def difference_matrix_product(k, M1, G1, lmbda1, M2, G2, lmbda2, check=True):
r"""
Return the product of the ``(G1,k,lmbda1)`` and ``(G2,k,lmbda2)`` difference
matrices ``M1`` and ``M2``.
The result is a `(G1 \times G2, k, \lambda_1 \lambda_2)`-difference matrix.
INPUT:
- ``k,lmbda1,lmbda2`` -- positive integer
- ``G1, G2`` -- groups
- ``M1, M2`` -- ``(G1,k,lmbda1)`` and ``(G,k,lmbda2)`` difference
matrices
- ``check`` (boolean) -- if ``True`` (default), the output is checked before
being returned.
EXAMPLES::
sage: from sage.combinat.designs.difference_matrices import (
....: difference_matrix_product,
....: is_difference_matrix)
sage: G1,M1 = designs.difference_matrix(11,6)
sage: G2,M2 = designs.difference_matrix(7,6)
sage: G,M = difference_matrix_product(6,M1,G1,1,M2,G2,1)
sage: G1
Finite Field of size 11
sage: G2
Finite Field of size 7
sage: G
The cartesian product of (Finite Field of size 11, Finite Field of size 7)
sage: is_difference_matrix(M,G,6,1)
True
"""
g1 = G1.cardinality()
g2 = G2.cardinality()
g = g1 * g2
lmbda = lmbda1 * lmbda2
from sage.categories.cartesian_product import cartesian_product
G = cartesian_product([G1, G2])
M = [[G((M1[j1][i], M2[j2][i])) for i in range(k)]
for j1 in range(lmbda1 * g1) for j2 in range(lmbda2 * g2)]
if check and not is_difference_matrix(M, G, k, lmbda, True):
raise RuntimeError(
"In the product construction, Sage built something which is not a ({},{},{})-DM!"
.format(g, k, lmbda))
return G, M
def difference_matrix_product(k, M1, G1, lmbda1, M2, G2, lmbda2, check=True):
r"""
Return the product of the ``(G1,k,lmbda1)`` and ``(G2,k,lmbda2)`` difference
matrices ``M1`` and ``M2``.
The result is a `(G1 \times G2, k, \lambda_1 \lambda_2)`-difference matrix.
INPUT:
- ``k,lmbda1,lmbda2`` -- positive integer
- ``G1, G2`` -- groups
- ``M1, M2`` -- ``(G1,k,lmbda1)`` and ``(G,k,lmbda2)`` difference
matrices
- ``check`` (boolean) -- if ``True`` (default), the output is checked before
being returned.
EXAMPLES::
sage: from sage.combinat.designs.difference_matrices import (
....: difference_matrix_product,
....: is_difference_matrix)
sage: G1,M1 = designs.difference_matrix(11,6)
sage: G2,M2 = designs.difference_matrix(7,6)
sage: G,M = difference_matrix_product(6,M1,G1,1,M2,G2,1)
sage: G1
Finite Field of size 11
sage: G2
Finite Field of size 7
sage: G
The Cartesian product of (Finite Field of size 11, Finite Field of size 7)
sage: is_difference_matrix(M,G,6,1)
True
"""
g1 = G1.cardinality()
g2 = G2.cardinality()
g = g1 * g2
lmbda = lmbda1 * lmbda2
from sage.categories.cartesian_product import cartesian_product
G = cartesian_product([G1, G2])
M = [[G((M1[j1][i], M2[j2][i])) for i in range(k)] for j1 in range(lmbda1 * g1) for j2 in range(lmbda2 * g2)]
if check and not is_difference_matrix(M, G, k, lmbda, True):
raise RuntimeError(
"In the product construction, Sage built something which is not a ({},{},{})-DM!".format(g, k, lmbda)
)
return G, M
def difference_matrix(g, k, lmbda=1, existence=False, check=True):
r"""
Return a `(g,k,\lambda)`-difference matrix
A matrix `M` is a `(g,k,\lambda)`-difference matrix if it has size `\lambda
g\times k`, its entries belong to the group `G` of cardinality `g`, and
for any two rows `R,R'` of `M` and `x\in G` there are exactly `\lambda`
values `i` such that `R_i-R'_i=x`.
INPUT:
- ``k`` -- (integer) number of columns. If ``k=None`` it is set to the
largest value available.
- ``g`` -- (integer) cardinality of the group `G`
- ``lmbda`` -- (integer; default: 1) -- number of times each element of `G`
appears as a difference.
