def 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.
"""
from sage.combinat.matrices.hadamard_matrix import hadamard_matrix
from sage.matrix.constructor import matrix
H = hadamard_matrix(n + 1)
H1 = H.matrix_from_columns(range(1, n + 1))
H2 = H1.matrix_from_rows(range(1, n + 1))
J = matrix(ZZ, n, n, [1] * n * n)
MS = J.parent()
A = MS((H2 + J) / 2) # convert -1's to 0's; coerce entries to ZZ
# A is the incidence matrix of the block design
return IncidenceStructureFromMatrix(A, name="HadamardDesign")
def 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]]>
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.
"""
from sage.combinat.matrices.hadamard_matrix import hadamard_matrix
from sage.matrix.constructor import matrix
H = hadamard_matrix(n + 1) #assumed to be normalised.
H1 = H.matrix_from_columns(range(1, n + 1))
H2 = H1.matrix_from_rows(range(1, n + 1))
J = matrix(ZZ, n, n, [1] * n * n)
MS = J.parent()
A = MS((H2 + J) / 2) # convert -1's to 0's; coerce entries to ZZ
# A is the incidence matrix of the block design
return IncidenceStructureFromMatrix(A, name="HadamardDesign")
def 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
:meth:`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.
"""
if n == 1 or n == 4:
raise ValueError(
"The Hadamard design with n = %s does not extend to a three design."
% n)
from sage.combinat.matrices.hadamard_matrix import hadamard_matrix
from sage.matrix.constructor import matrix, block_matrix
H = hadamard_matrix(n) #assumed to be normalised.
H1 = H.matrix_from_columns(range(1, n))
J = matrix(ZZ, n, n - 1, [1] * (n - 1) * n)
A1 = (H1 + J) / 2
A2 = (J - H1) / 2
A = block_matrix(1, 2, [A1, A2]) #the incidence matrix of the design.
return IncidenceStructureFromMatrix(A, name="HadamardThreeDesign")