(31,10,3) block plan
The (31,10,3) block plan is a special symmetrical block plan . To be able to construct it, a combinatorial problem had to be solved:
An empty 31 × 31 matrix was filled with ones in such a way that each row of the matrix contains exactly 10 ones and any two rows have exactly 3 ones in the same column. The solution to this problem is not trivial. There are only certain combinations of parameters (like here v = 31, k = 10, λ = 3) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .
designation
This symmetrical 2- (31,10,3) block plan is called a 7th order triplane .
properties
This symmetrical block diagram has the parameters v = 31, k = 10, λ = 3 and thus the following properties:
- It consists of 31 blocks and 31 points.
- Each block contains exactly 10 points.
- Every 2 blocks intersect in exactly 3 points.
- Each point lies on exactly 10 blocks.
- Each 2 points are connected by exactly 3 blocks.
Existence and characterization
There are exactly 151 non-isomorphic 2- (31,10,3) 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 3 4 8 12 16 20 24 28 1 2 3 5 9 13 17 21 25 29 1 2 3 6 10 14 18 22 26 30 7 8 9 10 12 13 14 20 21 22 11 12 13 14 16 17 18 24 25 26 15 16 17 18 20 21 22 28 29 30 4 5 6 19 20 21 22 24 25 26 8 9 10 23 24 25 26 28 29 30 4 5 6 12 13 14 27 28 29 30 4 5 6 8 9 10 16 17 18 31 1 8 13 18 19 22 25 27 28 31 1 4 7 12 17 22 23 26 29 31 1 5 7 8 11 16 21 26 27 30 1 6 9 11 12 15 20 25 30 31 1 6 7 10 13 15 16 19 24 29 1 5 10 11 14 17 19 20 23 28 1 4 9 14 15 18 21 23 24 27 2 10 12 17 19 21 24 27 30 31 2 6 7 14 16 21 23 25 28 31 2 4 7 10 11 18 20 25 27 29 2 5 8 11 14 15 22 24 29 31 2 5 7 9 12 15 18 19 26 28 2 4 9 11 13 16 19 22 23 30 2 6 8 13 15 17 20 23 26 27 3 9 14 16 19 20 26 27 29 31 3 5 7 13 18 20 23 24 30 31 3 6 7 9 11 17 22 24 27 28 3 4 10 11 13 15 21 26 28 31 3 4 7 8 14 15 17 19 25 30 3 6 8 11 12 18 19 21 23 29 3 5 10 12 15 16 22 23 25 27
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 . . . .
oval
An oval of the block plan is a set of its points, no three of which are on a block. Here are all 7 ovals of maximum order of this block diagram (in each line an oval is represented by the number of its points):
- Solution 1 (all ovals)
7 11 15 23 7 11 19 31 7 15 27 31 7 19 23 27 11 15 19 27 11 23 27 31 15 19 23 31
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 .
Web links
Individual evidence
- ↑ Edward Spence: A Complete Classification of Symmetric (31,10,3) Designs. In: Designs, Codes and Cryptography. Vol. 2, No. 2, 1992, pp. 127-136, doi : 10.1007 / BF00124892 .
- ^ 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.