Block designs.

A module to help with constructions and computations of block designs and other incidence structures.

A block design is an incidence structure consisting of a set of points P and a set of blocks B, where each block is considered as a subset of P. More precisely, a block design B is a class of k-element subsets of P such that the number r of blocks that contain any point x in P is independent of x, and the number \lambda of blocks that contain any given t-element subset T is independent of the choice of T (see [1] for more). Such a block design is also called a t-(v,k,\lambda)-design, and v (the number of points), b (the number of blocks), k, r, and \lambda are the parameters of the design. (In Python, lambda is reserved, so we sometimes use lmbda or L instead.)

In Sage, sets are replaced by (ordered) lists and the standard representation of a block design uses P = [0,1,..., v-1], so a block design is specified by (v,B).

REFERENCES:

[1]Block design from wikipedia, Wikipedia article Block_design
[2]What is a block design?, http://designtheory.org/library/extrep/extrep-1.1-html/node4.html (in ‘The External Representation of Block Designs’ by Peter J. Cameron, Peter Dobcsanyi, John P. Morgan, Leonard H. Soicher)

AUTHORS:

  • Peter Dobcsanyi and David Joyner (2007-2008)

    This is a significantly modified form of the module block_design.py (version 0.6) written by Peter Dobcsanyi peter@designtheory.org. Thanks go to Robert Miller for lots of good design suggestions.

Functions and methods

sage.combinat.designs.block_design.AffineGeometryDesign(n, d, F)

Returns an Affine Geometry Design.

INPUT:

  • n (integer) – the Euclidean dimension. The number of points is v=|F^n|.
  • d (integer) – the dimension of the (affine) subspaces of P = GF(q)^n which make up the blocks.
  • F – a Finite Field (i.e. FiniteField(17)), or a prime power (i.e. an integer)

AG_{n,d} (F), as it is sometimes denoted, is a 2 - (v, k, \lambda) design of points and d- flats (cosets of dimension n) in the affine geometry AG_n (F), where

v = q^n,\  k = q^d ,
\lambda =\frac{(q^{n-1}-1) \cdots (q^{n+1-d}-1)}{(q^{n-1}-1) \cdots (q-1)}.

Wraps some functions used in GAP Design’s PGPointFlatBlockDesign. Does not require GAP’s Design package.

EXAMPLES:

sage: BD = designs.AffineGeometryDesign(3, 1, GF(2))
sage: BD.parameters()
(2, 8, 2, 1)
sage: BD.is_block_design()
(True, [2, 8, 2, 1])
sage: BD = designs.AffineGeometryDesign(3, 2, GF(2))
sage: BD.parameters()
(2, 8, 4, 3)
sage: BD.is_block_design()
(True, [3, 8, 4, 1])

With an integer instead of a Finite Field:

sage: BD = designs.AffineGeometryDesign(3, 2, 4)
sage: BD.parameters()
(2, 64, 16, 5)
sage.combinat.designs.block_design.BlockDesign(max_pt, blks, name=None, test=True)

Returns an instance of the IncidenceStructure class.

Requires each B in blks to be contained in range(max_pt). Does not test if the result is a block design.

EXAMPLES:

sage: BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]], name="Fano plane")
Incidence structure with 7 points and 7 blocks
sage: print BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]], name="Fano plane")
Fano plane<points=[0, 1, 2, 3, 4, 5, 6], blocks=[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]>
sage.combinat.designs.block_design.HadamardDesign(n)

As described in Section 1, p. 10, in [CvL]. The input n must have the property that there is a Hadamard matrix of order n+1 (and that a construction of that Hadamard matrix has been implemented...).

EXAMPLES:

sage: designs.HadamardDesign(7)
Incidence structure with 7 points and 7 blocks
sage: print designs.HadamardDesign(7)
HadamardDesign<points=[0, 1, 2, 3, 4, 5, 6], blocks=[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]>

REFERENCES:

  • [CvL] P. Cameron, J. H. van Lint, Designs, graphs, codes and their links, London Math. Soc., 1991.
sage.combinat.designs.block_design.ProjectiveGeometryDesign(n, d, F, algorithm=None)

Returns a projective geometry design.

A projective geometry design of parameters n,d,F has for points the lines of F^{n+1}, and for blocks the d+1-dimensional subspaces of F^{n+1}, each of which contains \frac {|F|^{d+1}-1} {|F|-1} lines.

INPUT:

  • n is the projective dimension
  • d is the dimension of the subspaces of P = PPn(F) which make up the blocks.
  • F is a finite field.
  • algorithm – set to None by default, which results in using Sage’s own implementation. In order to use GAP’s implementation instead (i.e. its PGPointFlatBlockDesign function) set algorithm="gap". Note that GAP’s “design” package must be available in this case.

