def get_s(self):
"""Returns a single value of s as required for the Elligator map while there may even be two of them, in which case the returned values is the smaller of the two
"""
try:
return self._s
except AttributeError:
pass
s = []
d = self.__E.get_d()
if is_square(-d):
c = ( 2*(d-1) + 4*sqrt(-d) ) / ( 2*(d+1) )
s_2 = 2/c
if is_square(s_2):
s.append(sqrt(s_2))
c = ( 2*(d-1) - 4*sqrt(-d) ) / ( 2*(d+1) )
s_2 = 2/c
if is_square(s_2):
s.append(sqrt(s_2))
if len(s) == 0:
return None
elif len(s) == 1:
return s[0]
else:
if s[0] < s[1]:
return s[0]
else:
return s[1]
def random_point(self):
while(1):
x = self.base_ring().random_element()
test = (x**2 - 1)/(self._d*(x**2) - 1)
if is_square(test):
y = sqrt(test)
return self([x,y,1])
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 n_random_points(self,n):
v = []
while(len(v) < n):
x = self.base_ring().random_element()
test = (x**2 - 1)/(self._d*(x**2) - 1)
if is_square(test):
y = sqrt(test)
v.append(self([x,y,1]))
v.sort()
return v
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 is_image_point(self, point):
s = self._s
c = self._c
r = self._r
x = point[0]
y = point[1]
eta = (y-1) / (2 * (y+1))
if y + 1 == 0:
return False
if not is_square( (1 + eta * r)**2 - 1):
return False
if eta*r == -2:
temp = 2*s*(c - 1)*self.chi(c)/r
if not x == temp:
return False
return True
def points(self, limit = False):
try:
return self.__points
except AttributeError: pass
if not self.base_ring().is_finite():
raise TypeError("points() is defined only for Edwards curve over finite fields")
v = []
for x in self.base_ring():
test = (x**2 - 1)/(self._d*(x**2) - 1)
if is_square(test):
y = sqrt(test)
v.append(self([x,y,1]))
v.append(self([x,-y,1]))
v.sort()
self.__points = Sequence(v,immutable = True)
return self.__points
def __init__( self, q):
"""
We initialize by a list of integers $[a_1,...,a_N]$. The
lattice self is then $L = (L,\beta) = (G^{-1}\ZZ^n/\ZZ^n, G[x])$,
where $G$ is the symmetric matrix
$G=[a_1,...,a_n;*,a_{n+1},....;..;* ... * a_N]$,
i.e.~the $a_j$ denote the elements above the diagonal of $G$.
"""
self.__form = q
# We compute the rank
N = len(q)
assert is_square( 1+8*N)
n = Integer( (-1+isqrt(1+8*N))/2)
self.__rank = n
# We set up the Gram matrix
self.__G = matrix( IntegerRing(), n, n)
i = j = 0
for a in q:
self.__G[i,j] = self.__G[j,i] = Integer(a)
if j < n-1:
j += 1
else:
i += 1
j = i
# We compute the level
Gi = self.__G**-1
self.__Gi = Gi
a = lcm( map( lambda x: x.denominator(), Gi.list()))
I = Gi.diagonal()
b = lcm( map( lambda x: (x/2).denominator(), I))
self.__level = lcm( a, b)
# We define the undelying module and the ambient space
self.__module = FreeModule( IntegerRing(), n)
self.__space = self.__module.ambient_vector_space()
# We compute a shadow vector
self.__shadow_vector = self.__space([(a%2)/2 for a in self.__G.diagonal()])*Gi
# We define a basis
M = Matrix( IntegerRing(), n, n, 1)
self.__basis = self.__module.basis()
# We prepare a cache
self.__dual_vectors = None
self.__values = None
self.__chi = {}
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 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 (v,k,l) in [(3,2,1), (7,3,1), (7,4,2), (11,5,2), (11,6,3),
....: (13,4,1), (16,6,2), (19,9,4), (19,10,5)]:
....:
....: assert radical_difference_set(GF(v,'a'), k, l, existence=True), "pb with v={} k={} l={}".format(v,k,l)
"""
v = K.cardinality()
one = K.one()
x = K.multiplicative_generator()
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
return K.cyclotomic_cosets(x, [one])
# q = 3 mod 4
elif v%4 == 3 and k == (v-1)//2:
if existence:
return True
D = K.cyclotomic_cosets(x**2, [one])
# q = 3 mod 4
elif v%4 == 3 and k == (v+1)//2:
if existence:
return True
D = K.cyclotomic_cosets(x**2, [one])
D[0].insert(0, K.zero())
# 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
D = K.cyclotomic_cosets(x**4, [one])
# 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
D = K.cyclotomic_cosets(x**4, [one])
D[0].insert(0,K.zero())
# one case with k-1 = (v-1)/3
elif (v,k,l) == (16,6,2):
if existence:
return True
D = K.cyclotomic_cosets(x**3, [one])
D[0].insert(0,K.zero())
# one case with k = (v-1)/8
elif (v,k,l) == (73,9,1):
if existence:
return True
D = K.cyclotomic_cosets(x**8, [one])
else:
if existence:
return Unknown
raise NotImplementedError("no radical difference set is "
"implemented for the parameters (v,k,l) = ({},{},{}".format(v,k,l))
if check and not is_difference_family(K, D, v, k, l):
raise RuntimeError("Sage tried to build a cyclotomic coset with "
"parameters ({},{},{}) but it seems that it failed! Please "
"e-mail [email protected]".format(v,k,l))
return D
def difference_family(v, k, l=1, existence=False, check=True):
r"""
Return a (``k``, ``l``)-difference family on an Abelian group of size ``v``.
