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:

  Block design from wikipedia, Wikipedia article Block_design
  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)
 [Hu57] Daniel R. Hughes, “A class of non-Desarguesian projective planes”, The Canadian Journal of Mathematics (1957), http://cms.math.ca/cjm/v9/p378
 [We07] Charles Weibel, “Survey of Non-Desarguesian planes” (2007), notices of the AMS, vol. 54 num. 10, pages 1294–1303

AUTHORS:

• Quentin Honoré (2015): construction of Hughes plane trac ticket #18527

• 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 more 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, point_coordinates=True, check=True)

Return an affine geometry design.

The affine geometry design $$AG_d(n,q)$$ is the 2-design whose blocks are the $$d$$-vector subspaces in $$\GF{q}^n$$. It has parameters

$v = q^n,\ k = q^d,\ \lambda = \binom{n-1}{d-1}_q$

where the $$q$$-binomial coefficient $$\binom{m}{r}_q$$ is defined by

$\binom{m}{r}_q = \frac{(q^m - 1)(q^{m-1} - 1) \cdots (q^{m-r+1}-1)} {(q^r-1)(q^{r-1}-1)\cdots (q-1)}$

INPUT:

• n (integer) – the Euclidean dimension. The number of points of the design is $$v=|\GF{q}^n|$$.
• d (integer) – the dimension of the (affine) subspaces of $$\GF{q}^n$$ which make up the blocks.
• F – a finite field or a prime power.
• point_coordinates – (optional, default True) whether we use coordinates in $$\GF{q}^n$$ or plain integers for the points of the design.
• check – (optional, default True) whether to check the output.

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(4))
sage: BD.is_t_design(return_parameters=True)
(True, (2, 64, 16, 5))
sage: BD = designs.AffineGeometryDesign(4, 2, GF(3))
sage: BD.is_t_design(return_parameters=True)
(True, (2, 81, 9, 13))

With F 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))

Testing the option point_coordinates:

sage: designs.AffineGeometryDesign(3, 1, GF(2), point_coordinates=True).blocks()
[(0, 0, 0), (0, 0, 1)]
sage: designs.AffineGeometryDesign(3, 1, GF(2), point_coordinates=False).blocks()
[0, 1]
sage.combinat.designs.block_design.CremonaRichmondConfiguration()

Return the Cremona-Richmond configuration

The Cremona-Richmond configuration is a set system whose incidence graph is equal to the TutteCoxeterGraph(). It is a generalized quadrangle of parameters $$(2,2)$$.

For more information, see the Wikipedia article Cremona-Richmond_configuration.

EXAMPLES:

sage: H = designs.CremonaRichmondConfiguration(); H
Incidence structure with 15 points and 15 blocks
sage: g = graphs.TutteCoxeterGraph()
sage: H.incidence_graph().is_isomorphic(g)
True
sage.combinat.designs.block_design.DesarguesianProjectivePlaneDesign(n, point_coordinates=True, 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
• point_coordinates (boolean) – whether to label the points with their homogeneous coordinates (default) or with integers.
• 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)
(7,3,1)-Balanced Incomplete Block Design
sage: designs.DesarguesianProjectivePlaneDesign(3)
(13,4,1)-Balanced Incomplete Block Design
sage: designs.DesarguesianProjectivePlaneDesign(4)
(21,5,1)-Balanced Incomplete Block Design
sage: designs.DesarguesianProjectivePlaneDesign(5)
(31,6,1)-Balanced Incomplete Block Design
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:

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:

Incidence structure with 7 points and 7 blocks
Incidence structure with 7 points and 7 blocks

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, *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.HughesPlane(q2, check=True)

Return the Hughes projective plane of order q2.

Let $$q$$ be an odd prime, the Hughes plane of order $$q^2$$ is a finite projective plane of order $$q^2$$ introduced by D. Hughes in [Hu57]. Its construction is as follows.

Let $$K = GF(q^2)$$ be a finite field with $$q^2$$ elements and $$F = GF(q) \subset K$$ be its unique subfield with $$q$$ elements. We define a twisted multiplication on $$K$$ as

$\begin{split}x \circ y = \begin{cases} x\ y & \text{if y is a square in K}\\ x^q\ y & \text{otherwise} \end{cases}\end{split}$

The points of the Hughes plane are the triples $$(x, y, z)$$ of points in $$K^3 \backslash \{0,0,0\}$$ up to the equivalence relation $$(x,y,z) \sim (x \circ k, y \circ k, z \circ k)$$ where $$k \in K$$.

For $$a = 1$$ or $$a \in (K \backslash F)$$ we define a block $$L(a)$$ as the set of triples $$(x,y,z)$$ so that $$x + a \circ y + z = 0$$. The rest of the blocks are obtained by letting act the group $$GL(3, F)$$ by its standard action.

For more information, see Wikipedia article Hughes_plane and [We07].

