(47,23,11) block plan

The (47,23,11) 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 47 × 47 matrix was filled with ones in such a way that each row of the matrix contains exactly 23 ones and any two rows have exactly 11 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 = 47, k = 23, λ = 11) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .

designation

This symmetrical 2- (47,23,11) block diagram is called the Hadamard block diagram of the 12th order.

properties

This symmetrical block diagram has the parameters v = 47, k = 23, λ = 11 and thus the following properties:

It consists of 47 blocks and 47 points.
Each block contains exactly 23 points.
Every 2 blocks intersect in exactly 11 points.
Each point lies on exactly 23 blocks.
Each 2 points are connected by exactly 11 blocks.

Existence and characterization

There are at least 55 non-isomorphic 2- (47,23,11) block plans. One of these solutions is:

Solution 1 with the signature 47 92. It contains 1081 ovals of the 2nd order.

List of blocks

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

Solution 1

  2   3   4   5   7   8   9  10  13  15  17  18  19  22  25  26  28  29  33  35  37  38  43
  3   4   5   6   8   9  10  11  14  16  18  19  20  23  26  27  29  30  34  36  38  39  44
  4   5   6   7   9  10  11  12  15  17  19  20  21  24  27  28  30  31  35  37  39  40  45
  5   6   7   8  10  11  12  13  16  18  20  21  22  25  28  29  31  32  36  38  40  41  46
  6   7   8   9  11  12  13  14  17  19  21  22  23  26  29  30  32  33  37  39  41  42  47
  1   7   8   9  10  12  13  14  15  18  20  22  23  24  27  30  31  33  34  38  40  42  43
  2   8   9  10  11  13  14  15  16  19  21  23  24  25  28  31  32  34  35  39  41  43  44
  3   9  10  11  12  14  15  16  17  20  22  24  25  26  29  32  33  35  36  40  42  44  45
  4  10  11  12  13  15  16  17  18  21  23  25  26  27  30  33  34  36  37  41  43  45  46
  5  11  12  13  14  16  17  18  19  22  24  26  27  28  31  34  35  37  38  42  44  46  47
  1   6  12  13  14  15  17  18  19  20  23  25  27  28  29  32  35  36  38  39  43  45  47
  1   2   7  13  14  15  16  18  19  20  21  24  26  28  29  30  33  36  37  39  40  44  46
  2   3   8  14  15  16  17  19  20  21  22  25  27  29  30  31  34  37  38  40  41  45  47
  1   3   4   9  15  16  17  18  20  21  22  23  26  28  30  31  32  35  38  39  41  42  46
  2   4   5  10  16  17  18  19  21  22  23  24  27  29  31  32  33  36  39  40  42  43  47
  1   3   5   6  11  17  18  19  20  22  23  24  25  28  30  32  33  34  37  40  41  43  44
  2   4   6   7  12  18  19  20  21  23  24  25  26  29  31  33  34  35  38  41  42  44  45
  3   5   7   8  13  19  20  21  22  24  25  26  27  30  32  34  35  36  39  42  43  45  46
  4   6   8   9  14  20  21  22  23  25  26  27  28  31  33  35  36  37  40  43  44  46  47
  1   5   7   9  10  15  21  22  23  24  26  27  28  29  32  34  36  37  38  41  44  45  47
  1   2   6   8  10  11  16  22  23  24  25  27  28  29  30  33  35  37  38  39  42  45  46
  2   3   7   9  11  12  17  23  24  25  26  28  29  30  31  34  36  38  39  40  43  46  47
  1   3   4   8  10  12  13  18  24  25  26  27  29  30  31  32  35  37  39  40  41  44  47
  1   2   4   5   9  11  13  14  19  25  26  27  28  30  31  32  33  36  38  40  41  42  45
  2   3   5   6  10  12  14  15  20  26  27  28  29  31  32  33  34  37  39  41  42  43  46
  3   4   6   7  11  13  15  16  21  27  28  29  30  32  33  34  35  38  40  42  43  44  47
  1   4   5   7   8  12  14  16  17  22  28  29  30  31  33  34  35  36  39  41  43  44  45
  2   5   6   8   9  13  15  17  18  23  29  30  31  32  34  35  36  37  40  42  44  45  46
  3   6   7   9  10  14  16  18  19  24  30  31  32  33  35  36  37  38  41  43  45  46  47
  1   4   7   8  10  11  15  17  19  20  25  31  32  33  34  36  37  38  39  42  44  46  47
  1   2   5   8   9  11  12  16  18  20  21  26  32  33  34  35  37  38  39  40  43  45  47
  1   2   3   6   9  10  12  13  17  19  21  22  27  33  34  35  36  38  39  40  41  44  46
  2   3   4   7  10  11  13  14  18  20  22  23  28  34  35  36  37  39  40  41  42  45  47
  1   3   4   5   8  11  12  14  15  19  21  23  24  29  35  36  37  38  40  41  42  43  46
  2   4   5   6   9  12  13  15  16  20  22  24  25  30  36  37  38  39  41  42  43  44  47
  1   3   5   6   7  10  13  14  16  17  21  23  25  26  31  37  38  39  40  42  43  44  45
  2   4   6   7   8  11  14  15  17  18  22  24  26  27  32  38  39  40  41  43  44  45  46
  3   5   7   8   9  12  15  16  18  19  23  25  27  28  33  39  40  41  42  44  45  46  47
  1   4   6   8   9  10  13  16  17  19  20  24  26  28  29  34  40  41  42  43  45  46  47
  1   2   5   7   9  10  11  14  17  18  20  21  25  27  29  30  35  41  42  43  44  46  47
  1   2   3   6   8  10  11  12  15  18  19  21  22  26  28  30  31  36  42  43  44  45  47
  1   2   3   4   7   9  11  12  13  16  19  20  22  23  27  29  31  32  37  43  44  45  46
  2   3   4   5   8  10  12  13  14  17  20  21  23  24  28  30  32  33  38  44  45  46  47
  1   3   4   5   6   9  11  13  14  15  18  21  22  24  25  29  31  33  34  39  45  46  47
  1   2   4   5   6   7  10  12  14  15  16  19  22  23  25  26  30  32  34  35  40  46  47
  1   2   3   5   6   7   8  11  13  15  16  17  20  23  24  26  27  31  33  35  36  41  47
  1   2   3   4   6   7   8   9  12  14  16  17  18  21  24  25  27  28  32  34  36  37  42

Incidence matrix

This is a representation of the incidence matrix of this block diagram; see this illustration to understand this matrix

Solution 1

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. . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O . O O . . . . O . .
. . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O . O O . . . . O .
. . . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O . O O . . . . O
O . . . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O . O O . . . .
. O . . . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O . O O . . .
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. . . . O . . . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O . O O
O . . . . O . . . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O . O
O O . . . . O . . . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O .
. O O . . . . O . . . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O . O
O . O O . . . . O . . . . . O O O O . O O O O . . O . O . O O O . . O . . O O . O O . . . O .
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O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O . . O . O . O O O . . O . . O
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O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O . . O . O . O O
O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O . . O . O . O
O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O . . O . O .
. O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O . . O . O
O . O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O . . O .
. O . O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O . . O
O . O . O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O . .
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. . O . O . O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O O
O . . O . O . O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O O
O O . . O . O . O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O O
O O O . . O . O . O O O . . O . . O O . O O . . . O . O . O O . . . . O . . . . . O O O O . O
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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

  2   3   4   5   7   8   9  10  13  15  17  18  19  22  25  26  28  29  33  35  37  38  43

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 from this block diagram:

Solution 1

  1   2

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.

[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.