# Block designs¶

A block design is a set together with a family of subsets (repeated subsets are allowed) whose members are chosen to satisfy some set of properties that are deemed useful for a particular application. See Wikipedia article Block_design.

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)
 [We07] Charles Weibel, “Survey of Non-Desarguesian planes” (2007), notices of the AMS, vol. 54 num. 10, pages 1294–1303

AUTHORS:

• Vincent Delecroix (2014): rewrite the part on projective planes trac ticket #16281

• 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.

Todo

Implement finite non-Desarguesian plane as in [We07] and Wikipedia article Non-Desarguesian_plane.

## Functions and methods¶

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

Return 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.is_t_design(return_parameters=True)
(True, (2, 8, 2, 1))
sage: BD = designs.AffineGeometryDesign(3, 2, GF(2))
sage: BD.is_t_design(return_parameters=True)
(True, (3, 8, 4, 1))


With an integer instead of a Finite Field:

sage: BD = designs.AffineGeometryDesign(3, 2, 4)
sage: BD.is_t_design(return_parameters=True)
(True, (2, 64, 16, 5))

sage.combinat.designs.block_design.DesarguesianProjectivePlaneDesign(n, check=True)

Return the Desarguesian projective plane of order n as a 2-design.

The Desarguesian projective plane of order $$n$$ can also be defined as the projective plane over a field of order $$n$$. For more information, have a look at Wikipedia article Projective_plane.

INPUT:

• n – an integer which must be a power of a prime number
• check – (boolean) Whether to check that output is correct before returning it. As this is expected to be useless (but we are cautious guys), you may want to disable it whenever you want speed. Set to True by default.

EXAMPLES:

sage: designs.DesarguesianProjectivePlaneDesign(2)
Incidence structure with 7 points and 7 blocks
sage: designs.DesarguesianProjectivePlaneDesign(3)
Incidence structure with 13 points and 13 blocks
sage: designs.DesarguesianProjectivePlaneDesign(4)
Incidence structure with 21 points and 21 blocks
sage: designs.DesarguesianProjectivePlaneDesign(5)
Incidence structure with 31 points and 31 blocks
sage: designs.DesarguesianProjectivePlaneDesign(6)
Traceback (most recent call last):
...
ValueError: the order of a finite field must be a prime power


Return the Hadamard 3-design with parameters $$3-(n, \frac n 2, \frac n 4 - 1)$$.

This is the unique extension of the Hadamard $$2$$-design (see HadamardDesign()). We implement the description from pp. 12 in [CvL].

INPUT:

• n (integer) – a multiple of 4 such that $$n>4$$.

EXAMPLES:

sage: designs.Hadamard3Design(12)
Incidence structure with 12 points and 22 blocks


We verify that any two blocks of the Hadamard $$3$$-design $$3-(8, 4, 1)$$ design meet in $$0$$ or $$2$$ points. More generally, it is true that any two blocks of a Hadamard $$3$$-design meet in $$0$$ or $$\frac{n}{4}$$ points (for $$n > 4$$).

sage: D = designs.Hadamard3Design(8)
sage: N = D.incidence_matrix()
sage: N.transpose()*N
[4 2 2 2 2 2 2 2 2 2 2 2 2 0]
[2 4 2 2 2 2 2 2 2 2 2 2 0 2]
[2 2 4 2 2 2 2 2 2 2 2 0 2 2]
[2 2 2 4 2 2 2 2 2 2 0 2 2 2]
[2 2 2 2 4 2 2 2 2 0 2 2 2 2]
[2 2 2 2 2 4 2 2 0 2 2 2 2 2]
[2 2 2 2 2 2 4 0 2 2 2 2 2 2]
[2 2 2 2 2 2 0 4 2 2 2 2 2 2]
[2 2 2 2 2 0 2 2 4 2 2 2 2 2]
[2 2 2 2 0 2 2 2 2 4 2 2 2 2]
[2 2 2 0 2 2 2 2 2 2 4 2 2 2]
[2 2 0 2 2 2 2 2 2 2 2 4 2 2]
[2 0 2 2 2 2 2 2 2 2 2 2 4 2]
[0 2 2 2 2 2 2 2 2 2 2 2 2 4]


REFERENCES:

 [CvL] P. Cameron, J. H. van Lint, Designs, graphs, codes and their links, London Math. Soc., 1991.

