def __init__(self, parent, rdict):
r"""
Create an eta product object. Usually called implicitly via
EtaGroup_class.__call__ or the EtaProduct factory function.
EXAMPLES::
sage: EtaGroupElement(EtaGroup(8), {1:24, 2:-24})
Eta product of level 8 : (eta_1)^24 (eta_2)^-24
sage: g = _; g == loads(dumps(g))
True
"""
MultiplicativeGroupElement.__init__(self, parent)
self._N = self.parent().level()
N = self._N
if isinstance(rdict, EtaGroupElement):
rdict = rdict._rdict
# Note: This is needed because the "x in G" test tries to call G(x)
# and see if it returns an error. So sometimes this will be getting
# called with rdict being an eta product, not a dictionary.
if rdict == 1:
rdict = {}
# Check Ligozat criteria
sumR = sumDR = sumNoverDr = 0
prod = 1
for d in rdict.keys():
if N % d:
raise ValueError("%s does not divide %s" % (d, N))
for d in rdict.keys():
if rdict[d] == 0:
rdict.pop(d)
continue
sumR += rdict[d]
sumDR += rdict[d] * d
sumNoverDr += rdict[d] * N / d
prod *= (N / d)**rdict[d]
if sumR != 0:
raise ValueError("sum r_d (=%s) is not 0" % sumR)
if (sumDR % 24) != 0:
raise ValueError("sum d r_d (=%s) is not 0 mod 24" % sumDR)
if (sumNoverDr % 24) != 0:
raise ValueError("sum (N/d) r_d (=%s) is not 0 mod 24" %
sumNoverDr)
if not is_square(prod):
raise ValueError("product (N/d)^(r_d) (=%s) is not a square" %
prod)
self._sumDR = sumDR # this is useful to have around
self._rdict = rdict
self._keys = rdict.keys() # avoid factoring N every time
def __init__(self, parent, rdict):
r"""
Create an eta product object. Usually called implicitly via
EtaGroup_class.__call__ or the EtaProduct factory function.
EXAMPLE::
sage: EtaGroupElement(EtaGroup(8), {1:24, 2:-24})
Eta product of level 8 : (eta_1)^24 (eta_2)^-24
sage: g = _; g == loads(dumps(g))
True
"""
MultiplicativeGroupElement.__init__(self, parent)
self._N = self.parent().level()
N = self._N
if isinstance(rdict, EtaGroupElement):
rdict = rdict._rdict
# Note: This is needed because the "x in G" test tries to call G(x)
# and see if it returns an error. So sometimes this will be getting
# called with rdict being an eta product, not a dictionary.
if rdict == 1:
rdict = {}
# Check Ligozat criteria
sumR = sumDR = sumNoverDr = 0
prod = 1
for d in rdict.keys():
if N % d:
raise ValueError("%s does not divide %s" % (d, N))
for d in rdict.keys():
if rdict[d] == 0:
rdict.pop(d)
continue
sumR += rdict[d]
sumDR += rdict[d]*d
sumNoverDr += rdict[d]*N/d
prod *= (N/d)**rdict[d]
if sumR != 0:
raise ValueError("sum r_d (=%s) is not 0" % sumR)
if (sumDR % 24) != 0:
raise ValueError("sum d r_d (=%s) is not 0 mod 24" % sumDR)
if (sumNoverDr % 24) != 0:
raise ValueError("sum (N/d) r_d (=%s) is not 0 mod 24" % sumNoverDr)
if not is_square(prod):
raise ValueError("product (N/d)^(r_d) (=%s) is not a square" % prod)
self._sumDR = sumDR # this is useful to have around
self._rdict = rdict
self._keys = rdict.keys() # avoid factoring N every time
def an(self, use_database=False, descent_second_limit=12):
r"""
Returns the Birch and Swinnerton-Dyer conjectural order of `Sha`
as a provably correct integer, unless the analytic rank is > 1,
in which case this function returns a numerical value.
INPUT:
- ``use_database`` -- bool (default: ``False``); if ``True``, try
to use any databases installed to lookup the analytic order of
`Sha`, if possible. The order of `Sha` is computed if it cannot
be looked up.
- ``descent_second_limit`` -- int (default: 12); limit to use on
point searching for the quartic twist in the hard case
This result is proved correct if the order of vanishing is 0
and the Manin constant is <= 2.