EXAMPLES:

The points of the following design are the \frac {2^{2+1}-1} {2-1}=7 lines of \mathbb{Z}_2^{2+1}. It has 7 blocks, corresponding to each 2-dimensional subspace of \mathbb{Z}_2^{2+1}:

sage: designs.ProjectiveGeometryDesign(2, 1, GF(2))
Incidence structure with 7 points and 7 blocks
sage: BD = designs.ProjectiveGeometryDesign(2, 1, GF(2), algorithm="gap") # optional - gap_packages (design package)
sage: BD.is_block_design()                                     # optional - gap_packages (design package)
(True, [2, 7, 3, 1])
sage.combinat.designs.block_design.ProjectivePlaneDesign(n, type='Desarguesian')

Returns a projective plane of order n.

A finite projective plane is a 2-design with n^2+n+1 lines (or blocks) and n^2+n+1 points. For more information on finite projective planes, see the Wikipedia article Projective_plane#Finite_projective_planes.

INPUT:

  • n – the finite projective plane’s order

  • type – When set to "Desarguesian", the method returns Desarguesian projective planes, i.e. a finite projective plane obtained by considering the 1- and 2- dimensional spaces of F_n^3.

    For the moment, no other value is available for this parameter.

EXAMPLES:

sage: designs.ProjectivePlaneDesign(2)
Incidence structure with 7 points and 7 blocks

Non-existent ones:

sage: designs.ProjectivePlaneDesign(10)
Traceback (most recent call last):
...
ValueError: No projective plane design of order 10 exists.
sage: designs.ProjectivePlaneDesign(14)
Traceback (most recent call last):
...
ValueError: By the Bruck-Ryser-Chowla theorem, no projective plane of order 14 exists.

An unknown one:

sage: designs.ProjectivePlaneDesign(12)
Traceback (most recent call last):
...
ValueError: If such a projective plane exists, we do not know how to build it.

TESTS:

sage: designs.ProjectivePlaneDesign(10, type="AnyThingElse")
Traceback (most recent call last):
...
ValueError: The value of 'type' must be 'Desarguesian'.
sage.combinat.designs.block_design.WittDesign(n)

INPUT:

  • n is in 9,10,11,12,21,22,23,24.

Wraps GAP Design’s WittDesign. If n=24 then this function returns the large Witt design W_{24}, the unique (up to isomorphism) 5-(24,8,1) design. If n=12 then this function returns the small Witt design W_{12}, the unique (up to isomorphism) 5-(12,6,1) design. The other values of n return a block design derived from these.

EXAMPLES:

sage: BD = designs.WittDesign(9)   # optional - gap_packages (design package)
sage: BD.parameters()      # optional - gap_packages (design package)
(2, 9, 3, 1)
sage: BD                   # optional - gap_packages (design package)
Incidence structure with 9 points and 12 blocks
sage: print BD             # optional - gap_packages (design package)
WittDesign<points=[0, 1, 2, 3, 4, 5, 6, 7, 8], blocks=[[0, 1, 7], [0, 2, 5], [0, 3, 4], [0, 6, 8], [1, 2, 6], [1, 3, 5], [1, 4, 8], [2, 3, 8], [2, 4, 7], [3, 6, 7], [4, 5, 6], [5, 7, 8]]>
sage: BD = designs.WittDesign(12)  # optional - gap_packages (design package)
sage: BD.parameters(t=5)   # optional - gap_packages (design package)
(5, 12, 6, 1)
sage.combinat.designs.block_design.steiner_triple_system(n)

Returns a Steiner Triple System

A Steiner Triple System (STS) of a set \{0,...,n-1\} is a family S of 3-sets such that for any i \not = j there exists exactly one set of S in which they are both contained.

It can alternatively be thought of as a factorization of the complete graph K_n with triangles.

A Steiner Triple System of a n-set exists if and only if n \equiv 1 \pmod 6 or n \equiv 3 \pmod 6, in which case one can be found through Bose’s and Skolem’s constructions, respectively [AndHon97].

INPUT:

  • n returns a Steiner Triple System of \{0,...,n-1\}

EXAMPLE:

A Steiner Triple System on 9 elements

sage: sts = designs.steiner_triple_system(9)
sage: sts
Incidence structure with 9 points and 12 blocks
sage: list(sts)
[[0, 1, 5], [0, 2, 4], [0, 3, 6], [0, 7, 8], [1, 2, 3], [1, 4, 7], [1, 6, 8], [2, 5, 8], [2, 6, 7], [3, 4, 8], [3, 5, 7], [4, 5, 6]]

As any pair of vertices is covered once, its parameters are

sage: sts.parameters()
(2, 9, 3, 1)

An exception is raised for invalid values of n

sage: designs.steiner_triple_system(10)
Traceback (most recent call last):
...
ValueError: Steiner triple systems only exist for n = 1 mod 6 or n = 3 mod 6

REFERENCE:

[AndHon97]A short course in Combinatorial Designs, Ian Anderson, Iiro Honkala, Internet Editions, Spring 1997, http://www.utu.fi/~honkala/designs.ps
sage.combinat.designs.block_design.tdesign_params(t, v, k, L)

Return the design’s parameters: (t, v, b, r , k, L). Note that t must be given.

EXAMPLES:

sage: BD = BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: from sage.combinat.designs.block_design import tdesign_params
sage: tdesign_params(2,7,3,1)
(2, 7, 7, 3, 3, 1)

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