Let `G` be a finite Abelian group. For a given subset `D` of `G`, we define
`\Delta D` to be the multi-set of differences `\Delta D = \{x - y; x \in D,
y \in D, x \not= y\}`. A `(G,k,\lambda)`-*difference family* is a collection
of `k`-subsets of `G`, `D = \{D_1, D_2, \ldots, D_b\}` such that the union
of the difference sets `\Delta D_i` for `i=1,...b`, seen as a multi-set,
contains each element of `G \backslash \{0\}` exactly `\lambda`-times.
When there is only one block, i.e. `\lambda(v - 1) = k(k-1)`, then a
`(G,k,\lambda)`-difference family is also called a *difference set*.
See also :wikipedia:`Difference_set`.
If there is no such difference family, an ``EmptySetError`` is raised and if
there is no construction at the moment ``NotImplementedError`` is raised.
EXAMPLES::
sage: K,D = designs.difference_family(73,4)
sage: D
[[0, 1, 8, 64],
[0, 25, 54, 67],
[0, 41, 36, 69],
[0, 3, 24, 46],
[0, 2, 16, 55],
[0, 50, 35, 61]]
sage: K,D = designs.difference_family(337,7)
sage: D
[[1, 175, 295, 64, 79, 8, 52],
[326, 97, 125, 307, 142, 249, 102],
[121, 281, 310, 330, 123, 294, 226],
[17, 279, 297, 77, 332, 136, 210],
[150, 301, 103, 164, 55, 189, 49],
[35, 59, 215, 218, 69, 280, 135],
[289, 25, 331, 298, 252, 290, 200],
[191, 62, 66, 92, 261, 180, 159]]
For `k=6,7` we look at the set of small prime powers for which a
construction is available::
sage: def prime_power_mod(r,m):
....: k = m+r
....: while True:
....: if is_prime_power(k):
....: yield k
....: k += m
sage: from itertools import islice
sage: l6 = {True:[], False: [], Unknown: []}
sage: for q in islice(prime_power_mod(1,30), 60):
....: l6[designs.difference_family(q,6,existence=True)].append(q)
sage: l6[True]
[31, 151, 181, 211, ..., 3061, 3121, 3181]
sage: l6[Unknown]
[61, 121]
sage: l6[False]
[]
sage: l7 = {True: [], False: [], Unknown: []}
sage: for q in islice(prime_power_mod(1,42), 60):
....: l7[designs.difference_family(q,7,existence=True)].append(q)
sage: l7[True]
[337, 421, 463, 883, 1723, 3067, 3319, 3529, 3823, 3907, 4621, 4957, 5167]
sage: l7[Unknown]
[43, 127, 169, 211, ..., 4999, 5041, 5209]
sage: l7[False]
[]
Other constructions for `\lambda > 1`::
sage: for v in xrange(2,100):
....: constructions = []
....: for k in xrange(2,10):
....: for l in xrange(2,10):
....: if designs.difference_family(v,k,l,existence=True):
....: constructions.append((k,l))
....: _ = designs.difference_family(v,k,l)
....: if constructions:
....: print "%2d: %s"%(v, ', '.join('(%d,%d)'%(k,l) for k,l in constructions))
4: (3,2)
5: (4,3)
7: (3,2), (6,5)
8: (7,6)
9: (4,3), (8,7)
11: (5,2), (5,4)
13: (3,2), (4,3), (6,5)
15: (7,3)
16: (3,2), (5,4)
17: (4,3), (8,7)
19: (3,2), (6,5), (9,4), (9,8)
25: (3,2), (4,3), (6,5), (8,7)
29: (4,3), (7,6)
31: (3,2), (5,4), (6,5)
37: (3,2), (4,3), (6,5), (9,2), (9,8)
41: (4,3), (5,4), (8,7)
43: (3,2), (6,5), (7,6)
49: (3,2), (4,3), (6,5), (8,7)
53: (4,3)
61: (3,2), (4,3), (5,4), (6,5)
64: (3,2), (7,6), (9,8)
67: (3,2), (6,5)
71: (5,4), (7,6)
73: (3,2), (4,3), (6,5), (8,7), (9,8)
79: (3,2), (6,5)
81: (4,3), (5,4), (8,7)
89: (4,3), (8,7)
97: (3,2), (4,3), (6,5), (8,7)
TESTS:
Check more of the Wilson constructions from [Wi72]_::
sage: Q5 = [241, 281,421,601,641, 661, 701, 821,881]
sage: Q9 = [73, 1153, 1873, 2017]
sage: Q15 = [76231]
sage: Q4 = [13, 73, 97, 109, 181, 229, 241, 277, 337, 409, 421, 457]
sage: Q8 = [1009, 3137, 3697]
sage: for Q,k in [(Q4,4),(Q5,5),(Q8,8),(Q9,9),(Q15,15)]:
....: for q in Q:
....: assert designs.difference_family(q,k,1,existence=True) is True
....: _ = designs.difference_family(q,k,1)
Check Singer difference sets::
sage: sgp = lambda q,d: ((q**(d+1)-1)//(q-1), (q**d-1)//(q-1), (q**(d-1)-1)//(q-1))
sage: for q in range(2,10):
....: if is_prime_power(q):
....: for d in [2,3,4]:
....: v,k,l = sgp(q,d)
....: assert designs.difference_family(v,k,l,existence=True) is True
