(78,22,6) block plan
The (78,22,6) 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 78 × 78 matrix was filled with ones in such a way that each row of the matrix contains exactly 22 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 = 78, k = 22, 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 = 78, k = 22, λ = 6 and thus the following properties:
- It consists of 78 blocks and 78 points.
- Each block contains exactly 22 points.
- Every 2 blocks intersect in exactly 6 points.
- Each point lies on exactly 22 blocks.
- Each 2 points are connected by exactly 6 blocks.
Existence and characterization
There are at least three non-isomorphic 2- (78,22,6) block plans. One of these solutions 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 5 10 11 13 16 19 20 29 32 33 44 50 52 57 63 65 70 76 78 2 4 10 14 16 19 20 21 22 25 31 37 39 42 45 46 60 61 64 73 74 77 2 4 10 18 24 26 27 29 32 33 34 35 38 47 48 51 55 58 59 73 74 77 8 9 12 16 19 20 34 35 38 40 42 45 46 47 48 51 54 56 62 70 76 78 8 9 12 21 22 25 29 32 33 41 43 49 53 55 58 59 60 61 64 70 76 78 2 4 10 15 17 23 28 30 36 42 45 46 55 58 59 66 67 69 70 75 76 78 1 2 3 5 6 11 12 17 20 21 30 33 34 40 45 51 53 58 64 66 71 77 3 5 11 15 17 20 21 22 23 26 27 32 38 43 46 47 61 62 65 74 75 78 3 5 11 14 19 25 28 30 33 34 35 36 39 48 49 52 56 59 60 74 75 78 9 10 13 17 20 21 35 36 39 41 43 46 47 48 49 52 55 57 63 66 71 77 9 10 13 22 23 26 30 33 34 42 44 50 54 56 59 60 61 62 65 66 71 77 3 5 11 16 18 24 29 31 37 43 46 47 56 59 60 66 67 68 70 71 76 77 2 3 4 6 7 12 13 18 21 22 31 34 35 41 46 52 54 59 65 67 72 78 4 6 12 14 16 18 21 22 23 24 28 33 39 44 47 48 53 62 63 66 75 76 4 6 12 15 20 26 27 29 31 34 35 36 37 40 49 50 57 60 61 66 75 76 1 10 11 18 21 22 27 36 37 40 42 44 47 48 49 50 56 58 64 67 72 78 1 10 11 14 23 24 31 34 35 43 45 51 53 55 57 60 61 62 63 67 72 78 4 6 12 17 19 25 30 32 38 44 47 48 57 60 61 67 68 69 71 72 77 78 1 3 4 5 7 8 13 19 22 23 32 35 36 40 42 47 53 55 60 66 68 73 5 7 13 15 17 19 22 23 24 25 27 29 34 45 48 49 54 63 64 67 76 77 5 7 13 14 16 21 28 30 32 35 36 37 38 41 50 51 58 61 62 67 76 77 2 11 12 19 22 23 28 37 38 41 43 45 48 49 50 51 57 59 65 66 68 73 2 11 12 15 24 25 32 35 36 44 46 52 54 56 58 61 62 63 64 66 68 73 5 7 13 18 20 26 31 33 39 45 48 49 58 61 62 66 68 69 70 72 73 78 1 2 4 5 6 8 9 20 23 24 33 36 37 41 43 48 54 56 61 67 69 74 1 6 8 16 18 20 23 24 25 26 28 30 35 46 49 50 55 64 65 68 77 78 1 6 8 15 17 22 29 31 33 36 37 38 39 42 51 52 59 62 63 68 77 78 3 12 13 20 23 24 29 38 39 42 44 46 49 50 51 52 53 58 60 67 69 74 3 12 13 16 25 26 33 36 37 40 45 47 55 57 59 62 63 64 65 67 69 74 1 6 8 14 19 21 27 32 34 46 49 50 59 62 63 66 67 69 70 71 73 74 2 3 5 6 7 9 10 21 24 25 34 37 38 42 44 49 55 57 62 68 70 75 2 7 9 14 17 19 21 24 25 26 29 31 36 47 50 51 53 56 65 66 69 78 2 7 9 16 18 23 27 30 32 34 37 38 39 40 43 52 60 63 64 66 69 78 1 4 13 21 24 25 27 30 39 40 43 45 47 50 51 52 54 59 61 68 70 75 1 4 13 14 17 26 34 37 38 41 46 48 53 56 58 60 63 64 65 68 70 75 2 7 9 15 20 22 28 33 35 47 50 51 60 63 64 67 68 70 71 72 74 75 3 4 6 7 8 10 11 22 25 26 35 38 39 43 45 50 56 58 63 69 71 76 3 8 10 14 15 18 20 22 25 26 30 32 37 48 51 52 53 54 57 66 67 70 3 8 10 17 19 24 27 28 31 33 35 38 39 40 41 44 61 64 65 66 67 70 1 2 5 22 25 26 