 12 13 28 30 32 34 44 47 54 55 56 57 60 63 64 68 71 73 78 82 93 96 101 1 2 8 13 14 29 31 33 35 45 48 55 56 57 58 61 64 65 69 72 74 79 83 94 97 2 3 9 14 15 30 32 34 36 46 49 56 57 58 59 62 65 66 70 73 75 80 84 95 98 3 4 10 15 16 31 33 35 37 47 50 57 58 59 60 63 66 67 71 74 76 81 85 96 99 4 5 11 16 17 32 34 36 38 48 51 58 59 60 61 64 67 68 72 75 77 82 86 97 100 5 6 12 17 18 33 35 37 39 49 52 59 60 61 62 65 68 69 73 76 78 83 87 98 101 1 6 7 13 18 19 34 36 38 40 50 53 60 61 62 63 66 69 70 74 77 79 84 88 99 2 7 8 14 19 20 35 37 39 41 51 54 61 62 63 64 67 70 71 75 78 80 85 89 100 3 8 9 15 20 21 36 38 40 42 52 55 62 63 64 65 68 71 72 76 79 81 86 90 101 1 4 9 10 16 21 22 37 39 41 43 53 56 63 64 65 66 69 72 73 77 80 82 87 91 2 5 10 11 17 22 23 38 40 42 44 54 57 64 65 66 67 70 73 74 78 81 83 88 92 3 6 11 12 18 23 24 39 41 43 45 55 58 65 66 67 68 71 74 75 79 82 84 89 93 4 7 12 13 19 24 25 40 42 44 46 56 59 66 67 68 69 72 75 76 80 83 85 90 94 5 8 13 14 20 25 26 41 43 45 47 57 60 67 68 69 70 73 76 77 81 84 86 91 95 6 9 14 15 21 26 27 42 44 46 48 58 61 68 69 70 71 74 77 78 82 85 87 92 96 7 10 15 16 22 27 28 43 45 47 49 59 62 69 70 71 72 75 78 79 83 86 88 93 97 8 11 16 17 23 28 29 44 46 48 50 60 63 70 71 72 73 76 79 80 84 87 89 94 98 9 12 17 18 24 29 30 45 47 49 51 61 64 71 72 73 74 77 80 81 85 88 90 95 99 10 13 18 19 25 30 31 46 48 50 52 62 65 72 73 74 75 78 81 82 86 89 91 96 100 11 14 19 20 26 31 32 47 49 51 53 63 66 73 74 75 76 79 82 83 87 90 92 97 101 1 12 15 20 21 27 32 33 48 50 52 54 64 67 74 75 76 77 80 83 84 88 91 93 98 2 13 16 21 22 28 33 34 49 51 53 55 65 68 75 76 77 78 81 84 85 89 92 94 99 3 14 17 22 23 29 34 35 50 52 54 56 66 69 76 77 78 79 82 85 86 90 93 95 100 4 15 18 23 24 30 35 36 51 53 55 57 67 70 77 78 79 80 83 86 87 91 94 96 101
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 5 16 19 24 25 31 36 37 52 54 56 58 68 71 78 79 80 81 84 87 88 92 95 97
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 for solution 1 of this block diagram:
- Solution 1
1 2 12
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.