....: _ = designs.difference_family(v,k,l)
.. TODO::
There is a slightly more general version of difference families where
the stabilizers of the blocks are taken into account. A block is *short*
if the stabilizer is not trivial. The more general version is called a
*partial difference family*. It is still possible to construct BIBD from
this more general version (see the chapter 16 in the Handbook
[DesignHandbook]_).
Implement recursive constructions from Buratti "Recursive for difference
matrices and relative difference families" (1998) and Jungnickel
"Composition theorems for difference families and regular planes" (1978)
"""
if (l * (v - 1)) % (k * (k - 1)) != 0:
if existence:
return False
raise EmptySetError(
"A (v,%d,%d)-difference family may exist only if %d*(v-1) = mod %d"
% (k, l, l, k * (k - 1)))
from block_design import are_hyperplanes_in_projective_geometry_parameters
from database import DF_constructions
if (v, k, l) in DF_constructions:
if existence:
return True
return DF_constructions[(v, k, l)]()
e = k * (k - 1)
t = l * (v - 1) // e # number of blocks
D = None
if arith.is_prime_power(v):
from sage.rings.finite_rings.constructor import GF
G = K = GF(v, 'z')
x = K.multiplicative_generator()
if l == (k - 1):
if existence:
return True
return K, K.cyclotomic_cosets(x**((v - 1) // k))[1:]
if t == 1:
# some of the difference set constructions VI.18.48 from the
# Handbook of combinatorial designs
# q = 3 mod 4
if v % 4 == 3 and k == (v - 1) // 2:
if existence:
return True
D = K.cyclotomic_cosets(x**2, [1])
# 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
D = K.cyclotomic_cosets(x**4, [1])
# 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
D = K.cyclotomic_cosets(x**4, [1])
D[0].insert(0, K.zero())
if D is None and l == 1:
one = K.one()
# Wilson (1972), Theorem 9
if k % 2 == 1:
m = (k - 1) // 2
xx = x**m
to_coset = {
x**i * xx**j: i
for i in xrange(m) for j in xrange((v - 1) / m)
}
r = x**((v - 1) // k) # primitive k-th root of unity
if len(set(to_coset[r**j - one]
for j in xrange(1, m + 1))) == m:
if existence:
return True
B = [r**j for j in xrange(k)
] # = H^((k-1)t) whose difference is
# H^(mt) (r^i - 1, i=1,..,m)
# Now pick representatives a translate of R for by a set of
# representatives of H^m / H^(mt)
D = [[x**(i * m) * b for b in B] for i in xrange(t)]
# Wilson (1972), Theorem 10
else:
m = k // 2
xx = x**m
to_coset = {
x**i * xx**j: i
for i in xrange(m) for j in xrange((v - 1) / m)
}
r = x**((v - 1) // (k - 1)) # primitive (k-1)-th root of unity
if (all(to_coset[r**j - one] != 0 for j in xrange(1, m))
and len(set(to_coset[r**j - one]
for j in xrange(1, m))) == m - 1):
if existence:
return True
B = [K.zero()] + [r**j for j in xrange(k - 1)]
D = [[x**(i * m) * b for b in B] for i in xrange(t)]
# Wilson (1972), Theorem 11
if D is None and k == 6:
r = x**((v - 1) // 3) # primitive cube root of unity
r2 = r * r
xx = x**5
to_coset = {
x**i * xx**j: i
for i in xrange(5) for j in xrange((v - 1) / 5)
}
for c in to_coset:
if c == 1 or c == r or c == r2:
continue
if len(
set(to_coset[elt]
for elt in (r - 1, c * (r - 1), c - 1, c - r,
c - r**2))) == 5:
if existence:
return True
B = [one, r, r**2, c, c * r, c * r**2]
D = [[x**(i * 5) * b for b in B] for i in xrange(t)]
break
if D is None and are_hyperplanes_in_projective_geometry_parameters(
v, k, l):
_, (q, d) = are_hyperplanes_in_projective_geometry_parameters(
v, k, l, True)
if existence:
return True
else:
G, D = singer_difference_set(q, d)
if D is None:
if existence:
return Unknown
raise NotImplementedError("No constructions for these parameters")
if check and not is_difference_family(G, D, verbose=False):
raise RuntimeError
return G, 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
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.
