(85,21,5) block plan
The (85,21,5) block plan is a special symmetrical block plan . In order to be able to construct it, this combinatorial problem had to be solved: an empty 85 × 85 matrix was filled with ones in such a way that each row of the matrix contains exactly 21 ones and any two rows have exactly 5 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 = 85, k = 21, λ = 5) 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 = 85, k = 21, λ = 5 and thus the following properties:
- It consists of 85 blocks and 85 points.
- Each block contains exactly 21 points.
- Every 2 blocks intersect in exactly 5 points.
- Each point lies on exactly 21 blocks.
- Each 2 points are connected by exactly 5 blocks.
Existence and characterization
There are at least 213964 non-isomorphic 2- (85,21,5) block plans. Two of these solutions are:
- Solution 1 with the signature 85 · 84. It contains 3570 ovals of the 2nd order.
- Solution 2 with the signature 73x20, 6x21, 1x22, 5x84. It contains 72 ovals of the 4th order.
List of blocks
All the blocks of this block plan are listed here; See this illustration to understand this list
- Solution 1
1 2 8 12 20 23 25 26 28 30 41 42 50 59 66 72 73 76 78 82 85 1 2 3 9 13 21 24 26 27 29 31 42 43 51 60 67 73 74 77 79 83 2 3 4 10 14 22 25 27 28 30 32 43 44 52 61 68 74 75 78 80 84 3 4 5 11 15 23 26 28 29 31 33 44 45 53 62 69 75 76 79 81 85 1 4 5 6 12 16 24 27 29 30 32 34 45 46 54 63 70 76 77 80 82 2 5 6 7 13 17 25 28 30 31 33 35 46 47 55 64 71 77 78 81 83 3 6 7 8 14 18 26 29 31 32 34 36 47 48 56 65 72 78 79 82 84 4 7 8 9 15 19 27 30 32 33 35 37 48 49 57 66 73 79 80 83 85 1 5 8 9 10 16 20 28 31 33 34 36 38 49 50 58 67 74 80 81 84 2 6 9 10 11 17 21 29 32 34 35 37 39 50 51 59 68 75 81 82 85 1 3 7 10 11 12 18 22 30 33 35 36 38 40 51 52 60 69 76 82 83 2 4 8 11 12 13 19 23 31 34 36 37 39 41 52 53 61 70 77 83 84 3 5 9 12 13 14 20 24 32 35 37 38 40 42 53 54 62 71 78 84 85 1 4 6 10 13 14 15 21 25 33 36 38 39 41 43 54 55 63 72 79 85 1 2 5 7 11 14 15 16 22 26 34 37 39 40 42 44 55 56 64 73 80 2 3 6 8 12 15 16 17 23 27 35 38 40 41 43 45 56 57 65 74 81 3 4 7 9 13 16 17 18 24 28 36 39 41 42 44 46 57 58 66 75 82 4 5 8 10 14 17 18 19 25 29 37 40 42 43 45 47 58 59 67 76 83 5 6 9 11 15 18 19 20 26 30 38 41 43 44 46 48 59 60 68 77 84 6 7 10 12 16 19 20 21 27 31 39 42 44 45 47 49 60 61 69 78 85 1 7 8 11 13 17 20 21 22 28 32 40 43 45 46 48 50 61 62 70 79 2 8 9 12 14 18 21 22 23 29 33 41 44 46 47 49 51 62 63 71 80 3 9 10 13 15 19 22 23 24 30 34 42 45 47 48 50 52 63 64 72 81 4 10 11 14 16 20 23 24 25 31 35 43 46 48 49 51 53 64 65 73 82 5 11 12 15 17 21 24 25 26 32 36 44 47 49 50 52 54 65 66 74 83 6 12 13 16 18 22 25 26 27 33 37 45 48 50 51 53 55 66 67 75 84 7 13 14 17 19 23 26 27 28 34 38 46 49 51 52 54 56 67 68 76 85 1 8 14 15 18 20 24 27 28 29 35 39 47 50 52 53 55 57 68 69 77 2 9 15 16 19 21 25 28 29 30 36 40 48 51 53 54 56 58 69 70 78 3 10 16 17 20 22 26 29 30 31 37 41 49 52 54 55 57 59 70 71 79 4 