If the optional parameter ``use_database`` is ``True`` (default:
``False``), this function returns the analytic order of `Sha` as
listed in Cremona's tables, if this curve appears in Cremona's
tables.
NOTE:
If you come across the following error::
sage: E = EllipticCurve([0, 0, 1, -34874, -2506691])
sage: E.sha().an()
Traceback (most recent call last):
...
RuntimeError: Unable to compute the rank, hence generators, with certainty (lower bound=0, generators found=[]). This could be because Sha(E/Q)[2] is nontrivial.
Try increasing descent_second_limit then trying this command again.
You can increase the ``descent_second_limit`` (in the above example,
set to the default, 12) option to try again::
sage: E.sha().an(descent_second_limit=16) # long time (2s on sage.math, 2011)
1
EXAMPLES::
sage: E = EllipticCurve([0, -1, 1, -10, -20]) # 11A = X_0(11)
sage: E.sha().an()
1
sage: E = EllipticCurve([0, -1, 1, 0, 0]) # X_1(11)
sage: E.sha().an()
1
sage: EllipticCurve('14a4').sha().an()
1
sage: EllipticCurve('14a4').sha().an(use_database=True) # will be faster if you have large Cremona database installed
1
The smallest conductor curve with nontrivial `Sha`::
sage: E = EllipticCurve([1,1,1,-352,-2689]) # 66b3
sage: E.sha().an()
4
The four optimal quotients with nontrivial `Sha` and conductor <= 1000::
sage: E = EllipticCurve([0, -1, 1, -929, -10595]) # 571A
sage: E.sha().an()
4
sage: E = EllipticCurve([1, 1, 0, -1154, -15345]) # 681B
sage: E.sha().an()
9
sage: E = EllipticCurve([0, -1, 0, -900, -10098]) # 960D
sage: E.sha().an()
4
sage: E = EllipticCurve([0, 1, 0, -20, -42]) # 960N
sage: E.sha().an()
4
The smallest conductor curve of rank > 1::
sage: E = EllipticCurve([0, 1, 1, -2, 0]) # 389A (rank 2)
sage: E.sha().an()
1.00000000000000
The following are examples that require computation of the Mordell-
Weil group and regulator::
sage: E = EllipticCurve([0, 0, 1, -1, 0]) # 37A (rank 1)
sage: E.sha().an()
1
sage: E = EllipticCurve("1610f3")
sage: E.sha().an()
4
In this case the input curve is not minimal, and if this function did
not transform it to be minimal, it would give nonsense::
sage: E = EllipticCurve([0,-432*6^2])
sage: E.sha().an()
1
See :trac:`10096`: this used to give the wrong result 6.0000
before since the minimal model was not used::
sage: E = EllipticCurve([1215*1216,0]) # non-minimal model
sage: E.sha().an() # long time (2s on sage.math, 2011)
1.00000000000000
sage: E.minimal_model().sha().an() # long time (1s on sage.math, 2011)
1.00000000000000
"""
if hasattr(self, '__an'):
return self.__an
if use_database:
d = self.Emin.database_curve()
if hasattr(d, 'db_extra'):
self.__an = Integer(round(float(d.db_extra[4])))
return self.__an
# it's critical to switch to the minimal model.
E = self.Emin
eps = E.root_number()
if eps == 1:
L1_over_omega = E.lseries().L_ratio()
if L1_over_omega == 0: # order of vanishing is at least 2
return self.an_numerical(use_database=use_database)
T = E.torsion_subgroup().order()
Sha = (L1_over_omega * T * T) / Q(E.tamagawa_product())
try:
Sha = Integer(Sha)
except ValueError:
raise RuntimeError("There is a bug in an, since the computed conjectural order of Sha is %s, which is not an integer."%Sha)
if not arith.is_square(Sha):
raise RuntimeError("There is a bug in an, since the computed conjectural order of Sha is %s, which is not a square."%Sha)
E.__an = Sha
self.__an = Sha
return Sha
else: # rank > 0 (Not provably correct)
L1, error_bound = E.lseries().deriv_at1(10*sqrt(E.conductor()) + 10)
if abs(L1) < error_bound:
s = self.an_numerical()
E.__an = s
self.__an = s
return s
regulator = E.regulator(use_database=use_database, descent_second_limit=descent_second_limit)
T = E.torsion_subgroup().order()
omega = E.period_lattice().omega()
Sha = Integer(round ( (L1 * T * T) / (E.tamagawa_product() * regulator * omega) ))
try:
Sha = Integer(Sha)
except ValueError:
raise RuntimeError("There is a bug in an, since the computed conjectural order of Sha is %s, which is not an integer."%Sha)
if not arith.is_square(Sha):
raise RuntimeError("There is a bug in an, since the computed conjectural order of Sha is %s, which is not a square."%Sha)
E.__an = Sha
self.__an = Sha
return Sha
def regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e,existence=False):
r"""
Return a Regular Symmetric Hadamard Matrix with Constant Diagonal.