27 28 31 40 41 44 46 48 51 52 55 60 62 69 71 76 1 2 5 14 15 18 35 38 39 42 47 49 53 54 57 59 61 64 65 69 71 76 3 8 10 16 21 23 29 34 36 48 51 52 61 64 65 68 69 71 72 73 75 76 4 5 7 8 9 11 12 14 23 26 27 36 39 44 46 51 57 59 64 70 72 77 4 9 11 14 15 16 19 21 23 26 31 33 38 40 49 52 54 55 58 67 68 71 4 9 11 18 20 25 27 28 29 32 34 36 39 41 42 45 53 62 65 67 68 71 2 3 6 14 23 26 28 29 32 40 41 42 45 47 49 52 56 61 63 70 72 77 2 3 6 15 16 19 27 36 39 43 48 50 53 54 55 58 60 62 65 70 72 77 4 9 11 17 22 24 30 35 37 40 49 52 53 62 65 69 70 72 73 74 76 77 5 6 8 9 10 12 13 14 15 24 27 28 37 45 47 52 58 60 65 71 73 78 5 10 12 14 15 16 17 20 22 24 32 34 39 40 41 50 55 56 59 68 69 72 5 10 12 19 21 26 27 28 29 30 33 35 37 42 43 46 53 54 63 68 69 72 3 4 7 14 15 24 29 30 33 40 41 42 43 46 48 50 57 62 64 71 73 78 3 4 7 16 17 20 27 28 37 44 49 51 53 54 55 56 59 61 63 71 73 78 5 10 12 18 23 25 31 36 38 40 41 50 53 54 63 70 71 73 74 75 77 78 1 6 7 9 10 11 13 15 16 25 28 29 38 40 46 48 53 59 61 66 72 74 6 11 13 15 16 17 18 21 23 25 27 33 35 41 42 51 56 57 60 69 70 73 6 11 13 14 20 22 28 29 30 31 34 36 38 43 44 47 54 55 64 69 70 73 4 5 8 15 16 25 30 31 34 41 42 43 44 47 49 51 58 63 65 66 72 74 4 5 8 17 18 21 28 29 38 45 50 52 54 55 56 57 60 62 64 66 72 74 6 11 13 19 24 26 32 37 39 41 42 51 54 55 64 66 71 72 74 75 76 78 1 2 7 8 10 11 12 16 17 26 29 30 39 41 47 49 54 60 62 67 73 75 1 7 12 16 17 18 19 22 24 26 28 34 36 42 43 52 57 58 61 70 71 74 1 7 12 15 21 23 29 30 31 32 35 37 39 44 45 48 55 56 65 70 71 74 5 6 9 16 17 26 31 32 35 42 43 44 45 48 50 52 53 59 64 67 73 75 5 6 9 18 19 22 29 30 39 40 46 51 55 56 57 58 61 63 65 67 73 75 1 7 12 14 20 25 27 33 38 42 43 52 55 56 65 66 67 72 73 75 76 77 2 3 8 9 11 12 13 14 17 18 27 30 31 42 48 50 55 61 63 68 74 76 2 8 13 14 17 18 19 20 23 25 29 35 37 40 43 44 58 59 62 71 72 75 2 8 13 16 22 24 27 30 31 32 33 36 38 45 46 49 53 56 57 71 72 75 6 7 10 14 17 18 32 33 36 40 43 44 45 46 49 51 54 60 65 68 74 76 6 7 10 19 20 23 27 30 31 41 47 52 53 56 57 58 59 62 64 68 74 76 2 8 13 15 21 26 28 34 39 40 43 44 53 56 57 67 68 73 74 76 77 78 1 3 4 9 10 12 13 15 18 19 28 31 32 43 49 51 56 62 64 69 75 77 1 3 9 15 18 19 20 21 24 26 30 36 38 41 44 45 59 60 63 72 73 76 1 3 9 17 23 25 28 31 32 33 34 37 39 46 47 50 54 57 58 72 73 76 7 8 11 15 18 19 33 34 37 41 44 45 46 47 50 52 53 55 61 69 75 77 7 8 11 20 21 24 28 31 32 40 42 48 54 57 58 59 60 63 65 69 75 77 1 3 9 14 16 22 27 29 35 41 44 45 54 57 58 66 68 69 74 75 77 78
oval
An oval of the block plan is a set of its points, no three of which are on a block. Here are all 13 ovals of maximum order for solution 1 of this block diagram (in each line an oval is represented by the set of its points):
- Solution 1 (all ovals)
14 27 40 53 15 28 41 54 16 29 42 55 17 30 43 56 18 31 44 57 19 32 45 58 20 33 46 59 21 34 47 60 22 35 48 61 23 36 49 62 24 37 50 63 25 38 51 64 26 39 52 65
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
- ^ Zvonimir Janko , Tran van Trung : Construction of an New Symmetric Block Design for (78,22,6) with the Help of Tactical Decompositions. In: Journal of Combinatorial Theory, Series A. Vol. 40, No. 2, 1985, pp. 451-455, doi : 10.1016 / 0097-3165 (85) 90107-4 .
- ^ 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.