11 17 18 21 23 27 30 31 32 38 42 50 53 55 56 58 60 71 72 80 5 12 18 19 22 24 28 31 32 33 39 43 51 54 56 57 59 61 72 73 81 6 13 19 20 23 25 29 32 33 34 40 44 52 55 57 58 60 62 73 74 82 7 14 20 21 24 26 30 33 34 35 41 45 53 56 58 59 61 63 74 75 83 8 15 21 22 25 27 31 34 35 36 42 46 54 57 59 60 62 64 75 76 84 9 16 22 23 26 28 32 35 36 37 43 47 55 58 60 61 63 65 76 77 85 1 10 17 23 24 27 29 33 36 37 38 44 48 56 59 61 62 64 66 77 78 2 11 18 24 25 28 30 34 37 38 39 45 49 57 60 62 63 65 67 78 79 3 12 19 25 26 29 31 35 38 39 40 46 50 58 61 63 64 66 68 79 80 4 13 20 26 27 30 32 36 39 40 41 47 51 59 62 64 65 67 69 80 81 5 14 21 27 28 31 33 37 40 41 42 48 52 60 63 65 66 68 70 81 82 6 15 22 28 29 32 34 38 41 42 43 49 53 61 64 66 67 69 71 82 83 7 16 23 29 30 33 35 39 42 43 44 50 54 62 65 67 68 70 72 83 84 8 17 24 30 31 34 36 40 43 44 45 51 55 63 66 68 69 71 73 84 85 1 9 18 25 31 32 35 37 41 44 45 46 52 56 64 67 69 70 72 74 85 1 2 10 19 26 32 33 36 38 42 45 46 47 53 57 65 68 70 71 73 75 2 3 11 20 27 33 34 37 39 43 46 47 48 54 58 66 69 71 72 74 76 3 4 12 21 28 34 35 38 40 44 47 48 49 55 59 67 70 72 73 75 77 4 5 13 22 29 35 36 39 41 45 48 49 50 56 60 68 71 73 74 76 78 5 6 14 23 30 36 37 40 42 46 49 50 51 57 61 69 72 74 75 77 79 6 7 15 24 31 37 38 41 43 47 50 51 52 58 62 70 73 75 76 78 80 7 8 16 25 32 38 39 42 44 48 51 52 53 59 63 71 74 76 77 79 81 8 9 17 26 33 39 40 43 45 49 52 53 54 60 64 72 75 77 78 80 82 9 10 18 27 34 40 41 44 46 50 53 54 55 61 65 73 76 78 79 81 83 10 11 19 28 35 41 42 45 47 51 54 55 56 62 66 74 77 79 80 82 84 11 12 20 29 36 42 43 46 48 52 55 56 57 63 67 75 78 80 81 83 85 1 12 13 21 30 37 43 44 47 49 53 56 57 58 64 68 76 79 81 82 84 2 13 14 22 31 38 44 45 48 50 54 57 58 59 65 69 77 80 82 83 85 1 3 14 15 23 32 39 45 46 49 51 55 58 59 60 66 70 78 81 83 84 2 4 15 16 24 33 40 46 47 50 52 56 59 60 61 67 71 79 82 84 85 1 3 5 16 17 25 34 41 47 48 51 53 57 60 61 62 68 72 80 83 85 1 2 4 6 17 18 26 35 42 48 49 52 54 58 61 62 63 69 73 81 84 2 3 5 7 18 19 27 36 43 49 50 53 55 59 62 63 64 70 74 82 85 1 3 4 6 8 19 20 28 37 44 50 51 54 56 60 63 64 65 71 75 83 2 4 5 7 9 20 21 29 38 45 51 52 55 57 61 64 65 66 72 76 84 3 5 6 8 10 21 22 30 39 46 52 53 56 58 62 65 66 67 73 77 85 1 4 6 7 9 11 22 23 31 40 47 53 54 57 59 63 66 67 68 74 78 2 5 7 8 10 12 23 24 32 41 48 54 55 58 60 64 67 68 69 75 79 3 6 8 9 11 13 24 25 33 42 49 55 56 59 61 65 68 69 70 76 80 4 7 9 10 12 14 25 26 34 43 50 56 57 60 62 66 69 70 71 77 81 5 8 10 11 13 15 26 27 35 44 51 57 58 61 63 67 70 71 72 78 82 6 9 11 12 14 16 27 28 36 45 52 58 59 62 64 68 71 72 73 79 83 7 10 12 13 15 17 28 29 37 46 53 59 60 63 65 69 72 73 74 80 84 8 11 13 14 16 18 29 30 38 47 54 60 61 64 66 70 73 74 75 81 85 1 9 12 14 15 17 19 30 31 39 48 55 61 62 65 67 71 74 75 76 82 2 10 13 15 16 18 20 31 32 40 49 56 62 63 66 68 72 75 76 77 83 3 11 14 16 17 19 21 32 33 41 50 57 63 64 67 69 73 76 77 78 84 4 12 15 17 18 20 22 33 34 42 51 58 64 65 68 70 74 77 78 79 85 1 5 13 16 18 19 21 23 34 35 43 52 59 65 66 69 71 75 78 79 80 2 6 14 17 19 20 22 24 35 36 44 53 60 66 67 70 72 76 79 80 81 3 7 15 18 20 21 23 25 36 37 45 54 61 67 68 71 73 77 80 81 82 4 8 16 19 21 22 24 26 37 38 46 55 62 68 69 72 74 78 81 82 83 5 9 17 20 22 23 25 27 38 39 47 56 63 69 70 73 75 79 82 83 84 6 10 18 21 23 24 26 28 39 40 48 57 64 70 71 74 76 80 83 84 85 1 7 11 19 22 24 25 27 29 40 41 49 58 65 71 72 75 77 81 84 85