A Hadamard matrix is said to be *regular* if its rows all sum to the same
value.
For `\epsilon\in\{-1,+1\}`, we say that `M` is a `(n,\epsilon)-RSHCD` if
`M` is a regular symmetric Hadamard matrix with constant diagonal
`\delta\in\{-1,+1\}` and row sums all equal to `\delta \epsilon
\sqrt(n)`. For more information, see [HX10]_ or 10.5.1 in
[BH12]_. For the case `n=324`, see :func:`RSHCD_324` and [CP16]_.
INPUT:
- ``n`` (integer) -- side of the matrix
- ``e`` -- one of `-1` or `+1`, equal to the value of `\epsilon`
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import regular_symmetric_hadamard_matrix_with_constant_diagonal
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,1)
[ 1 1 1 -1]
[ 1 1 -1 1]
[ 1 -1 1 1]
[-1 1 1 1]
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,-1)
[ 1 -1 -1 -1]
[-1 1 -1 -1]
[-1 -1 1 -1]
[-1 -1 -1 1]
Other hardcoded values::
sage: for n,e in [(36,1),(36,-1),(100,1),(100,-1),(196, 1)]:
....: print regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)
36 x 36 dense matrix over Integer Ring
36 x 36 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
196 x 196 dense matrix over Integer Ring
sage: for n,e in [(324,1),(324,-1)]: # not tested - long time, tested in RSHCD_324
....: print regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e) # not tested - long time
324 x 324 dense matrix over Integer Ring
324 x 324 dense matrix over Integer Ring
From two close prime powers::
sage: print regular_symmetric_hadamard_matrix_with_constant_diagonal(64,-1)
64 x 64 dense matrix over Integer Ring
Recursive construction::
sage: print regular_symmetric_hadamard_matrix_with_constant_diagonal(144,-1)
144 x 144 dense matrix over Integer Ring
REFERENCE:
.. [BH12] \A. Brouwer and W. Haemers,
Spectra of graphs,
Springer, 2012,
http://homepages.cwi.nl/~aeb/math/ipm/ipm.pdf
.. [HX10] \W. Haemers and Q. Xiang,
Strongly regular graphs with parameters `(4m^4,2m^4+m^2,m^4+m^2,m^4+m^2)` exist for all `m>1`,
European Journal of Combinatorics,
Volume 31, Issue 6, August 2010, Pages 1553-1559,
http://dx.doi.org/10.1016/j.ejc.2009.07.009.