- ``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.
- ``existence`` (boolean) -- instead of building the design, return:
- ``True`` -- meaning that Sage knows how to build the design
- ``Unknown`` -- meaning that Sage does not know how to build the
design, but that the design may exist (see :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the design does not exist.
.. NOTE::
When ``k=None`` and ``existence=True`` the function returns an
integer, i.e. the largest `k` such that we can build a
`(g,k,\lambda)`-DM.
EXAMPLES::
sage: G,M = designs.difference_matrix(25,10); G
Finite Field in x of size 5^2
sage: designs.difference_matrix(993,None,existence=1)
32
Here we print for each `g` the maximum possible `k` for which Sage knows
how to build a `(g,k,1)`-difference matrix::
sage: for g in range(2,30):
....: k_max = designs.difference_matrix(g=g,k=None,existence=True)
....: print "{:2} {}".format(g, k_max)
....: _ = designs.difference_matrix(g,k_max)
2 2
3 3
4 4
5 5
6 2
7 7
8 8
9 9
10 2
11 11
12 6
13 13
14 2
15 3
16 16
17 17
18 2
19 19
20 4
21 6
22 2
23 23
24 8
25 25
26 2
27 27
28 6
29 29
TESTS::
sage: designs.difference_matrix(10,12,1,existence=True)
False
sage: designs.difference_matrix(10,12,1)
Traceback (most recent call last):
...
EmptySetError: No (10,12,1)-Difference Matrix exists as k(=12)>g(=10)
sage: designs.difference_matrix(10,9,1,existence=True)
Unknown
sage: designs.difference_matrix(10,9,1)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build a (10,9,1)-Difference Matrix!
"""
if lmbda == 1 and k is not None and k > g:
if existence:
return False
raise EmptySetError(
"No ({},{},{})-Difference Matrix exists as k(={})>g(={})".format(
g, k, lmbda, k, g))
# Prime powers
elif lmbda == 1 and is_prime_power(g):
if k is None:
if existence:
return g
else:
k = g
elif existence:
return True
F = FiniteField(g, 'x')
F_set = list(F)
F_k_set = F_set[:k]
G = F
M = [[x * y for y in F_k_set] for x in F_set]
# Treat the case k=None
# (find the max k such that there exists a DM)
elif k is None:
i = 2
while difference_matrix(g=g, k=i, lmbda=lmbda, existence=True):
i += 1
return i - 1
# From the database
elif (g, lmbda) in DM_constructions and DM_constructions[g, lmbda][0] >= k:
if existence:
return True
_, f = DM_constructions[g, lmbda]
G, M = f()
M = [R[:k] for R in M]
# Product construction
elif find_product_decomposition(g, k, lmbda):
if existence:
return True
(g1, lmbda1), (g2, lmbda2) = find_product_decomposition(g, k, lmbda)
G1, M1 = difference_matrix(g1, k, lmbda1)
G2, M2 = difference_matrix(g2, k, lmbda2)
G, M = difference_matrix_product(k,
M1,
G1,
lmbda1,
M2,
G2,
lmbda2,
check=False)
else:
if existence:
return Unknown
raise NotImplementedError(
"I don't know how to build a ({},{},{})-Difference Matrix!".format(
g, k, lmbda))
if check and not is_difference_matrix(M, G, k, lmbda, 1):
raise RuntimeError(
"Sage built something which is not a ({},{},{})-DM!".format(
g, k, lmbda))
return G, M
def difference_matrix(g,k,lmbda=1,existence=False,check=True):
r"""
Return a `(g,k,\lambda)`-difference matrix
A matrix `M` is a `(g,k,\lambda)`-difference matrix if it has size `\lambda
g\times k`, its entries belong to the group `G` of cardinality `g`, and
for any two rows `R,R'` of `M` and `x\in G` there are exactly `\lambda`
values `i` such that `R_i-R'_i=x`.
INPUT:
- ``k`` -- (integer) number of columns. If ``k=None`` it is set to the
largest value available.
- ``g`` -- (integer) cardinality of the group `G`
- ``lmbda`` -- (integer; default: 1) -- number of times each element of `G`
appears as a difference.
- ``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.
- ``existence`` (boolean) -- instead of building the design, return:
- ``True`` -- meaning that Sage knows how to build the design
- ``Unknown`` -- meaning that Sage does not know how to build the
design, but that the design may exist (see :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the design does not exist.
