(13,4,1) block plan

The (13,4,1) block plan is a special symmetrical block plan in finite geometry and combinatorics . In order to be able to construct it, this combinatorial problem had to be solved: an empty 13 × 13 matrix was filled with ones in such a way that each row of the matrix contains exactly 4 ones and any two rows have exactly 1 one 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 = 13, k = 4, λ = 1) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .

designation

This symmetrical 2- (13,4,1) block plan is called the projective plane or Desarguean plane of order 3.

properties

This symmetrical block diagram has the parameters v = 13, k = 4, λ = 1 and thus the following properties:

It consists of 13 blocks and 13 points.
Each block contains exactly 4 points.
Every 2 blocks intersect in exactly 1 point.
Each point lies on exactly 4 blocks.
Each 2 points are connected by exactly 1 block.

Existence and characterization

Exactly one 2- (13,4,1) block plan exists (except for isomorphism ). It is self-dual and has the signature 13 · 4. It contains 234 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

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

Incidence matrix

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

O O O O . . . . . . . . .
O . . . O O O . . . . . .
O . . . . . . O O O . . .
O . . . . . . . . . O O O
. O . . O . . O . . O . .
. O . . . O . . O . . O .
. O . . . . O . . O . . O
. . 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 ) 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.

  1   2   4  10

Orthogonal Latin Squares (MOLS)

This order 3 projective plane is equivalent to these 2 MOLS of order 3:

{\ displaystyle {\ begin {bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 3 & 2 \\ 2 & 1 & 3 \\ 3 & 2 & 1 \ end {bmatrix}}}

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:

  1   2   5   9

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.