(15,7,3) block plan
The (15,7,3) 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 15 × 15 matrix was filled with ones in such a way that each row of the matrix contains exactly 7 ones and any two rows have exactly 3 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 = 15, k = 7, λ = 3) for which such a construction is at all feasible. The smallest of these (v, k, λ) are listed in this overview .
designation
This symmetrical 2- (15,7,3) block diagram is called a triplane of order 4. At the same time it is the Hadamard block diagram of the 4th order.
properties
This symmetrical block diagram has the parameters v = 15, k = 7, λ = 3 and thus the following properties:
- It consists of 15 blocks and 15 points.
- Each block contains exactly 7 points.
- Every 2 blocks intersect in exactly 3 points.
- Each point lies on exactly 7 blocks.
- Each 2 points are connected by exactly 3 blocks.
Existence and characterization
There are exactly five non-isomorphic 2- (15,7,3) block plans. These solutions are:
- Solution 1 ( self-dual ) with the signature 15 · 112. It contains 105 ovals of the 2nd order.
- Solution 2 ( self-dual ) with the signature 8 · 18, 6 · 20, 1 · 24. It contains 24 ovals of the 3rd order.
- Solution 3 ( self-dual ) with the signature 12 · 12, 3 · 16. It contains 16 ovals of the 3rd order.
- Solution 4 ( dual to solution 5) with the signature 8 · 21, 7 · 24. It contains 28 ovals of the 3rd order.
- Solution 5 ( dual to solution 4) with the signature 14 · 22, 1 · 28. It contains 28 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
1 2 3 5 6 9 11 2 3 4 6 7 10 12 3 4 5 7 8 11 13 4 5 6 8 9 12 14 5 6 7 9 10 13 15 1 6 7 8 10 11 14 2 7 8 9 11 12 15 1 3 8 9 10 12 13 2 4 9 10 11 13 14 3 5 10 11 12 14 15 1 4 6 11 12 13 15 1 2 5 7 12 13 14 2 3 6 8 13 14 15 1 3 4 7 9 14 15 1 2 4 5 8 10 15
- Solution 2
1 2 5 7 9 10 11 5 6 8 10 11 13 15 2 4 8 9 10 14 15 1 3 4 10 11 12 15 1 3 4 5 6 8 9 1 2 6 7 8 12 15 2 3 4 6 7 10 13 3 5 7 8 10 12 14 2 4 5 6 11 12 14 3 6 7 9 11 14 15 1 2 3 8 11 13 14 1 4 5 7 13 14 15 2 3 5 9 12 13 15 4 7 8 9 11 12 13 1 6 9 10 12 13 14
- Solution 3
1 2 4 5 6 8 15 2 4 8 11 12 13 14 1 2 3 6 10 12 14 1 5 8 9 10 11 14 1 3 5 7 8 12 13 2 5 7 9 12 14 15 1 3 4 9 11 12 15 3 4 5 6 9 13 14 3 6 7 8 11 14 15 4 6 7 8 9 10 12 1 2 6 7 9 11 13 5 6 10 11 12 13 15 2 3 8 9 10 13 15 2 3 4 5 7 10 11 1 4 7 10 13 14 15
- Solution 4
1 2 3 4 5 8 10 2 3 7 8 9 11 13 3 6 8 10 11 12 14 1 3 4 6 7 9 12 1 5 6 8 9 13 14 2 5 6 7 8 12 15 1 2 3 12 13 14 15 3 4 5 6 11 13 15 4 8 9 10 12 13 15 2 4 5 9 11 12 14 1 4 7 8 11 14 15 1 2 6 9 10 11 15 1 5 7 10 11 12 13 2 4 6 7 10 13 14 3 5 7 9 10 14 15
- Solution 5
1 4 5 7 11 12 13 1 2 6 7 10 12 14 1 2 3 4 7 8 15 1 4 8 9 10 11 14 1 5 6 8 10 13 15 3 4 5 6 8 12 14 2 4 6 11 13 14 15 1 2 3 5 6 9 11 2 4 5 9 10 12 15 1 3 9 12 13 14 15 2 3 8 10 11 12 13 3 4 6 7 9 10 13 2 5 7 8 9 13 14 3 5 7 10 11 14 15 6 7 8 9 11 12 15
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
- Solution 2
O O . . O . O . O O O . . . . . . . . O O . O . O O . O . O . O . O . . . O O O . . . O O O . O O . . . . . O O O . . O O . O O O O . O O . . . . . . O O . . . O O O . . . O . . O . O O O . O O . . O . . O . . . . O . O . O O . O . O . O . . O . O O O . . . . O O . O . . . O . . O O . O . O . . O O O O O . . . . O . . O . O O . O . . O O . O . . . . . O O O . O O . O . . . O . . O O . O . . . O . . O O O . O O O . . O . . . . O . . O O . O O O .
- Solution 3
O O . O O O . O . . . . . . O . O . O . . . O . . O O O O . O O O . . O . . . O . O . O . O . . . O . . O O O O . . O . O . O . O . O O . . . O O . . . O . . O . O . O . . O . O O O . O O . . . . O . O O . . O . . O O O O . . O . . . O O . . . O . . O O O . . O . . O O . . . O . O O O O O . O . . . O O . . . O O . O . O . O . . . . . . O O . . . O O O O . O . O O . . . . O O O . . O . O . O O O O . O . . O O . . . . O . . O . . O . . O . . O O O
- Solution 4
O O O O O . . O . O . . . . . . O O . . . O O O . O . O . . . . O . . O . O . O O O . O . O . O O . O O . O . . O . . . O . . . O O . O O . . . O O . . O . . O O O O . . . O . . O O O O . . . . . . . . O O O O . . O O O O . . . . O . O . O . . . O . . . O O O . O O . O . O . O O . . . O . O O . O . O . . O . . O O . . O . . O O O O . . . O . . O O O . . . O O . . . O . O . . O O O O . . . O . O . O O . . O . . O O . . . O . O . O . O O . . . O O
- Solution 5
O . . O O . O . . . O O O . . O O . . . O O . . O . O . O . O O O O . . O O . . . . . . O O . . O . . . O O O O . . O . O . . . O O . O . O . . O . O . . O O O O . O . . . O . O . . O . O . O . . . . O . O O O O O O . O O . . O . O . . . . . O . O O . . . O O . O . . O O . O . . . . . O . . O O O O . O O . . . . O . O O O O . . . . O O . O O . O O . . O . . . O . . O . O O O . . . O O . . . O . O . O . . O O . . O O . . . . . O O O O . O O . . O
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 3 5 6 9 11
oval
An oval of the block plan is a set of its points, no three of which are on a block. Here are examples of maximum order ovals from this block diagram (in each line an oval is represented by the number of its points):
- Solution 1
1 2
- Solution 2
1 2 4
- Solution 3 (all ovals)
2 4 9 2 5 13 2 7 8 2 11 15 3 4 8 3 5 15 3 7 9 3 11 13 4 5 12 4 6 11 5 6 7 6 8 13 6 9 15 7 11 12 8 12 15 9 12 13
- Solution 4
1 2 7
- Solution 5
1 2 13
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.