(25,9,3) block plan

The (25,9,3) 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 25 × 25 matrix was filled with ones in such a way that each row of the matrix contains exactly 9 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 = 25, k = 9, 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- (25,9,3) block diagram is called a 6-order triplane .

properties

This symmetrical block diagram has the parameters v = 25, k = 9, λ = 3 and thus the following properties:

• It consists of 25 blocks and 25 points.
• Each block contains exactly 9 points.
• Every 2 blocks intersect in exactly 3 points.
• Each point lies on exactly 9 blocks.
• Each 2 points are connected by exactly 3 blocks.

Existence and characterization

There are exactly 78 non-isomorphic 2- (25,9,3) block plans. Two of these solutions are:

• Solution 1 ( dual to solution 2) with the signature 2 x 2, 1 x 4, 3 x 5, 9 x 6, 5 x 7, 2 x 8, 2 x 9, 1 x 10. It contains 4 ovals of the 4th order.
• Solution 2 ( dual to solution 1) with the signature 1 x 1, 2 x 4, 2 x 5, 8 x 6, 7 x 7, 5 x 8 It contains 7 ovals of order 4.

List of blocks

All the blocks of this block plan are listed here; See this illustration to understand this list

• Solution 1

  1   5   8  12  16  19  22  24  25
  5   6   9  10  11  15  20  22  25
  1   4  12  14  15  17  20  23  25
  2   8   9  13  15  16  17  21  25
  3   4   5   9  14  15  19  21  24
  4   8  10  11  12  13  19  20  21
  4   7   8  10  15  17  18  22  24
  1   2   7  11  15  19  21  22  23
  9  10  12  14  16  18  21  22  23
  1   3   6   7   8  10  14  21  25
  2   5   7  10  14  16  17  19  20
  3   6   8  15  16  18  19  20  23
  3   5   7   8   9  11  12  17  23
  1   3   4   7   9  13  16  20  22
  1   4   5   6  11  16  17  18  21
  2   3   6  12  17  20  21  22  24
  1   2   8   9  11  14  18  20  24
  1   6   9  10  13  17  19  23  24
  6   7  11  12  13  14  15  16  24
  2   4   5   6   8  13  14  22  23
  3  11  13  14  17  18  19  22  25
  2   4   6   7   9  12  18  19  25
  5   7  13  18  20  21  23  24  25
  2   3   4  10  11  16  23  24  25
  1   2   3   5  10  12  13  15  18

• Solution 2

  1   3   8  10  14  15  17  18  25
  4   8  11  16  17  20  22  24  25
  5  10  12  13  14  16  21  24  25
  3   5   6   7  14  15  20  22  24
  1   2   5  11  13  15  20  23  25
  2  10  12  15  16  18  19  20  22
  7   8  10  11  13  14  19  22  23
  1   4   6   7  10  12  13  17  20
  2   4   5   9  13  14  17  18  22
  2   6   7   9  10  11  18  24  25
  2   6   8  13  15  17  19  21  24
  1   3   6   9  13  16  19  22  25
  4   6  14  18  19  20  21  23  25
  3   5   9  10  11  17  19  20  21
  2   3   4   5   7   8  12  19  25
  1   4   9  11  12  14  15  19  24
  3   4   7  11  13  15  16  18  21
  7   9  12  15  17  21  22  23  25
  1   5   6   8  11  12  18  21  22
  2   3   6  11  12  14  16  17  23
  4   5   6   8   9  10  15  16  23
  1   2   7   8   9  14  16  20  21
  3   8   9  12  13  18  20  23  24
  1   5   7  16  17  18  19  23  24
  1   2   3   4  10  21  22  23  24

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 . . . . . . .

• 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 . . O . . O . . O
. . . O . O . . . . . . . O . . . O O O O . O . O
. . O . O . . . O O O . . . . . O . O O O . . . .
. O O O O . O O . . . O . . . . . . O . . . . . O
O . . O . . . . O . O O . O O . . . O . . . . O .
. . O O . . O . . . O . O . O O . O . . O . . . .
. . . . . . O . O . . O . . O . O . . . O O O . O
O . . . O O . O . . O O . . . . . O . . O O . . .
. O O . . O . . . . O O . O . O O . . . . . O . .
. . . O O O . O O O . . . . O O . . . . . . O . .
O O . . . . O O O . . . . O . O . . . O O . . . .
. . O . . . . O O . . 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 the 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)

  2  17  18  23
  3   8  13  24
  4  21  22  25
  9  11  16  19

• Solution 2 (all ovals)

  1  10  11  16
  2   3   9  15
  2   9  19  23
  3  15  19  23
  5   6  17  25
  7  12  14  18
 13  20  21  22

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

• Listing of all 78 (25,9,3) block plans

Individual evidence

1. ↑ Ralph HF Denniston: Enumeration of Symmetric Designs (25,9,3). In: Eric Mendelsohn (Ed.): Algebraic and Geometric Combinatorics (= Annals of discrete mathematics. 15 = North Holland mathematics studies. 65). North-Holland, Amsterdam et al. 1982, ISBN 0-444-86365-6 , pp. 111-127, doi : 10.1016 / S0304-0208 (08) 73258-4 .
2. ^ 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.