"""
if existence and (n,e) in _rshcd_cache:
return _rshcd_cache[n,e]
from sage.graphs.strongly_regular_db import strongly_regular_graph
def true():
_rshcd_cache[n,e] = True
return True
M = None
if abs(e) != 1:
raise ValueError
if n<0:
if existence:
return False
raise ValueError
elif n == 4:
if existence:
return true()
if e == 1:
M = J(4)-2*matrix(4,[[int(i+j == 3) for i in range(4)] for j in range(4)])
else:
M = -J(4)+2*I(4)
elif n == 36:
if existence:
return true()
if e == 1:
M = strongly_regular_graph(36, 15, 6, 6).adjacency_matrix()
M = J(36) - 2*M
else:
M = strongly_regular_graph(36,14,4,6).adjacency_matrix()
M = -J(36) + 2*M + 2*I(36)
elif n == 100:
if existence:
return true()
if e == -1:
M = strongly_regular_graph(100,44,18,20).adjacency_matrix()
M = 2*M - J(100) + 2*I(100)
else:
M = strongly_regular_graph(100,45,20,20).adjacency_matrix()
M = J(100) - 2*M
elif n == 196 and e == 1:
if existence:
return true()
M = strongly_regular_graph(196,91,42,42).adjacency_matrix()
M = J(196) - 2*M
elif n == 324:
if existence:
return true()
M = RSHCD_324(e)
elif ( e == 1 and
n%16 == 0 and
is_square(n) and
is_prime_power(sqrt(n)-1) and
is_prime_power(sqrt(n)+1)):
if existence:
return true()
M = -rshcd_from_close_prime_powers(int(sqrt(n)))
# Recursive construction: the kronecker product of two RSHCD is a RSHCD
else:
from itertools import product
for n1,e1 in product(divisors(n)[1:-1],[-1,1]):
e2 = e1*e
n2 = n//n1
if (regular_symmetric_hadamard_matrix_with_constant_diagonal(n1,e1,existence=True) and
regular_symmetric_hadamard_matrix_with_constant_diagonal(n2,e2,existence=True)):
if existence:
return true()
M1 = regular_symmetric_hadamard_matrix_with_constant_diagonal(n1,e1)
M2 = regular_symmetric_hadamard_matrix_with_constant_diagonal(n2,e2)
M = M1.tensor_product(M2)
break
if M is None:
from sage.misc.unknown import Unknown
_rshcd_cache[n,e] = Unknown
if existence:
return Unknown
raise ValueError("I do not know how to build a {}-RSHCD".format((n,e)))
assert M*M.transpose() == n*I(n)
assert set(map(sum,M)) == {e*sqrt(n)}
return M
def an(self, use_database=False, descent_second_limit=12):
r"""
Returns the Birch and Swinnerton-Dyer conjectural order of `Sha`
as a provably correct integer, unless the analytic rank is > 1,
in which case this function returns a numerical value.
INPUT:
- ``use_database`` -- bool (default: ``False``); if ``True``, try
to use any databases installed to lookup the analytic order of
`Sha`, if possible. The order of `Sha` is computed if it cannot
be looked up.
- ``descent_second_limit`` -- int (default: 12); limit to use on
point searching for the quartic twist in the hard case
This result is proved correct if the order of vanishing is 0
and the Manin constant is <= 2.
If the optional parameter ``use_database`` is ``True`` (default:
``False``), this function returns the analytic order of `Sha` as
listed in Cremona's tables, if this curve appears in Cremona's
tables.
NOTE:
If you come across the following error::
sage: E = EllipticCurve([0, 0, 1, -34874, -2506691])
sage: E.sha().an()
Traceback (most recent call last):
...
RuntimeError: Unable to compute the rank, hence generators, with certainty (lower bound=0, generators found=[]). This could be because Sha(E/Q)[2] is nontrivial.
Try increasing descent_second_limit then trying this command again.
You can increase the ``descent_second_limit`` (in the above example,
set to the default, 12) option to try again::
sage: E.sha().an(descent_second_limit=16) # long time (2s on sage.math, 2011)
1
EXAMPLES::
sage: E = EllipticCurve([0, -1, 1, -10, -20]) # 11A = X_0(11)
sage: E.sha().an()
1
sage: E = EllipticCurve([0, -1, 1, 0, 0]) # X_1(11)
sage: E.sha().an()
1
sage: EllipticCurve('14a4').sha().an()
1
sage: EllipticCurve('14a4').sha().an(use_database=True) # will be faster if you have large Cremona database installed
1
The smallest conductor curve with nontrivial `Sha`::
sage: E = EllipticCurve([1,1,1,-352,-2689]) # 66b3
sage: E.sha().an()
4
The four optimal quotients with nontrivial `Sha` and conductor <= 1000::
sage: E = EllipticCurve([0, -1, 1, -929, -10595]) # 571A
sage: E.sha().an()
4
sage: E = EllipticCurve([1, 1, 0, -1154, -15345]) # 681B
sage: E.sha().an()
9
sage: E = EllipticCurve([0, -1, 0, -900, -10098]) # 960D
sage: E.sha().an()
4
sage: E = EllipticCurve([0, 1, 0, -20, -42]) # 960N
sage: E.sha().an()
4
The smallest conductor curve of rank > 1::
sage: E = EllipticCurve([0, 1, 1, -2, 0]) # 389A (rank 2)
sage: E.sha().an()
1.00000000000000
The following are examples that require computation of the Mordell-
Weil group and regulator::
sage: E = EllipticCurve([0, 0, 1, -1, 0]) # 37A (rank 1)
sage: E.sha().an()
1
sage: E = EllipticCurve("1610f3")
sage: E.sha().an()
4
In this case the input curve is not minimal, and if this function did
not transform it to be minimal, it would give nonsense::
sage: E = EllipticCurve([0,-432*6^2])
sage: E.sha().an()
1
See :trac:`10096`: this used to give the wrong result 6.0000
before since the minimal model was not used::
sage: E = EllipticCurve([1215*1216,0]) # non-minimal model
sage: E.sha().an() # long time (2s on sage.math, 2011)
1.00000000000000
sage: E.minimal_model().sha().an() # long time (1s on sage.math, 2011)
1.00000000000000
"""
if hasattr(self, '__an'):
return self.__an
if use_database:
d = self.Emin.database_curve()
if hasattr(d, 'db_extra'):
self.__an = Integer(round(float(d.db_extra[4])))
