(36,15,6) block plan
The (36,15,6) 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 36 × 36 matrix was filled with ones in such a way that each row of the matrix contains exactly 15 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 = 36, k = 15, 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 = 36, k = 15, λ = 6 and thus the following properties:
- It consists of 36 blocks and 36 points.
- Each block contains exactly 15 points.
- Every 2 blocks intersect in exactly 6 points.
- Each point lies on exactly 15 blocks.
- Each 2 points are connected by exactly 6 blocks.
Existence and characterization
There are at least 25634 non-isomorphic 2- (36,15,6) -block plans. One of these solutions is:
- Solution 1 with the signature 9 x 2, 6 x 3, 3 x 6, 3 x 8, 6 x 9, 4 x 12, 3 x 20, 2 x 51. It contains 16 ovals of the 3rd order.
List of blocks
All the blocks of this block plan are listed here; See this illustration to understand this list
- Solution 1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 3 4 5 6 7 8 17 18 19 20 21 22 23 24 1 2 4 5 6 7 8 25 26 27 28 29 30 31 32 1 2 3 5 9 10 11 17 18 19 25 26 27 33 34 1 2 3 4 9 10 11 20 21 22 28 29 30 35 36 1 2 3 7 12 13 14 17 18 23 28 29 31 33 35 1 2 3 6 12 13 14 20 21 24 25 26 32 34 36 1 2 3 9 12 15 16 19 22 23 24 27 30 31 32 1 4 5 8 12 13 14 19 22 27 30 33 34 35 36 1 4 5 11 12 15 16 17 20 23 25 28 31 34 36 1 4 5 10 12 15 16 18 21 24 26 29 32 33 35 1 6 7 8 9 10 11 23 24 31 32 33 34 35 36 1 6 7 9 14 15 16 17 20 22 26 29 30 33 34 1 6 7 9 13 15 16 18 19 21 25 27 28 35 36 1 8 10 11 13 14 16 17 19 21 22 25 29 31 32 1 8 10 11 13 14 15 18 20 23 24 26 27 28 30 2 4 6 10 13 15 19 21 22 23 26 28 31 33 34 2 4 6 11 14 16 19 20 23 24 25 27 29 33 35 2 4 8 9 14 15 17 18 21 23 25 30 32 33 36 2 5 7 10 13 16 18 22 23 24 25 29 30 34 36 2 5 7 11 14 15 17 19 22 24 26 28 32 35 36 2 5 8 9 13 15 17 20 21 24 27 29 31 34 35 2 6 8 10 12 16 17 18 19 20 28 30 32 34 35 2 7 8 11 12 16 18 20 21 22 26 27 31 33 36 3 4 7 10 14 15 18 19 20 27 29 31 32 34 36 3 4 7 11 13 16 17 21 24 27 28 30 32 33 34 3 4 8 9 14 16 18 22 24 25 26 28 31 34 35 3 5 6 10 14 16 17 21 23 26 27 30 31 35 36 3 5 6 11 13 15 18 20 22 25 30 31 32 33 35 3 5 8 9 13 16 19 20 23 26 28 29 32 33 36 3 6 8 10 12 15 17 22 24 25 27 28 29 33 36 3 7 8 11 12 15 19 21 23 25 26 29 30 34 35 4 6 9 11 12 13 17 18 19 24 26 29 30 31 36 4 7 9 10 12 13 17 20 22 23 25 26 27 32 35 5 6 9 11 12 14 18 21 22 23 27 28 29 32 34 5 7 9 10 12 14 19 20 21 24 25 28 30 31 33
Incidence matrix
This is a representation of the incidence matrix of this block diagram; see this illustration to understand this matrix
- Solution 1
. O O O O O O O O O O O O O O O . . . . . . . . . . . . . . . . . . . . O . O O O O O O . . . . . . . . O O O O O O O O . . . . . . . . . . . . O O . O O O O O . . . . . . . . . . . . . . . . O O O O O O O O . . . . O O O . O . . . O O O . . . . . O O O . . . . . O O O . . . . . O O . . O O O O . . . . O O O . . . . . . . . O O O . . . . . O O O . . . . O O O O O . . . O . . . . O O O . . O O . . . . O . . . . O O . O . O . O . O O O . . O . . . . . O O O . . . . . O O . . O O O . . . . . O . O . O O O O . . . . . O . . O . . O O . . O . . O O O . . O . . O O O . . . . O . . O O . . O . . . O O O . . . . O . . O . . . . O . . O . . O O O O O . . O O . . . . . O O . . O O O . . O . . O . O . . O . . O . . O . O O . . O O . . . . O . O . . O O . O . . O . . O . O . . O . . O O . O . O . . . . O O O O O O . . . . . . . . . . . O O . . . . . . O O O O O O O . . . . O O . O . . . . O O O O . . O . O . . . O . . O O . . O O . . O . . . . O O . O . . . O . O O . O O . O . . . O . O O . . . . . . O O O . . . . . . O . O O . O O . O O . O . O O . . O . . . O . O O . . . . O . . . . . . O . O O . O O O . . O . O . . O O . O O O . O . . . . . . . O . O . O . . . O . . O . O . . . O . O O O . . O . O . . O . O O . . . O . O . O . . . . O . . O . O . . O O . . O O O . O . O . . . O . O . . O . O . . . O O . . . . O O . O O . . O . O . O . . . . O . O O . . O . O . . O . O . . O . . O . . O . O . . . O O O O . . . O O . . . O . O . O . . O . O . . . O . . O O . O . O . . O . O . O . O . . . O . . O O . O . . O . . O O . . . O . O . O . . O O . . O . . O . O . O . . O O . . O . . . O . O . O . O . . . O O O O O . . . . . . . O . O . O . O O . . O . . . . O O . . O O . . . O . O . O O O . . . O O . . . O . O . . O . . O O . . O . . O . . . O O . . O O O . . . . . . O . O . O O . O . O . . O O . . O . . . O . O . . O O . . . O . . O . . O O . O . O O O . . . . O O . . . O O . . . . O . O . O . . . O . O O O . O . . O . . O O . . . O . O O . . . O . . . O . O O . . . O . O . . O O . . O O . . . O O . . O . O O . . . . O . O . O . . O . O . O . . O . . . . O O O O . O . . . O . O . . O O . . . O . . O . . O O . . O . . O . O O . . O O . . O . . O . . O . O . O . O . . O . O . . . . O . O O . O O O . . . O . . O . . O . . . O O . . O O . . O . . . O . O . O . O O . . O O . . . O O . . . . O . O . . O . O O O . . . O O O . . . . O . O . . O O O . . . . O . . . O . . O . O O . O O . . . O . . O . O O . O O O . . . . O . . O . . . . . O O . . O . O O . O . . . O . . O O O . . . O O O . . O . O . . . . . . O . O . O O . O . O . . . . O O O . . O O . . O . O O . O . . .
oval
An oval of the block plan is a set of its points, no three of which are on a block. Here are all 16 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)
8 17 26 8 18 29 8 20 25 8 21 28 9 17 28 9 18 20 9 21 26 9 25 29 12 17 21 12 18 25 12 20 29 12 26 28 19 24 34 19 31 35 24 30 35 30 31 34
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.