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)
Daniel R. Hughes, “A class of non-Desarguesian projective planes”, The Canadian Journal of Mathematics (1957), http://cms.math.ca/cjm/v9/p378
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 github issue #18527
Vincent Delecroix (2014): rewrite the part on projective planes github issue #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
\[\binom{m}{r}_q = \frac{(q^m - 1)(q^{m-1} - 1) \cdots (q^{m-r+1}-1)} {(q^r-1)(q^{r-1}-1)\cdots (q-1)}\]See also
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, defaultTrue
) whether we use coordinates in \(\GF{q}^n\) or plain integers for the points of the design.check
– (optional, defaultTrue
) whether to check the output.
EXAMPLES:
sage: # needs sage.combinat 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, 0, 1)] sage: designs.AffineGeometryDesign(3, 1, GF(2), ....: point_coordinates=False).blocks()[0] [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 # needs networkx Incidence structure with 15 points and 15 blocks sage: g = graphs.TutteCoxeterGraph() # needs networkx sage: H.incidence_graph().is_isomorphic(g) # needs networkx 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 numberpoint_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 toTrue
by default.
See also
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
- sage.combinat.designs.block_design.Hadamard3Design(n)#
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: # needs sage.combinat sage.modules 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: # needs sage.combinat sage.modules 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:
- 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: # needs sage.combinat sage.modules sage: designs.HadamardDesign(7) Incidence structure with 7 points and 7 blocks sage: print(designs.HadamardDesign(7)) 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: # needs sage.combinat sage.modules 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.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].
See also
DesarguesianProjectivePlaneDesign()
to build the Desarguesian projective planesINPUT:
q2
– an even power of an odd prime numbercheck
– (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 toTrue
by default.
EXAMPLES:
sage: H = designs.HughesPlane(9); 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
\[\binom{m}{r}_q = \frac{(q^m - 1)(q^{m-1} - 1) \cdots (q^{m-r+1}-1)} {(q^r-1)(q^{r-1}-1)\cdots (q-1)}\]See also
INPUT:
n
– the projective dimensiond
– the dimension of the subspaces which make up the blocks.F
– a finite field or a prime power.algorithm
– set toNone
by default, which results in using Sage’s own implementation. In order to use GAP’s implementation instead (i.e. itsPGPointFlatBlockDesign
function) setalgorithm="gap"
. Note that GAP’s “design” package must be available in this case, and that it can be installed with thegap_packages
spkg.point_coordinates
–True
by default. Ignored and assumed to beFalse
ifalgorithm="gap"
. IfTrue
, the ground set is indexed by coordinates in \(\GF{q}^{n+1}\). Otherwise the ground set is indexed by integers.check
– (optional, default toTrue
) 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)); PG # needs sage.combinat Incidence structure with 31 points and 155 blocks sage: PG.is_t_design(return_parameters=True) # needs sage.combinat (True, (2, 31, 7, 7)) sage: PG = designs.ProjectiveGeometryDesign(3, 1, GF(4)) # needs sage.combinat sage: PG.is_t_design(return_parameters=True) # needs sage.combinat (True, (2, 85, 5, 1))
Check with
F
being a prime power:sage: PG = designs.ProjectiveGeometryDesign(3, 2, 4); PG Incidence structure with 85 points and 85 blocks
Use coordinates:
sage: PG = designs.ProjectiveGeometryDesign(2, 1, GF(3)) sage: PG.blocks()[0] [(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] [0, 1, 2, 12]
Check that the constructor using gap also works:
sage: BD = designs.ProjectiveGeometryDesign(2, 1, GF(2), algorithm="gap") # optional - gap_package_design sage: BD.is_t_design(return_parameters=True) # optional - gap_package_design (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. Ifn=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: # optional - gap_package_design sage: BD = designs.WittDesign(9) sage: BD.is_t_design(return_parameters=True) (True, (2, 9, 3, 1)) sage: BD Incidence structure with 9 points and 12 blocks sage: print(BD) 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 integerd
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 toTrue
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)) # needs sage.combinat sage: PG.is_t_design(return_parameters=True) # needs sage.combinat (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 aNotImplementedError
.INPUT:
n
– the finite projective plane’s order
EXAMPLES:
sage: # needs sage.schemes 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 pointpt
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-designpt
– a point in the projective planepplane
. 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 toTrue
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)