return self.__an
# it's critical to switch to the minimal model.
E = self.Emin
eps = E.root_number()
if eps == 1:
L1_over_omega = E.lseries().L_ratio()
if L1_over_omega == 0: # order of vanishing is at least 2
return self.an_numerical(use_database=use_database)
T = E.torsion_subgroup().order()
Sha = (L1_over_omega * T * T) / Q(E.tamagawa_product())
try:
Sha = Integer(Sha)
except ValueError:
raise RuntimeError(
"There is a bug in an, since the computed conjectural order of Sha is %s, which is not an integer."
% Sha)
if not arith.is_square(Sha):
raise RuntimeError(
"There is a bug in an, since the computed conjectural order of Sha is %s, which is not a square."
% Sha)
E.__an = Sha
self.__an = Sha
return Sha
else: # rank > 0 (Not provably correct)
L1, error_bound = E.lseries().deriv_at1(10 * sqrt(E.conductor()) +
10)
if abs(L1) < error_bound:
s = self.an_numerical()
E.__an = s
self.__an = s
return s
regulator = E.regulator(use_database=use_database,
descent_second_limit=descent_second_limit)
T = E.torsion_subgroup().order()
omega = E.period_lattice().omega()
Sha = ((L1 * T * T) /
(E.tamagawa_product() * regulator * omega)).round()
try:
Sha = Integer(Sha)
except ValueError:
raise RuntimeError(
"There is a bug in an, since the computed conjectural order of Sha is %s, which is not an integer."
% Sha)
if not arith.is_square(Sha):
raise RuntimeError(
"There is a bug in an, since the computed conjectural order of Sha is %s, which is not a square."
% Sha)
E.__an = Sha
self.__an = Sha
return Sha
def regular_symmetric_hadamard_matrix_with_constant_diagonal(
n, e, existence=False):
r"""
Return a Regular Symmetric Hadamard Matrix with Constant Diagonal.
A Hadamard matrix is said to be *regular* if its rows all sum to the same
value.
For `\epsilon\in\{-1,+1\}`, we say that `M` is a `(n,\epsilon)-RSHCD` if
`M` is a regular symmetric Hadamard matrix with constant diagonal
`\delta\in\{-1,+1\}` and row sums all equal to `\delta \epsilon
\sqrt(n)`. For more information, see [HX10]_ or 10.5.1 in
[BH12]_. For the case `n=324`, see :func:`RSHCD_324` and [CP16]_.