.. NOTE::
When ``k=None`` and ``existence=True`` the function returns an
integer, i.e. the largest `k` such that we can build a
`(g,k,\lambda)`-DM.
EXAMPLES::
sage: G,M = designs.difference_matrix(25,10); G
Finite Field in x of size 5^2
sage: designs.difference_matrix(993,None,existence=1)
32
Here we print for each `g` the maximum possible `k` for which Sage knows
how to build a `(g,k,1)`-difference matrix::
sage: for g in range(2,30):
....: k_max = designs.difference_matrix(g=g,k=None,existence=True)
....: print "{:2} {}".format(g, k_max)
....: _ = designs.difference_matrix(g,k_max)
2 2
3 3
4 4
5 5
6 2
7 7
8 8
9 9
10 2
11 11
12 6
13 13
14 2
15 3
16 16
17 17
18 2
19 19
20 4
21 6
22 2
23 23
24 8
25 25
26 2
27 27
28 6
29 29
TESTS::
sage: designs.difference_matrix(10,12,1,existence=True)
False
sage: designs.difference_matrix(10,12,1)
Traceback (most recent call last):
...
EmptySetError: No (10,12,1)-Difference Matrix exists as k(=12)>g(=10)
sage: designs.difference_matrix(10,9,1,existence=True)
Unknown
sage: designs.difference_matrix(10,9,1)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build a (10,9,1)-Difference Matrix!
"""
if lmbda == 1 and k is not None and k>g:
if existence:
return False
raise EmptySetError("No ({},{},{})-Difference Matrix exists as k(={})>g(={})".format(g,k,lmbda,k,g))
# Prime powers
elif lmbda == 1 and is_prime_power(g):
if k is None:
if existence:
return g
else:
k = g
elif existence:
return True
F = FiniteField(g,'x')
F_set = list(F)
F_k_set = F_set[:k]
G = F
M = [[x*y for y in F_k_set] for x in F_set]
# Treat the case k=None
# (find the max k such that there exists a DM)
elif k is None:
i = 2
while difference_matrix(g=g,k=i,lmbda=lmbda,existence=True):
i += 1
return i-1
# From the database
elif (g,lmbda) in DM_constructions and DM_constructions[g,lmbda][0]>=k:
if existence:
return True
_,f = DM_constructions[g,lmbda]
G, M = f()
M = [R[:k] for R in M]
# Product construction
elif find_product_decomposition(g,k,lmbda):
if existence:
return True
(g1,lmbda1),(g2,lmbda2) = find_product_decomposition(g,k,lmbda)
G1,M1 = difference_matrix(g1,k,lmbda1)
G2,M2 = difference_matrix(g2,k,lmbda2)
G,M = difference_matrix_product(k,M1,G1,lmbda1,M2,G2,lmbda2,check=False)
else:
if existence:
return Unknown
raise NotImplementedError("I don't know how to build a ({},{},{})-Difference Matrix!".format(g,k,lmbda))
if check and not is_difference_matrix(M,G,k,lmbda,1):
raise RuntimeError("Sage built something which is not a ({},{},{})-DM!".format(g,k,lmbda))
return G,M
def difference_matrix(g,k,lmbda=1,existence=False,check=True):
r"""
Return a `(g,k,\lambda)`-difference matrix
A matrix `M` is a `(g,k,\lambda)`-difference matrix if its entries are
element of a group `G` of cardinality `g`, and if for any two rows `R,R'` of
`M` and `x\in G` there are exactly `\lambda` values `i` such that
`R_i-R'_i=x`.
INPUT:
- ``k`` -- (integer) number of columns. If ``k=None`` it is set to the
largest value available.
- ``g`` -- (integer) cardinality of the group `G`
- ``lmbda`` -- (integer; default: 1) -- number of times each element of `G`
appears as a difference.
- ``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.
- ``existence`` (boolean) -- instead of building the design, return:
- ``True`` -- meaning that Sage knows how to build the design
- ``Unknown`` -- meaning that Sage does not know how to build the
design, but that the design may exist (see :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the design does not exist.
.. NOTE::
When ``k=None`` and ``existence=True`` the function returns an
integer, i.e. the largest `k` such that we can build a
`(g,k,\lambda)`-DM.
EXAMPLES::
sage: G,M = designs.difference_matrix(25,10); G
Finite Field in x of size 5^2
sage: designs.difference_matrix(993,None,existence=1)
32
TESTS::
sage: designs.difference_matrix(10,12,1,existence=True)
False
sage: designs.difference_matrix(10,12,1)
Traceback (most recent call last):
...
