(16,6,2) block plan

The (16,6,2) 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 16 × 16 matrix was filled with ones in such a way that each row of the matrix contains exactly 6 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 = 16, k = 6, 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- (16,6,2) block plan is called a 4-order biplane .

properties

This symmetrical block diagram has the parameters v = 16, k = 6, λ = 2 and thus the following properties:

• It consists of 16 blocks and 16 points.
• Each block contains exactly 6 points.
• Every 2 blocks intersect in exactly 2 points.
• Each point lies on exactly 6 blocks.
• Every 2 points are connected by exactly 2 blocks.

Existence and characterization

There are exactly three non-isomorphic 2- (16,6,2) block plans. These solutions are:

• Solution 1 ( self-dual ) with the signature 16 · 20 and the λ-chains 320 · 3. It contains 60 ovals of the 4th order.
• Solution 2 ( self-dual ) with the signature 16 · 12 and the λ-chains 192 · 3, 64 · 6. It contains 28 ovals of the 4th order.
• Solution 3 ( self-dual ) with the signature 16 · 8 and the λ-chains 128 · 3, 96 · 6. It contains 12 ovals of the 4th 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   6   9  15
  2   3   4   7  10  16
  3   4   5   8   9  11
  1   4   5   6  10  12
  2   5   6   7  11  13
  3   6   7   8  12  14
  1   4   7   8  13  15
  1   2   5   8  14  16
  1   7   9  10  11  14
  2   8  10  11  12  15
  1   3  11  12  13  16
  2   4   9  12  13  14
  3   5  10  13  14  15
  4   6  11  14  15  16
  5   7   9  12  15  16
  6   8   9  10  13  16

• Solution 2

  1   2   3   5  10  15
  2   3   4   6  11  16
  3   4   5   7   9  12
  4   5   6   8  10  13
  1   5   6   7  11  14
  2   6   7   8  12  15
  1   3   7   8  13  16
  1   2   4   8   9  14
  2   7   9  10  11  13
  3   8  10  11  12  14
  1   4  11  12  13  15
  2   5  12  13  14  16
  3   6   9  13  14  15
  4   7  10  14  15  16
  5   8   9  11  15  16
  1   6   9  10  12  16

• Solution 3

  1   2   3   5   9  13
  2   3   4   6  10  14
  1   3   4   7  11  15
  1   2   4   8  12  16
  1   5   6   7  12  14
  2   6   7   8   9  15
  3   5   7   8  10  16
  4   5   6   8  11  13
  1   6   9  10  11  16
  2   7  10  11  12  13
  3   8   9  11  12  14
  4   5   9  10  12  15
  1   8  10  13  14  15
  2   5  11  14  15  16
  3   6  12  13  15  16
  4   7   9  13  14  16

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

• 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

• 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

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   4  11

• Solution 2

  1   2   6  13

• Solution 3 (all ovals)

  1   3   6   8  
  1   3  10  12  
  1   3  14  16   
  2   4   5   7  
  2   4   9  11 
  2   4  13  15  
  5   7   9  11   
  5   7  13  15  
  6   8  10  12   
  6   8  14  16   
  9  11  13  15  
 10  12  14  16  

literature

• Michael Klemm: Self-dual codes above the ring of whole numbers modulo 4 . In: Archives of Mathematics . tape 53 , no. 2 . Springer, 1989, p. 201-207 .
• 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

1. ^ Qazi M. Husain: On the totality of the solutions for the symmetrical incomplete block designs λ = 2, κ = 5 or 6. In: Sankhyā. Vol. 7, No. 2, 1945, pp. 204-208, ( JSTOR 25047845 ).
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.