- Solution 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 1 2 3 4 5 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 1 2 3 4 5 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 1 2 3 4 5 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 1 2 3 4 5 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 1 6 7 8 9 22 23 24 25 38 39 40 41 54 55 56 57 70 71 72 73 1 6 7 8 9 26 27 28 29 42 43 44 45 58 59 60 61 74 75 76 77 1 6 7 8 9 30 31 32 33 46 47 48 49 62 63 64 65 78 79 80 81 1 6 7 8 9 34 35 36 37 50 51 52 53 66 67 68 69 82 83 84 85 1 10 11 12 13 22 23 24 25 42 43 44 45 62 63 64 65 82 83 84 85 1 10 11 12 13 26 27 28 29 38 39 40 41 66 67 68 69 78 79 80 81 1 10 11 12 13 30 31 32 33 50 51 52 53 54 55 56 57 74 75 76 77 1 10 11 12 13 34 35 36 37 46 47 48 49 58 59 60 61 70 71 72 73 1 14 15 16 17 22 23 24 25 50 51 52 53 58 59 60 61 78 79 80 81 1 14 15 16 17 26 27 28 29 46 47 48 49 54 55 56 57 82 83 84 85 1 14 15 16 17 30 31 32 33 42 43 44 45 66 67 68 69 70 71 72 73 1 14 15 16 17 34 35 36 37 38 39 40 41 62 63 64 65 74 75 76 77 1 18 19 20 21 22 23 24 25 46 47 48 49 66 67 68 69 74 75 76 77 1 18 19 20 21 26 27 28 29 50 51 52 53 62 63 64 65 70 71 72 73 1 18 19 20 21 30 31 32 33 38 39 40 41 58 59 60 61 82 83 84 85 1 18 19 20 21 34 35 36 37 42 43 44 45 54 55 56 57 78 79 80 81 2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 2 6 10 14 18 23 29 33 37 39 43 47 51 55 59 63 67 71 75 79 83 2 6 10 14 18 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 2 6 10 14 18 25 27 31 35 41 45 49 53 57 61 65 69 73 77 81 85 2 7 11 15 19 22 26 30 34 39 43 47 51 56 60 64 68 73 77 81 85 2 7 11 15 19 25 27 31 35 40 44 48 52 55 59 63 67 70 74 78 82 2 7 11 15 19 24 28 32 36 41 45 49 53 54 58 62 66 71 75 79 83 2 7 11 15 19 23 29 33 37 38 42 46 50 57 61 65 69 72 76 80 84 2 8 12 16 20 22 26 30 34 40 44 48 52 57 61 65 69 71 75 79 83 2 8 12 16 20 25 27 31 35 39 43 47 51 54 58 62 66 72 76 80 84 2 8 12 16 20 24 28 32 36 38 42 46 50 55 59 63 67 73 77 81 85 2 8 12 16 20 23 29 33 37 41 45 49 53 56 60 64 68 70 74 78 82 2 9 13 17 21 22 26 30 34 41 45 49 53 55 59 63 67 72 76 80 84 2 9 13 17 21 25 27 31 35 38 42 46 50 56 60 64 68 71 75 79 83 2 9 13 17 21 24 28 32 36 39 43 47 51 57 61 65 69 70 74 78 82 2 9 13 17 21 23 29 33 37 40 44 48 52 54 58 62 66 73 77 81 85 3 6 11 17 20 22 27 32 37 40 43 46 53 56 59 62 69 72 75 78 85 3 6 11 17 20 23 28 31 34 39 44 49 50 55 60 65 66 71 76 81 82 3 6 11 17 20 24 29 30 35 38 45 48 51 54 61 64 67 70 77 80 83 3 6 11 17 20 25 26 33 36 41 42 47 52 57 58 63 68 73 74 79 84 3 7 10 16 21 22 27 32 37 39 