When `\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 values all equal to `\delta \epsilon
\sqrt(n)`. For more information, see [HX10]_ or 10.5.1 in
[BH12]_.
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
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 ( 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 difference_family(v, k, l=1, existence=False, check=True):
r"""
Return a (``k``, ``l``)-difference family on an Abelian group of cardinality ``v``.
Let `G` be a finite Abelian group. For a given subset `D` of `G`, we define
`\Delta D` to be the multi-set of differences `\Delta D = \{x - y; x \in D,
y \in D, x \not= y\}`. A `(G,k,\lambda)`-*difference family* is a collection
of `k`-subsets of `G`, `D = \{D_1, D_2, \ldots, D_b\}` such that the union
of the difference sets `\Delta D_i` for `i=1,...b`, seen as a multi-set,
contains each element of `G \backslash \{0\}` exactly `\lambda`-times.
When there is only one block, i.e. `\lambda(v - 1) = k(k-1)`, then a
`(G,k,\lambda)`-difference family is also called a *difference set*.
See also :wikipedia:`Difference_set`.
If there is no such difference family, an ``EmptySetError`` is raised and if
there is no construction at the moment ``NotImplementedError`` is raised.
EXAMPLES::
sage: K,D = designs.difference_family(73,4)
sage: D
[[0, 1, 8, 64],
[0, 25, 54, 67],
[0, 41, 36, 69],
[0, 3, 24, 46],
[0, 2, 16, 55],
[0, 50, 35, 61]]
sage: K,D = designs.difference_family(337,7)
sage: D
[[1, 175, 295, 64, 79, 8, 52],
[326, 97, 125, 307, 142, 249, 102],
[121, 281, 310, 330, 123, 294, 226],
[17, 279, 297, 77, 332, 136, 210],
[150, 301, 103, 164, 55, 189, 49],
[35, 59, 215, 218, 69, 280, 135],
[289, 25, 331, 298, 252, 290, 200],
[191, 62, 66, 92, 261, 180, 159]]
For `k=6,7` we look at the set of small prime powers for which a
construction is available::
sage: def prime_power_mod(r,m):
....: k = m+r
....: while True:
....: if is_prime_power(k):
....: yield k
....: k += m
sage: from itertools import islice
sage: l6 = {True:[], False: [], Unknown: []}
sage: for q in islice(prime_power_mod(1,30), 60):
....: l6[designs.difference_family(q,6,existence=True)].append(q)
sage: l6[True]
[31, 121, 151, 181, 211, ..., 3061, 3121, 3181]
sage: l6[Unknown]
[61]
sage: l6[False]
[]
sage: l7 = {True: [], False: [], Unknown: []}
sage: for q in islice(prime_power_mod(1,42), 60):
....: l7[designs.difference_family(q,7,existence=True)].append(q)
sage: l7[True]
[337, 421, 463, 883, 1723, 3067, 3319, 3529, 3823, 3907, 4621, 4957, 5167]
sage: l7[Unknown]
[43, 127, 169, 211, ..., 4999, 5041, 5209]
sage: l7[False]
[]
Other constructions for `\lambda > 1`::
sage: for v in xrange(2,100):
....: constructions = []
....: for k in xrange(2,10):
....: for l in xrange(2,10):
....: if designs.difference_family(v,k,l,existence=True):
....: constructions.append((k,l))
....: _ = designs.difference_family(v,k,l)
....: if constructions:
....: print "%2d: %s"%(v, ', '.join('(%d,%d)'%(k,l) for k,l in constructions))
2: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
3: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
4: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
5: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
7: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
8: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
9: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
11: (3,2), (4,3), (4,6), (5,2), (5,3), (5,4), (6,5), (7,6), (8,7), (9,8)
13: (3,2), (4,3), (5,4), (5,5), (6,5), (7,6), (8,7), (9,8)
15: (4,6), (5,6), (7,3)
16: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
17: (3,2), (4,3), (5,4), (5,5), (6,5), (7,6), (8,7), (9,8)
19: (3,2), (4,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,4), (9,5), (9,6), (9,7), (9,8)
21: (4,3), (6,3), (6,5)
22: (4,2), (6,5), (7,4), (8,8)
23: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
25: (3,2), (4,3), (5,4), (6,5), (7,6), (7,7), (8,7), (9,8)
