(79,13,2) block plan
The (79,13,2) 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 79 × 79 matrix was filled with ones in such a way that each row of the matrix contains exactly 13 ones and any two rows have exactly 2 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 = 79, k = 13, λ = 2) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .
designation
This symmetrical 2- (79,13,2) block diagram is called the 11th order biplane .
properties
This symmetrical block diagram has the parameters v = 79, k = 13, λ = 2 and thus the following properties:
- It consists of 79 blocks and 79 points.
- Each block contains exactly 13 points.
- Every 2 blocks intersect in exactly 2 points.
- Each point lies on exactly 13 blocks.
- Every 2 points are connected by exactly 2 blocks.
Existence and characterization
There are at least two non-isomorphic 2- (79,13,2) block plans. These solutions are:
- Solution 1 ( dual to solution 2) with the signature 2 11, 55 16, 22 31 and the λ-chains 1584 3, 605 4, 682 5, 825 6, 660 7, 330 8 , 275 x 9, 748 x 10, 2695 x 13. It contains 77 ovals of the 7th order.
- Solution 2 ( dual to solution 1) with the signature 11 · 1, 11 · 11, 55 · 24, 2 · 66 and the λ-chains 1584 · 3, 605 · 4, 682 · 5, 825 · 6, 660 · 7 , 330 x 8, 275 x 9, 748 x 10, 2695 x 13. It contains 77 ovals of the 7th 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 4 5 6 7 8 9 10 11 12 13 1 2 14 15 16 17 18 19 20 21 22 23 24 1 3 14 28 34 40 41 48 50 63 67 70 73 1 4 15 29 35 41 42 49 51 64 68 71 74 1 5 16 25 30 42 43 50 52 58 65 72 75 1 6 17 26 31 43 44 51 53 59 66 73 76 1 7 18 27 32 44 45 52 54 60 67 74 77 1 8 19 28 33 45 46 53 55 61 68 75 78 1 9 20 29 34 36 46 54 56 58 62 76 79 1 10 21 30 35 36 37 55 57 59 63 69 77 1 11 22 25 31 37 38 47 56 60 64 70 78 1 12 23 26 32 38 39 48 57 61 65 71 79 1 13 24 27 33 39 40 47 49 62 66 69 72 2 3 14 27 33 42 43 55 57 60 64 76 79 2 4 15 28 34 43 44 47 56 61 65 69 77 2 5 16 29 35 44 45 48 57 62 66 70 78 2 6 17 25 30 45 46 47 49 63 67 71 79 2 7 18 26 31 36 46 48 50 64 68 69 72 2 8 19 27 32 36 37 49 51 58 65 70 73 2 9 20 28 33 37 38 50 52 59 66 71 74 2 10 21 29 34 38 39 51 53 60 67 72 75 2 11 22 30 35 39 40 52 54 61 68 73 76 2 12 23 25 31 40 41 53 55 58 62 74 77 2 13 24 26 32 41 42 54 56 59 63 75 78 4 13 17 22 26 27 34 35 38 45 50 55 58 3 5 18 23 25 27 28 35 39 46 51 56 59 4 6 19 24 25 26 28 29 36 40 52 57 60 5 7 14 20 26 27 29 30 37 41 47 53 61 6 8 15 21 27 28 30 31 38 42 48 54 62 7 9 16 22 28 29 31 32 39 43 49 55 63 8 10 17 23 29 30 32 33 40 44 50 56 64 9 11 18 24 30 31 33 34 41 45 51 57 65 10 12 14 19 31 32 34 