EmptySetError: No (10,12,1)-Difference Matrix exists as k(=12)>g(=10)
sage: designs.difference_matrix(10,9,1,existence=True)
Unknown
sage: designs.difference_matrix(10,9,1)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build a (10,9,1)-Difference Matrix!
"""
if lmbda == 1 and k is not None and k>g:
if existence:
return False
raise EmptySetError("No ({},{},{})-Difference Matrix exists as k(={})>g(={})".format(g,k,lmbda,k,g))
# Prime powers
elif lmbda == 1 and is_prime_power(g):
if k is None:
if existence:
return g
else:
k = g
elif existence:
return True
F = FiniteField(g,'x')
F_set = list(F)
F_k_set = F_set[:k]
G = F
M = [[x*y for y in F_k_set] for x in F_set]
# From the database
elif (g,lmbda) in DM_constructions and (k is None or DM_constructions[g,lmbda][0]>=k):
if k is None:
k = DM_constructions[g,lmbda][0]
if existence:
return k
elif existence:
return True
_,f = DM_constructions[g,lmbda]
G, M = f()
M = [R[:k] for R in M]
else:
if existence:
return Unknown
raise NotImplementedError("I don't know how to build a ({},{},{})-Difference Matrix!".format(g,k,lmbda))
if check:
assert is_difference_matrix(M,G,k,lmbda,1), "Sage built something which is not a ({},{},{})-DM!".format(g,k,lmbda)
return G,M
def difference_matrix(g, k, lmbda=1, existence=False, check=True):
r"""
Return a `(g,k,\lambda)`-difference matrix
A matrix `M` is a `(g,k,\lambda)`-difference matrix if its entries are
element of a group `G` of cardinality `g`, and if for any two rows `R,R'` of
`M` and `x\in G` there are exactly `\lambda` values `i` such that
`R_i-R'_i=x`.
INPUT:
- ``k`` -- (integer) number of columns. If ``k=None`` it is set to the
largest value available.
- ``g`` -- (integer) cardinality of the group `G`
- ``lmbda`` -- (integer; default: 1) -- number of times each element of `G`
appears as a difference.
- ``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.
- ``existence`` (boolean) -- instead of building the design, return:
- ``True`` -- meaning that Sage knows how to build the design
- ``Unknown`` -- meaning that Sage does not know how to build the
design, but that the design may exist (see :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the design does not exist.
.. NOTE::
When ``k=None`` and ``existence=True`` the function returns an
integer, i.e. the largest `k` such that we can build a
`(g,k,\lambda)`-DM.
EXAMPLES::
sage: G,M = designs.difference_matrix(25,10); G
Finite Field in x of size 5^2
sage: designs.difference_matrix(993,None,existence=1)
32
TESTS::
sage: designs.difference_matrix(10,12,1,existence=True)
False
sage: designs.difference_matrix(10,12,1)
Traceback (most recent call last):
...
EmptySetError: No (10,12,1)-Difference Matrix exists as k(=12)>g(=10)
sage: designs.difference_matrix(10,9,1,existence=True)
Unknown
sage: designs.difference_matrix(10,9,1)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build a (10,9,1)-Difference Matrix!
"""
if lmbda == 1 and k is not None and k > g:
if existence:
return False
raise EmptySetError(
"No ({},{},{})-Difference Matrix exists as k(={})>g(={})".format(
g, k, lmbda, k, g))
# Prime powers
elif lmbda == 1 and is_prime_power(g):
if k is None:
if existence:
return g
else:
k = g
elif existence:
return True
F = FiniteField(g, 'x')
F_set = list(F)
F_k_set = F_set[:k]
G = F
M = [[x * y for y in F_k_set] for x in F_set]
# From the database
elif (g, lmbda) in DM_constructions and (
k is None or DM_constructions[g, lmbda][0] >= k):
if k is None:
k = DM_constructions[g, lmbda][0]
if existence:
return k
elif existence:
return True
_, f = DM_constructions[g, lmbda]
G, M = f()
M = [R[:k] for R in M]
else:
if existence:
return Unknown
raise NotImplementedError(
"I don't know how to build a ({},{},{})-Difference Matrix!".format(
g, k, lmbda))
if check:
assert is_difference_matrix(
M, G, k, lmbda,
1), "Sage built something which is not a ({},{},{})-DM!".format(
g, k, lmbda)
return G, M