INPUT:
- ``n`` (integer) -- side of the matrix
- ``e`` -- one of `-1` or `+1`, equal to the value of `\epsilon`
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import regular_symmetric_hadamard_matrix_with_constant_diagonal
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,1)
[ 1 1 1 -1]
[ 1 1 -1 1]
[ 1 -1 1 1]
[-1 1 1 1]
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,-1)
[ 1 -1 -1 -1]
[-1 1 -1 -1]
[-1 -1 1 -1]
[-1 -1 -1 1]
Other hardcoded values::
sage: for n,e in [(36,1),(36,-1),(100,1),(100,-1),(196, 1)]: # long time
....: print(repr(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)))
36 x 36 dense matrix over Integer Ring
36 x 36 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
196 x 196 dense matrix over Integer Ring
sage: for n,e in [(324,1),(324,-1)]: # not tested - long time, tested in RSHCD_324
....: print(repr(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)))
324 x 324 dense matrix over Integer Ring
324 x 324 dense matrix over Integer Ring
From two close prime powers::
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(64,-1)
64 x 64 dense matrix over Integer Ring (use the '.str()' method to see the entries)
From a prime power and a conference matrix::
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(676,1) # long time
676 x 676 dense matrix over Integer Ring (use the '.str()' method to see the entries)
Recursive construction::
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(144,-1)
144 x 144 dense matrix over Integer Ring (use the '.str()' method to see the entries)
REFERENCE:
.. [BH12] \A. Brouwer and W. Haemers,
Spectra of graphs,
Springer, 2012,
http://homepages.cwi.nl/~aeb/math/ipm/ipm.pdf
.. [HX10] \W. Haemers and Q. Xiang,
Strongly regular graphs with parameters `(4m^4,2m^4+m^2,m^4+m^2,m^4+m^2)` exist for all `m>1`,
European Journal of Combinatorics,
Volume 31, Issue 6, August 2010, Pages 1553-1559,
:doi:`10.1016/j.ejc.2009.07.009`
"""
if existence and (n, e) in _rshcd_cache:
return _rshcd_cache[n, e]
from sage.graphs.strongly_regular_db import strongly_regular_graph
def true():
_rshcd_cache[n, e] = True
return True
M = None
if abs(e) != 1:
raise ValueError
sqn = None
if is_square(n):
sqn = int(sqrt(n))
if n < 0:
if existence:
return False
raise ValueError
elif n == 4:
if existence:
return true()
if e == 1:
M = J(4) - 2 * matrix(4, [[int(i + j == 3) for i in range(4)]
for j in range(4)])
else:
M = -J(4) + 2 * I(4)
elif n == 36:
if existence:
return true()
if e == 1:
M = strongly_regular_graph(36, 15, 6, 6).adjacency_matrix()
M = J(36) - 2 * M
else:
M = strongly_regular_graph(36, 14, 4, 6).adjacency_matrix()
M = -J(36) + 2 * M + 2 * I(36)
elif n == 100:
if existence:
return true()
if e == -1:
M = strongly_regular_graph(100, 44, 18, 20).adjacency_matrix()
M = 2 * M - J(100) + 2 * I(100)
else:
M = strongly_regular_graph(100, 45, 20, 20).adjacency_matrix()
M = J(100) - 2 * M
elif n == 196 and e == 1:
if existence:
return true()
M = strongly_regular_graph(196, 91, 42, 42).adjacency_matrix()
M = J(196) - 2 * M
elif n == 324:
if existence:
return true()
M = RSHCD_324(e)
elif (e == 1 and n % 16 == 0 and not sqn is None
and is_prime_power(sqn - 1) and is_prime_power(sqn + 1)):
if existence:
return true()
M = -rshcd_from_close_prime_powers(sqn)
elif (e == 1 and not sqn is None and sqn % 4 == 2
and True == strongly_regular_graph(sqn - 1, (sqn - 2) // 2,
(sqn - 6) // 4,
existence=True)
and is_prime_power(ZZ(sqn + 1))):
if existence:
return true()
M = rshcd_from_prime_power_and_conference_matrix(sqn + 1)
# Recursive construction: the kronecker product of two RSHCD is a RSHCD
else:
from itertools import product
for n1, e1 in product(divisors(n)[1:-1], [-1, 1]):
e2 = e1 * e
n2 = n // n1
if (regular_symmetric_hadamard_matrix_with_constant_diagonal(
n1, e1, existence=True) and
regular_symmetric_hadamard_matrix_with_constant_diagonal(
n2, e2, existence=True)):
if existence:
return true()
M1 = regular_symmetric_hadamard_matrix_with_constant_diagonal(
n1, e1)
M2 = regular_symmetric_hadamard_matrix_with_constant_diagonal(
n2, e2)
M = M1.tensor_product(M2)
break
if M is None:
from sage.misc.unknown import Unknown
_rshcd_cache[n, e] = Unknown
if existence:
return Unknown
raise ValueError("I do not know how to build a {}-RSHCD".format(
(n, e)))
assert M * M.transpose() == n * I(n)
assert set(map(sum, M)) == {ZZ(e * sqn)}
return M
def radical_difference_set(K, k, l=1, existence=False, check=True):
r"""
Return a difference set made of a cyclotomic coset in the finite field
``K`` and with paramters ``k`` and ``l``.