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
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]]>


For example, the Hadamard 2-design with $$n = 11$$ is a design whose parameters are 2-(11, 5, 2). We verify that $$NJ = 5J$$ for this design.

sage: D = designs.HadamardDesign(11); N = D.incidence_matrix()
sage: J = matrix(ZZ, 11, 11, [1]*11*11); N*J
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 5]
[5 5 5 5 5 5 5 5 5 5 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, check=True)

Return 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, and that it can be installed with the gap_packages spkg.

EXAMPLES:

The set of $$d$$-dimensional subspaces in a $$n$$-dimensional projective space forms $$2$$-designs (or balanced incomplete block designs):

sage: PG = designs.ProjectiveGeometryDesign(4,2,GF(2))
sage: PG
Incidence structure with 31 points and 155 blocks
sage: PG.is_t_design(return_parameters=True)
(True, (2, 31, 7, 7))

sage: PG = designs.ProjectiveGeometryDesign(3,1,GF(4,'z'))
sage: PG.is_t_design(return_parameters=True)
(True, (2, 85, 5, 1))


Check that the constructor using gap also works:

sage: BD = designs.ProjectiveGeometryDesign(2, 1, GF(2), algorithm="gap") # optional - gap_packages (design package)
sage: BD.is_t_design(return_parameters=True)                              # optional - gap_packages (design package)
(True, (2, 7, 3, 1))

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.is_t_design(return_parameters=True) # optional - gap_packages (design package)
(True, (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.combinat.designs.block_design.are_hyperplanes_in_projective_geometry_parameters(v, k, lmbda, return_parameters=False)

Return True if the parameters (v,k,lmbda) are the one of hyperplanes in a (finite Desarguesian) projective space.

In other words, test whether there exists a prime power q and an integer d greater than two such that:

• $$v = (q^{d+1}-1)/(q-1) = q^d + q^{d-1} + ... + 1$$
• $$k = (q^d - 1)/(q-1) = q^{d-1} + q^{d-2} + ... + 1$$
• $$lmbda = (q^{d-1}-1)/(q-1) = q^{d-2} + q^{d-3} + ... + 1$$

If it exists, such a pair (q,d) is unique.

INPUT:

• v,k,lmbda (integers)

OUTPUT:

• a boolean or, if return_parameters is set to True a pair (True, (q,d)) or (False, (None,None)).

EXAMPLES:

sage: from sage.combinat.designs.block_design import are_hyperplanes_in_projective_geometry_parameters
sage: are_hyperplanes_in_projective_geometry_parameters(40,13,4)
True
sage: are_hyperplanes_in_projective_geometry_parameters(40,13,4,return_parameters=True)
(True, (3, 3))
sage: PG = designs.ProjectiveGeometryDesign(3,2,GF(3))
sage: PG.is_t_design(return_parameters=True)
(True, (2, 40, 13, 4))

sage: are_hyperplanes_in_projective_geometry_parameters(15,3,1)
False
sage: are_hyperplanes_in_projective_geometry_parameters(15,3,1,return_parameters=True)
(False, (None, None))