35 42 46 47 52 66 11 13 15 20 25 32 33 35 36 43 48 53 67 3 12 16 21 25 26 33 34 37 44 49 54 68 5 12 19 20 38 40 43 45 51 54 63 64 69 6 13 20 21 39 41 44 46 52 55 64 65 70 3 7 21 22 36 40 42 45 53 56 65 66 71 4 8 22 23 37 41 43 46 54 57 66 67 72 5 9 23 24 36 38 42 44 47 55 67 68 73 6 10 14 24 37 39 43 45 48 56 58 68 74 7 11 14 15 38 40 44 46 49 57 58 59 75 8 12 15 16 36 39 41 45 47 50 59 60 76 9 13 16 17 37 40 42 46 48 51 60 61 77 3 10 17 18 36 38 41 43 49 52 61 62 78 4 11 18 19 37 39 42 44 50 53 62 63 79 7 10 15 24 25 50 51 54 55 61 66 70 79 8 11 14 16 26 51 52 55 56 62 67 69 71 9 12 15 17 27 52 53 56 57 63 68 70 72 10 13 16 18 28 47 53 54 57 58 64 71 73 3 11 17 19 29 47 48 54 55 59 65 72 74 4 12 18 20 30 48 49 55 56 60 66 73 75 5 13 19 21 31 49 50 56 57 61 67 74 76 3 6 20 22 32 47 50 51 57 62 68 75 77 4 7 21 23 33 47 48 51 52 58 63 76 78 5 8 22 24 34 48 49 52 53 59 64 77 79 6 9 14 23 35 49 50 53 54 60 65 69 78 6 11 16 23 27 34 36 61 63 64 66 74 75 7 12 17 24 28 35 37 62 64 65 67 75 76 8 13 14 18 25 29 38 63 65 66 68 76 77 3 9 15 19 26 30 39 58 64 66 67 77 78 4 10 16 20 27 31 40 59 65 67 68 78 79 5 11 17 21 28 32 41 58 60 66 68 69 79 6 12 18 22 29 33 42 58 59 61 67 69 70 7 13 19 23 30 34 43 59 60 62 68 70 71 3 8 20 24 31 35 44 58 60 61 63 71 72 4 9 14 21 25 32 45 59 61 62 64 72 73 5 10 15 22 26 33 46 60 62 63 65 73 74 8 9 18 21 26 35 40 43 47 70 74 75 79 9 10 19 22 25 27 41 44 48 69 71 75 76 10 11 20 23 26 28 42 45 49 70 72 76 77 11 12 21 24 27 29 43 46 50 71 73 77 78 12 13 14 22 28 30 36 44 51 72 74 78 79 3 13 15 23 29 31 37 45 52 69 73 75 79 3 4 16 24 30 32 38 46 53 69 70 74 76 4 5 14 17 31 33 36 39 54 70 71 75 77 5 6 15 18 32 34 37 40 55 71 72 76 78 6 7 16 19 33 35 38 41 56 72 73 77 79 7 8 17 20 25 34 39 42 57 69 73 74 78
- Solution 2
1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 14 15 16 17 18 19 20 21 22 23 24 1 3 14 26 35 38 45 51 54 61 66 74 75 1 4 15 25 27 39 46 52 55 62 67 75 76 1 5 16 26 28 36 40 53 56 63 68 76 77 1 6 17 27 29 37 41 54 57 58 64 77 78 1 7 18 28 30 38 42 47 55 59 65 78 79 1 8 19 29 31 39 43 48 56 60 66 69 79 1 9 20 30 32 40 44 49 57 61 67 69 70 1 10 21 31 33 41 45 47 50 62 68 70 71 1 11 22 32 34 42 46 48 51 58 63 71 72 1 12 23 33 35 36 43 49 52 59 64 72 73 1 13 24 25 34 37 44 50 53 60 65 73 74 2 3 14 28 33 41 42 48 57 60 67 73 76 2 4 15 29 34 42 43 47 49 61 68 74 77 2 5 16 30 35 43 44 48 50 58 62 75 78 2 6 17 25 31 44 45 49 51 59 63 76 79 2 7 18 26 32 45 46 50 52 60 64 69 77 2 8 19 27 33 36 46 51 53 61 65 70 78 2 9 20 28 34 36 37 52 54 62 66 71 79 2 10 21 29 35 37 38 53 55 63 67 69 72 2 11 22 25 30 38 39 