(23,11,5) block plan

The (23,11,5) 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 23 × 23 matrix was filled with ones in such a way that each row of the matrix contains exactly 11 ones and any two rows have exactly 5 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 = 23, k = 11, λ = 5) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .

designation

This symmetrical 2- (23,11,5) block diagram is called the Hadamard block diagram of the 6th order.

properties

This symmetrical block diagram has the parameters v = 23, k = 11, λ = 5 and thus the following properties:

• It consists of 23 blocks and 23 points.
• Each block contains exactly 11 points.
• Every 2 blocks intersect in exactly 5 points.
• Each point lies on exactly 11 blocks.
• Each 2 points are connected by exactly 5 blocks.

Existence and characterization

There are exactly 1106 non-isomorphic 2- (23,11,5) -block plans. Six of these solutions are:

• Solution 1 with the signature 23 · 88. It contains 253 ovals of the 2nd order.
• Solution 2 with the signature 11 x 60, 11 x 100, 1 x 440. It contains 253 ovals of the 2nd order.
• Solution 3 with the signature 14 x 2, 4 x 3, 4 x 4, 1 x 52. It contains 2 ovals of the 3rd order.
• Solution 4 with the signature 4 x 1, 18 x 2, 1 x 74. It contains 2 ovals of the 3rd order.
• Solution 5 ( dual to solution 6) with the signature 6 · 2, 6 · 3, 9 · 4, 1 · 7/62, 1 · 7/90. It contains 4 ovals of the 3rd order.
• Solution 6 ( dual to solution 5) with the signature 6 · 2, 6 · 3, 9 · 4, 1 · 7/60, 1 · 7/88. It contains 4 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   4   6   8   9  12  13  16  18
  2   3   4   5   7   9  10  13  14  17  19
  3   4   5   6   8  10  11  14  15  18  20
  4   5   6   7   9  11  12  15  16  19  21
  5   6   7   8  10  12  13  16  17  20  22
  6   7   8   9  11  13  14  17  18  21  23
  1   7   8   9  10  12  14  15  18  19  22
  2   8   9  10  11  13  15  16  19  20  23
  1   3   9  10  11  12  14  16  17  20  21
  2   4  10  11  12  13  15  17  18  21  22
  3   5  11  12  13  14  16  18  19  22  23
  1   4   6  12  13  14  15  17  19  20  23
  1   2   5   7  13  14  15  16  18  20  21
  2   3   6   8  14  15  16  17  19  21  22
  3   4   7   9  15  16  17  18  20  22  23
  1   4   5   8  10  16  17  18  19  21  23
  1   2   5   6   9  11  17  18  19  20  22
  2   3   6   7  10  12  18  19  20  21  23
  1   3   4   7   8  11  13  19  20  21  22
  2   4   5   8   9  12  14  20  21  22  23
  1   3   5   6   9  10  13  15  21  22  23
  1   2   4   6   7  10  11  14  16  22  23
  1   2   3   5   7   8  11  12  15  17  23

• Solution 2

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

• Solution 3

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

• Solution 4

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

• Solution 5

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

• Solution 6

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

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

• 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 . . . O O . O . O . . . 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 O O . O O . . O .
O . . . O . . O . O O O O . . . O . . O . O O
. O O . . . O . O . O O . O O . . . . O . O O
. O O . . . . O O . O . O . . O O O . O O . .
. O . O . O . . . O O . O . O O . . O O . O .
. O . O . . . O O O . O . . O . O O O . . . O
. O . . O O . . . O O O . O . O . O . . O . O
. O . . O . O . O O . . O O . . O . O . O O .
. . O O . O . O . . O O . O . . O . O . O O .
. . O O . . O O . O . . O O . O . O . . . O O
. . O . O O O . . . O . O . O . O O O . . . O
. . O . O . O O . O . O . . O O . . O 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 O . . O . O . O O .
O . . . O . . . O O O O O . O . . O . O . . O
. O O . . . O . . O O O . O . . O O . . O . O
. O O . . . . O O . O O . . O . O . O O . O .
. O . O . O . . O . O . O . . O O O . O O . .
. O . O . . . O O O . . O O O . . . O . O . O
. O . . O O . . . O O . . O O O . O O . . O .
. O . . O . O O . O . . O . . O O . . O . O O
. . O O . O O . . . O . O O O . . . . O . O O
. . O O . . O . O O . O O . . O . O O . . O .
. . O . O O . O . . O O O . . O . . O . O . O
. . O . O . O O O . . . . O O O . O . 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 O . O O O . O . .
O . . . O . . O . O O O O . . . O . . O . O O
. O O . . . O . O . O O . O . . O . . O O O .
. O O . . . . O O . O . O . O O . O . O . . O
. O . O . O . . . O O . O O . . O O . . O . O
. O . O . . . O O O . O . . . O O O O . . O .
. O . . O O . . . O O O . . O O . . O O O . .
. O . . O . O . O O . . O O O . . . O . . O O
. . O O . O . O . . O O . O O . . . O . . O O
. . O O . . O O . O . . O . O . O . O O O . .
. . O . O O O . . . O . O . . O O O O . . O .
. . O . O . O O . O . O . O . O . O . . 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 6

O O O O O O O O O O O . . . . . . . . . . . .
O O O O O . . . . . . O O O O O O . . . . . .
O O O . . O O . . . . O O . . . . O O O O . .
O O O . . . . O O . . . . O O . . O O . . O O
O O O . . . . . . O O . . . . O O . . O O O O
O . . O O O O . . . . . . O . O . O . O . O O
O . . O O . . O O . . O . . . . O . O O O O .
O . . O O . . . . O O . O . O . . O O . O . O
O . . . . O O O . O . O O . O . O . . . . O O
O . . . . O O . O . O . . O O O O . O . O . .
O . . . . . . O O O O O O O . O . O . O . . .
. O . O . O . O . . O O . . O O . O . . O O .
. O . O . O . O . . O . O O . . O . O O . . O
. O . O . . O . O O . O . O . . O O . . O . O
. O . . O O . . O O . . O . . O O O O . . O .
. O . . O . O O O . . . O . O O . . . O O . O
. O . . O . O . . O O O . O O . . . O O . O .
. . O O . O . . O O . . O O O . . . . O O O .
. . O O . . O O . O . . . . O O O O O O . . .
. . O O . . O . O . O O O . . O . . O . . O O
. . O . O O . O . O . O . O . O . . O . 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   4   6   8   9  12  13  16  18

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

• Solution 3 (all ovals)

  2  13  23
 11  13  14

• Solution 4 (all ovals)

  6  14  21
 11  15  21

• Solution 5 (all ovals)

  2  12  23
  9  13  17
 10  13  16
 11  12  18

• Solution 6 (all ovals)

  1  15  20
  1  17  18 
  5   9  14 
  5  11  16

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 further (23,11,5) block plans

Individual evidence

1. ^ 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.