DesarguesianProjectivePlaneDesign() to build the Desarguesian projective planes

INPUT:

• q2 – an even power of an odd 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: H = designs.HughesPlane(9)
sage: H
(91,10,1)-Balanced Incomplete Block Design

We prove in the following computations that the Desarguesian plane H is not Desarguesian. Let us consider the two triangles $$(0,1,10)$$ and $$(57, 70, 59)$$. We show that the intersection points $$D_{0,1} \cap D_{57,70}$$, $$D_{1,10} \cap D_{70,59}$$ and $$D_{10,0} \cap D_{59,57}$$ are on the same line while $$D_{0,70}$$, $$D_{1,59}$$ and $$D_{10,57}$$ are not concurrent:

sage: blocks = H.blocks()
sage: line = lambda p,q: next(b for b in blocks if p in b and q in b)

sage: b_0_1 = line(0, 1)
sage: b_1_10 = line(1, 10)
sage: b_10_0 = line(10, 0)
sage: b_57_70 = line(57, 70)
sage: b_70_59 = line(70, 59)
sage: b_59_57 = line(59, 57)

sage: set(b_0_1).intersection(b_57_70)
{2}
sage: set(b_1_10).intersection(b_70_59)
{73}
sage: set(b_10_0).intersection(b_59_57)
{60}

sage: line(2, 73) == line(73, 60)
True

sage: b_0_57 = line(0, 57)
sage: b_1_70 = line(1, 70)
sage: b_10_59 = line(10, 59)

sage: p = set(b_0_57).intersection(b_1_70)
sage: q = set(b_1_70).intersection(b_10_59)
sage: p == q
False
sage.combinat.designs.block_design.ProjectiveGeometryDesign(n, d, F, algorithm=None, point_coordinates=True, check=True)

Return a projective geometry design.

The projective geometry design $$PG_d(n,q)$$ has for points the lines of $$\GF{q}^{n+1}$$, and for blocks the $$d+1$$-dimensional subspaces of $$\GF{q}^{n+1}$$, each of which contains $$\frac {|\GF{q}|^{d+1}-1} {|\GF{q}|-1}$$ lines. It is a $$2$$-design with parameters

$v = \binom{n+1}{1}_q,\ k = \binom{d+1}{1}_q,\ \lambda = \binom{n-1}{d-1}_q$

where the $$q$$-binomial coefficient $$\binom{m}{r}_q$$ is defined by

$\binom{m}{r}_q = \frac{(q^m - 1)(q^{m-1} - 1) \cdots (q^{m-r+1}-1)} {(q^r-1)(q^{r-1}-1)\cdots (q-1)}$

INPUT:

• n is the projective dimension
• d is the dimension of the subspaces which make up the blocks.
• F – a finite field or a prime power.
• 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.
• point_coordinatesTrue by default. Ignored and assumed to be False if algorithm="gap". If True, the ground set is indexed by coordinates in $$\GF{q}^{n+1}$$. Otherwise the ground set is indexed by integers.
• check – (optional, default to True) whether to check the output.

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))
sage: PG.is_t_design(return_parameters=True)
(True, (2, 85, 5, 1))

Check with F being a prime power:

sage: PG = designs.ProjectiveGeometryDesign(3, 2, 4)
sage: PG
Incidence structure with 85 points and 85 blocks

Use coordinates:

sage: PG = designs.ProjectiveGeometryDesign(2, 1, GF(3))
sage: PG.blocks()
[(1, 0, 0), (1, 0, 1), (1, 0, 2), (0, 0, 1)]

Use indexing by integers:

sage: PG = designs.ProjectiveGeometryDesign(2,1,GF(3),point_coordinates=0)
sage: PG.blocks()
[0, 1, 2, 12]

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.

Note

Requires GAP’s Design package (included in the gap_packages Sage spkg).

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)
Incidence structure with 9 points and 12 blocks
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))
sage.combinat.designs.block_design.normalize_hughes_plane_point(p, q)

Return the normalized form of point p as a 3-tuple.

In the Hughes projective plane over the finite field $$K$$, all triples $$(xk, yk, zk)$$ with $$k \in K$$ represent the same point (where the multiplication is over the nearfield built from $$K$$). This function chooses a canonical representative among them.

This function is used in HughesPlane().

INPUT:

• p - point with the coordinates (x,y,z) (a list, a vector, a tuple…)
• q - cardinality of the underlying finite field

EXAMPLES:

sage: from sage.combinat.designs.block_design import normalize_hughes_plane_point
sage: K = FiniteField(9,'x')
sage: x = K.gen()
sage: normalize_hughes_plane_point((x, x+1, x), 9)
(1, x, 1)
sage: normalize_hughes_plane_point(vector((x,x,x)), 9)
(1, 1, 1)
sage: zero = K.zero()
sage: normalize_hughes_plane_point((2*x+2, zero, zero), 9)
(1, 0, 0)
sage: one = K.one()
sage: normalize_hughes_plane_point((2*x, one, zero), 9)
(2*x, 1, 0)
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)
(7,3,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(3)
(13,4,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(4)
(21,5,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(5)
(31,6,1)-Balanced Incomplete Block Design
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.
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,point_coordinates=False)
sage: projective_plane_to_OA(p2)
[[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
sage: p3 = designs.DesarguesianProjectivePlaneDesign(3,point_coordinates=False)
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,point_coordinates=False)
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.q3_minus_one_matrix(K)

Return a companion matrix in $$GL(3, K)$$ whose multiplicative order is $$q^3 - 1$$.

This function is used in HughesPlane()

EXAMPLES:

sage: from sage.combinat.designs.block_design import q3_minus_one_matrix
sage: m = q3_minus_one_matrix(GF(3))
sage: m.multiplicative_order() == 3**3 - 1
True

sage: m = q3_minus_one_matrix(GF(4,'a'))
sage: m.multiplicative_order() == 4**3 - 1
True

sage: m = q3_minus_one_matrix(GF(5))
sage: m.multiplicative_order() == 5**3 - 1
True

sage: m = q3_minus_one_matrix(GF(9,'a'))
sage: m.multiplicative_order() == 9**3 - 1
True
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)