27: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
28: (3,2), (6,5)
29: (3,2), (4,3), (5,4), (6,5), (7,3), (7,6), (8,4), (8,6), (8,7), (9,8)
31: (3,2), (4,2), (4,3), (5,2), (5,4), (6,5), (7,6), (8,7), (9,8)
32: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
33: (5,5), (6,5)
34: (4,2)
35: (5,2), (8,4)
37: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,2), (9,3), (9,8)
39: (6,5)
40: (3,2)
41: (3,2), (4,3), (5,4), (6,3), (6,5), (7,6), (8,7), (9,8)
43: (3,2), (4,2), (4,3), (5,4), (6,5), (7,2), (7,3), (7,6), (8,4), (8,7), (9,8)
46: (4,2), (6,2)
47: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
49: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,3), (9,8)
51: (5,2), (6,3)
53: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
55: (9,4)
57: (7,3)
59: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
61: (3,2), (4,3), (5,4), (6,2), (6,3), (6,5), (7,6), (8,7), (9,8)
64: (3,2), (4,3), (5,4), (6,5), (7,2), (7,6), (8,7), (9,8)
67: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
71: (3,2), (4,3), (5,2), (5,4), (6,5), (7,3), (7,6), (8,4), (8,7), (9,8)
73: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
75: (5,2)
79: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
81: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
83: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
85: (7,2), (7,3), (8,2)
89: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
97: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,3), (9,8)
TESTS:
Check more of the Wilson constructions from [Wi72]_::
sage: Q5 = [241, 281,421,601,641, 661, 701, 821,881]
sage: Q9 = [73, 1153, 1873, 2017]
sage: Q15 = [76231]
sage: Q4 = [13, 73, 97, 109, 181, 229, 241, 277, 337, 409, 421, 457]
sage: Q8 = [1009, 3137, 3697]
sage: for Q,k in [(Q4,4),(Q5,5),(Q8,8),(Q9,9),(Q15,15)]:
....: for q in Q:
....: assert designs.difference_family(q,k,1,existence=True) is True
....: _ = designs.difference_family(q,k,1)
Check Singer difference sets::
sage: sgp = lambda q,d: ((q**(d+1)-1)//(q-1), (q**d-1)//(q-1), (q**(d-1)-1)//(q-1))
sage: for q in range(2,10):
....: if is_prime_power(q):
....: for d in [2,3,4]:
....: v,k,l = sgp(q,d)
....: assert designs.difference_family(v,k,l,existence=True) is True
....: _ = designs.difference_family(v,k,l)
Check twin primes difference sets::
sage: for p in [3,5,7,9,11]:
....: v = p*(p+2); k = (v-1)/2; lmbda = (k-1)/2
....: G,D = designs.difference_family(v,k,lmbda)
Check the database:
sage: from sage.combinat.designs.database import DF
sage: for v,k,l in DF:
....: df = designs.difference_family(v,k,l,check=True)
.. TODO::
Implement recursive constructions from Buratti "Recursive for difference
matrices and relative difference families" (1998) and Jungnickel
"Composition theorems for difference families and regular planes" (1978)
"""
from block_design import are_hyperplanes_in_projective_geometry_parameters
from database import DF
if (v, k, l) in DF:
if existence:
return True
vv, blocks = DF[v, k, l].iteritems().next()
# Build the group
from sage.rings.finite_rings.integer_mod_ring import Zmod
if len(vv) == 1:
G = Zmod(vv[0])
else:
from sage.categories.cartesian_product import cartesian_product
G = cartesian_product([Zmod(i) for i in vv])
df = [[G(i) for i in b] for b in blocks]
if check:
assert is_difference_family(
G, df, v=v, k=k,
l=l), "Sage built an invalid ({},{},{})-DF!".format(v, k, l)
return G, df
e = k * (k - 1)
t = l * (v - 1) // e # number of blocks
D = None
factorization = arith.factor(v)
if len(factorization) == 1: # i.e. is v a prime power
from sage.rings.finite_rings.constructor import GF
G = K = GF(v, 'z')
x = K.multiplicative_generator()
if l == (k - 1):
if existence:
return True
return K, K.cyclotomic_cosets(x**((v - 1) // k))[1:]
if t == 1:
# some of the difference set constructions VI.18.48 from the
# Handbook of combinatorial designs
# q = 3 mod 4
if v % 4 == 3 and k == (v - 1) // 2:
if existence:
return True
D = K.cyclotomic_cosets(x**2, [1])