TESTS:

sage: sgp = lambda q,d: ((q**(d+1)-1)//(q-1), (q**d-1)//(q-1), (q**(d-1)-1)//(q-1))
sage: for q in [3,4,5,7,8,9,11]:
....:     for d in [2,3,4,5]:
....:         v,k,l = sgp(q,d)
....:         assert are_hyperplanes_in_projective_geometry_parameters(v,k,l,True) == (True, (q,d))
....:         assert are_hyperplanes_in_projective_geometry_parameters(v+1,k,l) is False
....:         assert are_hyperplanes_in_projective_geometry_parameters(v-1,k,l) is False
....:         assert are_hyperplanes_in_projective_geometry_parameters(v,k+1,l) is False
....:         assert are_hyperplanes_in_projective_geometry_parameters(v,k-1,l) is False
....:         assert are_hyperplanes_in_projective_geometry_parameters(v,k,l+1) is False
....:         assert are_hyperplanes_in_projective_geometry_parameters(v,k,l-1) is False

sage.combinat.designs.block_design.projective_plane(n, check=True, existence=False)

Return a projective plane of order n as a 2-design.

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.

If no construction is possible, then the function raises a EmptySetError whereas if no construction is available the function raises a NotImplementedError.

INPUT:

• n – the finite projective plane’s order

EXAMPLES:

sage: designs.projective_plane(2)
Incidence structure with 7 points and 7 blocks
sage: designs.projective_plane(3)
Incidence structure with 13 points and 13 blocks
sage: designs.projective_plane(4)
Incidence structure with 21 points and 21 blocks
sage: designs.projective_plane(5)
Incidence structure with 31 points and 31 blocks
sage: designs.projective_plane(6)
Traceback (most recent call last):
...
EmptySetError: By the Bruck-Ryser theorem, no projective plane of order 6 exists.
sage: designs.projective_plane(10)
Traceback (most recent call last):
...
EmptySetError: No projective plane of order 10 exists by C. Lam, L. Thiel and S. Swiercz "The nonexistence of finite projective planes of order 10" (1989), Canad. J. Math.
sage: designs.projective_plane(12)
Traceback (most recent call last):
...
NotImplementedError: If such a projective plane exists, we do not know how to build it.
sage: designs.projective_plane(14)
Traceback (most recent call last):
...
EmptySetError: By the Bruck-Ryser theorem, no projective plane of order 14 exists.


TESTS:

sage: designs.projective_plane(2197, existence=True)
True
sage: designs.projective_plane(6, existence=True)
False
sage: designs.projective_plane(10, existence=True)
False
sage: designs.projective_plane(12, existence=True)
Unknown

sage.combinat.designs.block_design.projective_plane_to_OA(pplane, pt=None, check=True)

Return the orthogonal array built from the projective plane pplane.

The orthogonal array $$OA(n+1,n,2)$$ is obtained from the projective plane pplane by removing the point pt and the $$n+1$$ lines that pass through it. These $$n+1$$ lines form the $$n+1$$ groups while the remaining $$n^2+n$$ lines form the transversals.

INPUT:

• pplane - a projective plane as a 2-design
• pt - a point in the projective plane pplane. If it is not provided then it is set to $$n^2 + n$$.
• check – (boolean) Whether to check that output is correct before returning it. As this is expected to be useless (but we are cautious guys), you may want to disable it whenever you want speed. Set to True by default.

EXAMPLES:

sage: from sage.combinat.designs.block_design import projective_plane_to_OA
sage: p2 = designs.DesarguesianProjectivePlaneDesign(2)
sage: projective_plane_to_OA(p2)
[[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
sage: p3 = designs.DesarguesianProjectivePlaneDesign(3)
sage: projective_plane_to_OA(p3)
[[0, 0, 0, 0],
[0, 1, 2, 1],
[0, 2, 1, 2],
[1, 0, 2, 2],
[1, 1, 1, 0],
[1, 2, 0, 1],
[2, 0, 1, 1],
[2, 1, 0, 2],
[2, 2, 2, 0]]

sage: pp = designs.DesarguesianProjectivePlaneDesign(16)
sage: _ = projective_plane_to_OA(pp, pt=0)
sage: _ = projective_plane_to_OA(pp, pt=3)
sage: _ = projective_plane_to_OA(pp, pt=7)

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 = designs.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)
`

Covering designs: coverings of $$t$$-element subsets of a $$v$$-set by $$k$$-sets