Most of these difference sets appear in chapter VI.18.48 of the Handbook of
combinatorial designs.
EXAMPLES::
sage: from sage.combinat.designs.difference_family import radical_difference_set
sage: D = radical_difference_set(GF(7), 3, 1); D
[[1, 2, 4]]
sage: sorted(x-y for x in D[0] for y in D[0] if x != y)
[1, 2, 3, 4, 5, 6]
sage: D = radical_difference_set(GF(16,'a'), 6, 2)
sage: sorted(x-y for x in D[0] for y in D[0] if x != y)
[1,
1,
a,
a,
a + 1,
a + 1,
a^2,
a^2,
...
a^3 + a^2 + a + 1,
a^3 + a^2 + a + 1]
sage: for k in range(2,50):
....: for l in reversed(divisors(k*(k-1))):
....: v = k*(k-1)//l + 1
....: if is_prime_power(v) and radical_difference_set(GF(v,'a'),k,l,existence=True):
....: _ = radical_difference_set(GF(v,'a'),k,l)
....: print "{:3} {:3} {:3}".format(v,k,l)
3 2 1
4 3 2
7 3 1
5 4 3
7 4 2
13 4 1
11 5 2
7 6 5
11 6 3
16 6 2
8 7 6
9 8 7
19 9 4
37 9 2
73 9 1
11 10 9
19 10 5
23 11 5
13 12 11
23 12 6
27 13 6
27 14 7
16 15 14
31 15 7
...
41 40 39
79 40 20
83 41 20
43 42 41
83 42 21
47 46 45
49 48 47
197 49 12
"""
v = K.cardinality()
if l*(v-1) != k*(k-1):
if existence:
return False
raise EmptySetError("l*(v-1) is not equal to k*(k-1)")
# trivial case
if (v-1) == k:
if existence:
return True
add_zero = False
# q = 3 mod 4
elif v%4 == 3 and k == (v-1)//2:
if existence:
return True
add_zero = False
# q = 3 mod 4
elif v%4 == 3 and k == (v+1)//2:
if existence:
return True
add_zero = True
# q = 4t^2 + 1, t odd
elif v%8 == 5 and k == (v-1)//4 and arith.is_square((v-1)//4):
if existence:
return True
add_zero = False
# q = 4t^2 + 9, t odd
elif v%8 == 5 and k == (v+3)//4 and arith.is_square((v-9)//4):
if existence:
return True
add_zero = True
# exceptional case 1
elif (v,k,l) == (16,6,2):
if existence:
return True
add_zero = True
# exceptional case 2
elif (v,k,l) == (73,9,1):
if existence:
return True
add_zero = False
# are there more ??
else:
x = K.multiplicative_generator()
D = K.cyclotomic_cosets(x**((v-1)//k), [K.one()])
if is_difference_family(K, D, v, k, l):
print "** You found a new example of radical difference set **\n"\
"** for the parameters (v,k,l)=({},{},{}). **\n"\
"** Please contact [email protected] **\n".format(v,k,l)
if existence:
return True
add_zero = False
else:
D = K.cyclotomic_cosets(x**((v-1)//(k-1)), [K.one()])
D[0].insert(0,K.zero())
if is_difference_family(K, D, v, k, l):
print "** You found a new example of radical difference set **\n"\
"** for the parameters (v,k,l)=({},{},{}). **\n"\
"** Please contact [email protected] **\n".format(v,k,l)
if existence:
return True
add_zero = True
elif existence:
return False
else:
raise EmptySetError("no radical difference set exist "
"for the parameters (v,k,l) = ({},{},{}".format(v,k,l))
x = K.multiplicative_generator()
if add_zero:
r = x**((v-1)//(k-1))
D = K.cyclotomic_cosets(r, [K.one()])
D[0].insert(0, K.zero())
else:
r = x**((v-1)//k)
D = K.cyclotomic_cosets(r, [K.one()])
if check and not is_difference_family(K, D, v, k, l):
raise RuntimeError("Sage tried to build a radical difference set with "
"parameters ({},{},{}) but it seems that it failed! Please "
"e-mail [email protected]".format(v,k,l))
return D
def radical_difference_set(K, k, l=1, existence=False, check=True):
r"""
Return a difference set made of a cyclotomic coset in the finite field
``K`` and with paramters ``k`` and ``l``.