# 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
D = K.cyclotomic_cosets(x**4, [1])
# 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
D = K.cyclotomic_cosets(x**4, [1])
D[0].insert(0, K.zero())
if D is None and l == 1:
one = K.one()
# Wilson (1972), Theorem 9
if k % 2 == 1:
m = (k - 1) // 2
xx = x**m
to_coset = {
x**i * xx**j: i
for i in xrange(m) for j in xrange((v - 1) / m)
}
r = x**((v - 1) // k) # primitive k-th root of unity
if len(set(to_coset[r**j - one]
for j in xrange(1, m + 1))) == m:
if existence:
return True
B = [r**j for j in xrange(k)
] # = H^((k-1)t) whose difference is
# H^(mt) (r^i - 1, i=1,..,m)
# Now pick representatives a translate of R for by a set of
# representatives of H^m / H^(mt)
D = [[x**(i * m) * b for b in B] for i in xrange(t)]
# Wilson (1972), Theorem 10
else:
m = k // 2
xx = x**m
to_coset = {
x**i * xx**j: i
for i in xrange(m) for j in xrange((v - 1) / m)
}
r = x**((v - 1) // (k - 1)) # primitive (k-1)-th root of unity
if (all(to_coset[r**j - one] != 0 for j in xrange(1, m))
and len(set(to_coset[r**j - one]
for j in xrange(1, m))) == m - 1):
if existence:
return True
B = [K.zero()] + [r**j for j in xrange(k - 1)]
D = [[x**(i * m) * b for b in B] for i in xrange(t)]
# Wilson (1972), Theorem 11
if D is None and k == 6:
r = x**((v - 1) // 3) # primitive cube root of unity
r2 = r * r
xx = x**5
to_coset = {
x**i * xx**j: i
for i in xrange(5) for j in xrange((v - 1) / 5)
}
for c in to_coset:
if c == 1 or c == r or c == r2:
continue
if len(
set(to_coset[elt]
for elt in (r - 1, c * (r - 1), c - 1, c - r,
c - r**2))) == 5:
if existence:
return True
B = [one, r, r**2, c, c * r, c * r**2]
D = [[x**(i * 5) * b for b in B] for i in xrange(t)]
break
# Twin prime powers construction (see :wikipedia:`Difference_set`)
#
# i.e. v = p(p+2) where p and p+2 are prime powers
# k = (v-1)/2
# lambda = (k-1)/2
elif (len(factorization) == 2
and abs(pow(*factorization[0]) - pow(*factorization[1])) == 2
and k == (v - 1) // 2 and (l is None or 2 * l == (v - 1) // 2 - 1)):
# A difference set can be built from the set of elements
# (x,y) in GF(p) x GF(p+2) such that:
#
# - either y=0
# - x and y with x and y squares
# - x and y with x and y non-squares
if existence:
return True
from sage.rings.finite_rings.constructor import FiniteField
from sage.categories.cartesian_product import cartesian_product
from itertools import product
p, q = pow(*factorization[0]), pow(*factorization[1])
if p > q:
p, q = q, p
Fp = FiniteField(p, 'x')
Fq = FiniteField(q, 'x')
Fpset = set(Fp)
Fqset = set(Fq)
Fp_squares = set(x**2 for x in Fpset)
Fq_squares = set(x**2 for x in Fqset)
# Pairs of squares, pairs of non-squares
d = []
d.extend(
product(Fp_squares.difference([0]), Fq_squares.difference([0])))
d.extend(
product(Fpset.difference(Fp_squares),
Fqset.difference(Fq_squares)))
# All (x,0)
d.extend((x, 0) for x in Fpset)
G = cartesian_product([Fp, Fq])
D = [d]
if D is None and are_hyperplanes_in_projective_geometry_parameters(
v, k, l):
_, (q, d) = are_hyperplanes_in_projective_geometry_parameters(
v, k, l, True)
if existence:
return True
else:
G, D = singer_difference_set(q, d)
if D is None:
if existence:
return Unknown
raise NotImplementedError("No constructions for these parameters")
if check and not is_difference_family(G, D, verbose=False):
raise RuntimeError
return G, D
def difference_family(v, k, l=1, existence=False, check=True):
r"""
Return a (``k``, ``l``)-difference family on an Abelian group of size ``v``.
Let `G` be a finite Abelian group. For a given subset `D` of `G`, we define
`\Delta D` to be the multi-set of differences `\Delta D = \{x - y; x \in D,
y \in D, x \not= y\}`. A `(G,k,\lambda)`-*difference family* is a collection
of `k`-subsets of `G`, `D = \{D_1, D_2, \ldots, D_b\}` such that the union
of the difference sets `\Delta D_i` for `i=1,...b`, seen as a multi-set,
contains each element of `G \backslash \{0\}` exactly `\lambda`-times.