44 49 50 54 61 64 67 73 74 79 84 3 7 10 16 21 23 28 31 34 40 43 46 53 57 58 63 68 70 77 80 83 3 7 10 16 21 24 29 30 35 41 42 47 52 56 59 62 69 71 76 81 82 3 7 10 16 21 25 26 33 36 38 45 48 51 55 60 65 66 72 75 78 85 3 8 13 15 18 22 27 32 37 38 45 48 51 57 58 63 68 71 76 81 82 3 8 13 15 18 23 28 31 34 41 42 47 52 54 61 64 67 72 75 78 85 3 8 13 15 18 24 29 30 35 40 43 46 53 55 60 65 66 73 74 79 84 3 8 13 15 18 25 26 33 36 39 44 49 50 56 59 62 69 70 77 80 83 3 9 12 14 19 22 27 32 37 41 42 47 52 55 60 65 66 70 77 80 83 3 9 12 14 19 23 28 31 34 38 45 48 51 56 59 62 69 73 74 79 84 3 9 12 14 19 24 29 30 35 39 44 49 50 57 58 63 68 72 75 78 85 3 9 12 14 19 25 26 33 36 40 43 46 53 54 61 64 67 71 76 81 82 4 6 12 15 21 22 28 33 35 40 42 49 51 57 59 64 66 71 74 80 85 4 6 12 15 21 23 27 30 36 39 45 46 52 54 60 63 69 72 77 79 82 4 6 12 15 21 24 26 31 37 38 44 47 53 55 61 62 68 73 76 78 83 4 6 12 15 21 25 29 32 34 41 43 48 50 56 58 65 67 70 75 81 84 4 7 13 14 20 22 28 33 35 39 45 46 52 55 61 62 68 70 75 81 84 4 7 13 14 20 23 27 30 36 40 42 49 51 56 58 65 67 73 76 78 83 4 7 13 14 20 24 26 31 37 41 43 48 50 57 59 64 66 72 77 79 82 4 7 13 14 20 25 29 32 34 38 44 47 53 54 60 63 69 71 74 80 85 4 8 10 17 19 22 28 33 35 38 44 47 53 56 58 65 67 72 77 79 82 4 8 10 17 19 23 27 30 36 41 43 48 50 55 61 62 68 71 74 80 85 4 8 10 17 19 24 26 31 37 40 42 49 51 54 60 63 69 70 75 81 84 4 8 10 17 19 25 29 32 34 39 45 46 52 57 59 64 66 73 76 78 83 4 9 11 16 18 22 28 33 35 41 43 48 50 54 60 63 69 73 76 78 83 4 9 11 16 18 23 27 30 36 38 44 47 53 57 59 64 66 70 75 81 84 4 9 11 16 18 24 26 31 37 39 45 46 52 56 58 65 67 71 74 80 85 4 9 11 16 18 25 29 32 34 40 42 49 51 55 61 62 68 72 77 79 82 5 6 13 16 19 22 29 31 36 40 45 47 50 55 58 64 69 72 74 81 83 5 6 13 16 19 23 26 32 35 39 42 48 53 56 61 63 66 71 77 78 84 5 6 13 16 19 24 27 33 34 38 43 49 52 57 60 62 67 70 76 79 85 5 6 13 16 19 25 28 30 37 41 44 46 51 54 59 65 68 73 75 80 82 5 7 12 17 18 22 29 31 36 39 42 48 53 54 59 65 68 70 76 79 85 5 7 12 17 18 23 26 32 35 40 45 47 50 57 60 62 67 73 75 80 82 5 7 12 17 18 24 27 33 34 41 44 46 51 56 61 63 66 72 74 81 83 5 7 12 17 18 25 28 30 37 38 43 49 52 55 58 64 69 71 77 78 84 5 8 11 14 21 22 29 31 36 38 43 49 52 56 61 63 66 73 75 80 82 5 8 11 14 21 23 26 32 35 41 44 46 51 55 58 64 69 70 76 79 85 5 8 11 14 21 24 27 33 34 40 45 47 50 54 59 65 68 71 77 78 84 5 8 11 14 21 25 28 30 37 39 42 48 53 57 60 62 67 72 74 81 83 5 9 10 15 20 22 29 31 36 41 44 46 51 57 60 62 67 71 77 78 84 5 9 10 15 20 23 26 32 35 38 43 49 52 54 59 65 68 72 74 81 83 5 9 10 15 20 24 27 33 34 39 42 48 53 55 58 64 69 73 75 80 82 5 9 10 15 20 25 28 30 37 40 45 47 50 56 61 63 66 70 76 79 85
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 8 12 20 23 25 26 28 30 41 42 50 59 66 72 73 76 78 82 85
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 each solution to this block diagram:
- Solution 1
1 2
- Solution 2
7 30 53 61
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.