Most of these difference sets appear in chapter VI.18.48 of the Handbook of
combinatorial designs.
EXAMPLES::
sage: from sage.combinat.designs.difference_family import radical_difference_set
sage: D = radical_difference_set(GF(7), 3, 1); D
[[1, 2, 4]]
sage: sorted(x-y for x in D[0] for y in D[0] if x != y)
[1, 2, 3, 4, 5, 6]
sage: D = radical_difference_set(GF(16,'a'), 6, 2)
sage: sorted(x-y for x in D[0] for y in D[0] if x != y)
[1,
1,
a,
a,
a + 1,
a + 1,
a^2,
a^2,
...
a^3 + a^2 + a + 1,
a^3 + a^2 + a + 1]
sage: for k in range(2,50):
....: for l in reversed(divisors(k*(k-1))):
....: v = k*(k-1)//l + 1
....: if is_prime_power(v) and radical_difference_set(GF(v,'a'),k,l,existence=True):
....: _ = radical_difference_set(GF(v,'a'),k,l)
....: print "{:3} {:3} {:3}".format(v,k,l)
3 2 1
4 3 2
7 3 1
5 4 3
7 4 2
13 4 1
11 5 2
7 6 5
11 6 3
16 6 2
8 7 6
9 8 7
19 9 4
37 9 2
73 9 1
11 10 9
19 10 5
23 11 5
13 12 11
23 12 6
27 13 6
27 14 7
16 15 14
31 15 7
...
41 40 39
79 40 20
83 41 20
43 42 41
83 42 21
47 46 45
49 48 47
197 49 12
"""
v = K.cardinality()
if l * (v - 1) != k * (k - 1):
if existence:
return False
raise EmptySetError("l*(v-1) is not equal to k*(k-1)")
# trivial case
if (v - 1) == k:
if existence:
return True
add_zero = False
# q = 3 mod 4
elif v % 4 == 3 and k == (v - 1) // 2:
if existence:
return True
add_zero = False
# q = 3 mod 4
elif v % 4 == 3 and k == (v + 1) // 2:
if existence:
return True
add_zero = True
# q = 4t^2 + 1, t odd
elif v % 8 == 5 and k == (v - 1) // 4 and arith.is_square((v - 1) // 4):
if existence:
return True
add_zero = False
# q = 4t^2 + 9, t odd
elif v % 8 == 5 and k == (v + 3) // 4 and arith.is_square((v - 9) // 4):
if existence:
return True
add_zero = True
# exceptional case 1
elif (v, k, l) == (16, 6, 2):
if existence:
return True
add_zero = True
# exceptional case 2
elif (v, k, l) == (73, 9, 1):
if existence:
return True
add_zero = False
# are there more ??
else:
x = K.multiplicative_generator()
D = K.cyclotomic_cosets(x**((v - 1) // k), [K.one()])
if is_difference_family(K, D, v, k, l):
print "** You found a new example of radical difference set **\n"\
"** for the parameters (v,k,l)=({},{},{}). **\n"\
"** Please contact [email protected] **\n".format(v,k,l)
if existence:
return True
add_zero = False
else:
D = K.cyclotomic_cosets(x**((v - 1) // (k - 1)), [K.one()])
D[0].insert(0, K.zero())
if is_difference_family(K, D, v, k, l):
print "** You found a new example of radical difference set **\n"\
"** for the parameters (v,k,l)=({},{},{}). **\n"\
"** Please contact [email protected] **\n".format(v,k,l)
if existence:
return True
add_zero = True
elif existence:
return False
else:
raise EmptySetError(
"no radical difference set exist "
"for the parameters (v,k,l) = ({},{},{}".format(v, k, l))
x = K.multiplicative_generator()
if add_zero:
r = x**((v - 1) // (k - 1))
D = K.cyclotomic_cosets(r, [K.one()])
D[0].insert(0, K.zero())
else:
r = x**((v - 1) // k)
D = K.cyclotomic_cosets(r, [K.one()])
if check and not is_difference_family(K, D, v, k, l):
raise RuntimeError(
"Sage tried to build a radical difference set with "
"parameters ({},{},{}) but it seems that it failed! Please "
"e-mail [email protected]".format(v, k, l))
return D