When there is only one block, i.e. `\lambda(v - 1) = k(k-1)`, then a
`(G,k,\lambda)`-difference family is also called a *difference set*.
See also :wikipedia:`Difference_set`.
If there is no such difference family, an ``EmptySetError`` is raised and if
there is no construction at the moment ``NotImplementedError`` is raised.
EXAMPLES::
sage: K,D = designs.difference_family(73,4)
sage: D
[[0, 1, 8, 64],
[0, 25, 54, 67],
[0, 41, 36, 69],
[0, 3, 24, 46],
[0, 2, 16, 55],
[0, 50, 35, 61]]
sage: K,D = designs.difference_family(337,7)
sage: D
[[1, 175, 295, 64, 79, 8, 52],
[326, 97, 125, 307, 142, 249, 102],
[121, 281, 310, 330, 123, 294, 226],
[17, 279, 297, 77, 332, 136, 210],
[150, 301, 103, 164, 55, 189, 49],
[35, 59, 215, 218, 69, 280, 135],
[289, 25, 331, 298, 252, 290, 200],
[191, 62, 66, 92, 261, 180, 159]]
For `k=6,7` we look at the set of small prime powers for which a
construction is available::
sage: def prime_power_mod(r,m):
....: k = m+r
....: while True:
....: if is_prime_power(k):
....: yield k
....: k += m
sage: from itertools import islice
sage: l6 = {True:[], False: [], Unknown: []}
sage: for q in islice(prime_power_mod(1,30), 60):
....: l6[designs.difference_family(q,6,existence=True)].append(q)
sage: l6[True]
[31, 151, 181, 211, ..., 3061, 3121, 3181]
sage: l6[Unknown]
[61, 121]
sage: l6[False]
[]
sage: l7 = {True: [], False: [], Unknown: []}
sage: for q in islice(prime_power_mod(1,42), 60):
....: l7[designs.difference_family(q,7,existence=True)].append(q)
sage: l7[True]
[337, 421, 463, 883, 1723, 3067, 3319, 3529, 3823, 3907, 4621, 4957, 5167]
sage: l7[Unknown]
[43, 127, 169, 211, ..., 4999, 5041, 5209]
sage: l7[False]
[]
Other constructions for `\lambda > 1`::
sage: for v in xrange(2,100):
....: constructions = []
....: for k in xrange(2,10):
....: for l in xrange(2,10):
....: if designs.difference_family(v,k,l,existence=True):
....: constructions.append((k,l))
....: _ = designs.difference_family(v,k,l)
....: if constructions:
....: print "%2d: %s"%(v, ', '.join('(%d,%d)'%(k,l) for k,l in constructions))
4: (3,2)
5: (4,3)
7: (3,2), (6,5)
8: (7,6)
9: (4,3), (8,7)
11: (5,2), (5,4)
13: (3,2), (4,3), (6,5)
15: (7,3)
16: (3,2), (5,4)
17: (4,3), (8,7)
19: (3,2), (6,5), (9,4), (9,8)
25: (3,2), (4,3), (6,5), (8,7)
29: (4,3), (7,6)
31: (3,2), (5,4), (6,5)
37: (3,2), (4,3), (6,5), (9,2), (9,8)
41: (4,3), (5,4), (8,7)
43: (3,2), (6,5), (7,6)
49: (3,2), (4,3), (6,5), (8,7)
53: (4,3)
61: (3,2), (4,3), (5,4), (6,5)
64: (3,2), (7,6), (9,8)
67: (3,2), (6,5)
71: (5,4), (7,6)
73: (3,2), (4,3), (6,5), (8,7), (9,8)
79: (3,2), (6,5)
81: (4,3), (5,4), (8,7)
89: (4,3), (8,7)
97: (3,2), (4,3), (6,5), (8,7)
TESTS:
Check more of the Wilson constructions from [Wi72]_::
sage: Q5 = [241, 281,421,601,641, 661, 701, 821,881]
sage: Q9 = [73, 1153, 1873, 2017]
sage: Q15 = [76231]
sage: Q4 = [13, 73, 97, 109, 181, 229, 241, 277, 337, 409, 421, 457]
sage: Q8 = [1009, 3137, 3697]
sage: for Q,k in [(Q4,4),(Q5,5),(Q8,8),(Q9,9),(Q15,15)]:
....: for q in Q:
....: assert designs.difference_family(q,k,1,existence=True) is True
....: _ = designs.difference_family(q,k,1)
Check Singer difference sets::
sage: sgp = lambda q,d: ((q**(d+1)-1)//(q-1), (q**d-1)//(q-1), (q**(d-1)-1)//(q-1))
sage: for q in range(2,10):
....: if is_prime_power(q):
....: for d in [2,3,4]:
....: v,k,l = sgp(q,d)
....: assert designs.difference_family(v,k,l,existence=True) is True
....: _ = designs.difference_family(v,k,l)
.. TODO::
There is a slightly more general version of difference families where
the stabilizers of the blocks are taken into account. A block is *short*
if the stabilizer is not trivial. The more general version is called a
*partial difference family*. It is still possible to construct BIBD from
this more general version (see the chapter 16 in the Handbook
[DesignHandbook]_).