54 56 64 68 70 73 2 12 23 26 31 39 40 55 57 58 65 71 74 2 13 24 27 32 40 41 47 56 59 66 72 75 5 11 17 23 26 27 34 35 47 60 67 70 79 6 12 18 24 25 27 28 35 48 61 68 69 71 7 13 14 19 25 26 28 29 49 58 62 70 72 3 8 15 20 26 27 29 30 50 59 63 71 73 4 9 16 21 27 28 30 31 51 60 64 72 74 5 10 17 22 28 29 31 32 52 61 65 73 75 6 11 18 23 29 30 32 33 53 62 66 74 76 7 12 19 24 30 31 33 34 54 63 67 75 77 8 13 14 20 31 32 34 35 55 64 68 76 78 3 9 15 21 25 32 33 35 56 58 65 77 79 4 10 16 22 25 26 33 34 57 59 66 69 78 9 10 18 19 27 34 38 40 43 45 58 73 76 10 11 19 20 28 35 39 41 44 46 59 74 77 11 12 20 21 25 29 36 40 42 45 60 75 78 12 13 21 22 26 30 37 41 43 46 61 76 79 3 13 22 23 27 31 36 38 42 44 62 69 77 3 4 23 24 28 32 37 39 43 45 63 70 78 4 5 14 24 29 33 38 40 44 46 64 71 79 5 6 14 15 30 34 36 39 41 45 65 69 72 6 7 15 16 31 35 37 40 42 46 66 70 73 7 8 16 17 25 32 36 38 41 43 67 71 74 8 9 17 18 26 33 37 39 42 44 68 72 75 11 13 15 17 28 33 40 43 50 51 54 55 69 3 12 16 18 29 34 41 44 51 52 55 56 70 4 13 17 19 30 35 42 45 52 53 56 57 71 3 5 18 20 25 31 43 46 47 53 54 57 72 4 6 19 21 26 32 36 44 47 48 54 55 73 5 7 20 22 27 33 37 45 48 49 55 56 74 6 8 21 23 28 34 38 46 49 50 56 57 75 7 9 22 24 29 35 36 39 47 50 51 57 76 8 10 14 23 25 30 37 40 47 48 51 52 77 9 11 15 24 26 31 38 41 48 49 52 53 78 10 12 14 16 27 32 39 42 49 50 53 54 79 5 9 19 23 25 41 42 50 55 61 63 64 66 6 10 20 24 26 42 43 51 56 62 64 65 67 7 11 14 21 27 43 44 52 57 63 65 66 68 8 12 15 22 28 44 45 47 53 58 64 66 67 9 13 16 23 29 45 46 48 54 59 65 67 68 3 10 17 24 30 36 46 49 55 58 60 66 68 4 11 14 18 31 36 37 50 56 58 59 61 67 5 12 15 19 32 37 38 51 57 59 60 62 68 6 13 16 20 33 38 39 47 52 58 60 61 63 3 7 17 21 34 39 40 48 53 59 61 62 64 4 8 18 22 35 40 41 49 54 60 62 63 65 10 13 15 18 36 48 57 63 64 70 74 75 79 3 11 16 19 37 47 49 64 65 69 71 75 76 4 12 17 20 38 48 50 65 66 70 72 76 77 5 13 18 21 39 49 51 66 67 71 73 77 78 3 6 19 22 40 50 52 67 68 72 74 78 79 4 7 20 23 41 51 53 58 68 69 73 75 79 5 8 21 24 42 52 54 58 59 69 70 74 76 6 9 14 22 43 53 55 59 60 70 71 75 77 7 10 15 23 44 54 56 60 61 71 72 76 78 8 11 16 24 45 55 57 61 62 72 73 77 79 9 12 14 17 46 47 56 62 63 69 73 74 78
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 3 31 46 49 65 77
- Solution 2
1 2 25 36 47 58 69
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
- ↑ Michael Aschbacher : On collineation groups of symmetric block designs. In: Journal of Combinatorial Theory, Series A. Vol. 11, No. 3, 1971, pp. 272-281, doi : 10.1016 / 0097-3165 (71) 90054-9 .
- ^ 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.