Implement recursive constructions from Buratti "Recursive for difference
matrices and relative difference families" (1998) and Jungnickel
"Composition theorems for difference families and regular planes" (1978)
"""
if (l * (v - 1)) % (k * (k - 1)) != 0:
if existence:
return False
raise EmptySetError(
"A (v,%d,%d)-difference family may exist only if %d*(v-1) = mod %d" % (k, l, l, k * (k - 1))
)
from block_design import are_hyperplanes_in_projective_geometry_parameters
from database import DF_constructions
if (v, k, l) in DF_constructions:
if existence:
return True
return DF_constructions[(v, k, l)]()
e = k * (k - 1)
t = l * (v - 1) // e # number of blocks
D = None
if arith.is_prime_power(v):
from sage.rings.finite_rings.constructor import GF
G = K = GF(v, "z")
x = K.multiplicative_generator()
if l == (k - 1):
if existence:
return True
return K, K.cyclotomic_cosets(x ** ((v - 1) // k))[1:]
if t == 1:
# some of the difference set constructions VI.18.48 from the
# Handbook of combinatorial designs
# q = 3 mod 4
if v % 4 == 3 and k == (v - 1) // 2:
if existence:
return True
D = K.cyclotomic_cosets(x ** 2, [1])
# 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
D = K.cyclotomic_cosets(x ** 4, [1])
# 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
D = K.cyclotomic_cosets(x ** 4, [1])
D[0].insert(0, K.zero())
if D is None and l == 1:
one = K.one()
# Wilson (1972), Theorem 9
if k % 2 == 1:
m = (k - 1) // 2
xx = x ** m
to_coset = {x ** i * xx ** j: i for i in xrange(m) for j in xrange((v - 1) / m)}
r = x ** ((v - 1) // k) # primitive k-th root of unity
if len(set(to_coset[r ** j - one] for j in xrange(1, m + 1))) == m:
if existence:
return True
B = [r ** j for j in xrange(k)] # = H^((k-1)t) whose difference is
# H^(mt) (r^i - 1, i=1,..,m)
# Now pick representatives a translate of R for by a set of
# representatives of H^m / H^(mt)
D = [[x ** (i * m) * b for b in B] for i in xrange(t)]
# Wilson (1972), Theorem 10
else:
m = k // 2
xx = x ** m
to_coset = {x ** i * xx ** j: i for i in xrange(m) for j in xrange((v - 1) / m)}
r = x ** ((v - 1) // (k - 1)) # primitive (k-1)-th root of unity
if (
all(to_coset[r ** j - one] != 0 for j in xrange(1, m))
and len(set(to_coset[r ** j - one] for j in xrange(1, m))) == m - 1
):
if existence:
return True
B = [K.zero()] + [r ** j for j in xrange(k - 1)]
D = [[x ** (i * m) * b for b in B] for i in xrange(t)]
# Wilson (1972), Theorem 11
if D is None and k == 6:
r = x ** ((v - 1) // 3) # primitive cube root of unity
r2 = r * r
xx = x ** 5
to_coset = {x ** i * xx ** j: i for i in xrange(5) for j in xrange((v - 1) / 5)}
for c in to_coset:
if c == 1 or c == r or c == r2:
continue
if len(set(to_coset[elt] for elt in (r - 1, c * (r - 1), c - 1, c - r, c - r ** 2))) == 5:
if existence:
return True
B = [one, r, r ** 2, c, c * r, c * r ** 2]
D = [[x ** (i * 5) * b for b in B] for i in xrange(t)]
break
if D is None and are_hyperplanes_in_projective_geometry_parameters(v, k, l):
_, (q, d) = are_hyperplanes_in_projective_geometry_parameters(v, k, l, True)
if existence:
return True
else:
G, D = singer_difference_set(q, d)
if D is None:
if existence:
return Unknown
raise NotImplementedError("No constructions for these parameters")
if check and not is_difference_family(G, D, verbose=False):
raise RuntimeError
return G, D
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.
When `\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 values all equal to `\delta \epsilon
\sqrt(n)`. For more information, see [HX10]_ or 10.5.1 in
[BH12]_.
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
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 ( 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