def BIBD_5q_5_for_q_prime_power(q):
r"""
Return a `(5q,5,1)`-BIBD with `q\equiv 1\pmod 4` a prime power.
See Theorem 24 [ClaytonSmith]_.
INPUT:
- ``q`` (integer) -- a prime power such that `q\equiv 1\pmod 4`.
EXAMPLES::
sage: from sage.combinat.designs.bibd import BIBD_5q_5_for_q_prime_power
sage: for q in [25, 45, 65, 85, 125, 145, 185, 205, 305, 405, 605]: # long time
....: _ = BIBD_5q_5_for_q_prime_power(q/5) # long time
"""
from sage.rings.finite_rings.finite_field_constructor import FiniteField
if q%4 != 1 or not is_prime_power(q):
raise ValueError("q is not a prime power or q%4!=1.")
d = (q-1)/4
B = []
F = FiniteField(q,'x')
a = F.primitive_element()
L = {b:i for i,b in enumerate(F)}
for b in L:
B.append([i*q + L[b] for i in range(5)])
for i in range(5):
for j in range(d):
B.append([ i*q + L[b ],
((i+1)%5)*q + L[ a**j+b ],
((i+1)%5)*q + L[-a**j+b ],
((i+4)%5)*q + L[ a**(j+d)+b],
((i+4)%5)*q + L[-a**(j+d)+b],
])
return B
def v_4_1_rbibd(v, existence=False):
r"""
Return a `(v,4,1)`-RBIBD.
INPUT:
- `n` (integer)
- ``existence`` (boolean; ``False`` by default) -- whether to build the
design or only answer whether it exists.
.. SEEALSO::
- :meth:`IncidenceStructure.is_resolvable`
- :func:`resolvable_balanced_incomplete_block_design`
.. NOTE::
A resolvable `(v,4,1)`-BIBD exists whenever `1\equiv 4\pmod(12)`. This
function, however, only implements a construction of `(v,4,1)`-BIBD such
that `v=3q+1\equiv 1\pmod{3}` where `q` is a prime power (see VII.7.5.a
from [BJL99]_).
EXAMPLE::
sage: rBIBD = designs.resolvable_balanced_incomplete_block_design(28,4)
sage: rBIBD.is_resolvable()
True
sage: rBIBD.is_t_design(return_parameters=True)
(True, (2, 28, 4, 1))
TESTS::
sage: for q in prime_powers(2,30):
....: if (3*q+1)%12 == 4:
....: _ = designs.resolvable_balanced_incomplete_block_design(3*q+1,4) # indirect doctest
"""
# Volume 1, VII.7.5.a from [BJL99]_
if v % 3 != 1 or not is_prime_power((v - 1) // 3):
if existence:
return Unknown
raise NotImplementedError(
"I don't know how to build a ({},{},1)-RBIBD!".format(v, 4))
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
q = (v - 1) // 3
nn = (q - 1) // 4
G = GF(q, 'x')
w = G.primitive_element()
e = w**(nn)
assert e**2 == -1
first_class = [[(w**i, j), (-w**i, j), (e * w**i, j + 1),
(-e * w**i, j + 1)] for i in range(nn) for j in range(3)]
first_class.append([(0, 0), (0, 1), (0, 2), 'inf'])
label = {p: i for i, p in enumerate(G)}
classes = [[[
v - 1 if x == 'inf' else (x[1] % 3) * q + label[x[0] + g] for x in S
] for S in first_class] for g in G]
BIBD = BalancedIncompleteBlockDesign(v,
blocks=sum(classes, []),
k=4,
check=True,
copy=False)
BIBD._classes = classes
assert BIBD.is_resolvable()
return BIBD
def kirkman_triple_system(v, existence=False):
r"""
Return a Kirkman Triple System on `v` points.
A Kirkman Triple System `KTS(v)` is a resolvable Steiner Triple System. It
exists if and only if `v\equiv 3\pmod{6}`.
INPUT:
- `n` (integer)
- ``existence`` (boolean; ``False`` by default) -- whether to build the
`KTS(n)` or only answer whether it exists.
.. SEEALSO::
:meth:`IncidenceStructure.is_resolvable`
EXAMPLES:
A solution to Kirkmman's original problem::
sage: kts = designs.kirkman_triple_system(15)
sage: classes = kts.is_resolvable(1)[1]
sage: names = '0123456789abcde'
sage: def to_name(r_s_t):
....: r, s, t = r_s_t
....: return ' ' + names[r] + names[s] + names[t] + ' '
sage: rows = [' '.join(('Day {}'.format(i) for i in range(1,8)))]
sage: rows.extend(' '.join(map(to_name,row)) for row in zip(*classes))
sage: print('\n'.join(rows))
Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7
07e 18e 29e 3ae 4be 5ce 6de
139 24a 35b 46c 05d 167 028
26b 03c 14d 257 368 049 15a
458 569 06a 01b 12c 23d 347
acd 7bd 78c 89d 79a 8ab 9bc
TESTS::
sage: for i in range(3,300,6):
....: _ = designs.kirkman_triple_system(i)
"""
if v % 6 != 3:
if existence:
return False
raise ValueError("There is no KTS({}) as v!=3 mod(6)".format(v))
if existence:
return False
elif v == 3:
return BalancedIncompleteBlockDesign(3, [[0, 1, 2]], k=3, lambd=1)
elif v == 9:
classes = [[[0, 1, 5], [2, 6, 7], [3, 4, 8]],
[[1, 6, 8], [3, 5, 7], [0, 2, 4]],
[[1, 4, 7], [0, 3, 6], [2, 5, 8]],
[[4, 5, 6], [0, 7, 8], [1, 2, 3]]]
KTS = BalancedIncompleteBlockDesign(
v, [tr for cl in classes for tr in cl], k=3, lambd=1, copy=False)
KTS._classes = classes
return KTS
# Construction 1.1 from [Stinson91] (originally Theorem 6 from [RCW71])
#
# For all prime powers q=1 mod 6, there exists a KTS(2q+1)
elif ((v - 1) // 2) % 6 == 1 and is_prime_power((v - 1) // 2):
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
q = (v - 1) // 2
K = GF(q, 'x')
a = K.primitive_element()
t = (q - 1) // 6
# m is the solution of a^m=(a^t+1)/2
from sage.groups.generic import discrete_log
m = discrete_log((a**t + 1) / 2, a)
assert 2 * a**m == a**t + 1
# First parallel class
first_class = [[(0, 1), (0, 2), 'inf']]
b0 = K.one()
b1 = a**t
b2 = a**m
first_class.extend([(b0 * a**i, 1), (b1 * a**i, 1), (b2 * a**i, 2)]
for i in list(range(t)) +
list(range(2 * t, 3 * t)) +
list(range(4 * t, 5 * t)))
b0 = a**(m + t)
b1 = a**(m + 3 * t)
b2 = a**(m + 5 * t)
first_class.extend([[(b0 * a**i, 2), (b1 * a**i, 2), (b2 * a**i, 2)]
for i in range(t)])
# Action of K on the points
action = lambda v, x: (v + x[0], x[1]) if len(x) == 2 else x
# relabel to integer
relabel = {(p, x): i + (x - 1) * q
for i, p in enumerate(K) for x in [1, 2]}
relabel['inf'] = 2 * q
classes = [[[relabel[action(p, x)] for x in tr] for tr in first_class]
for p in K]
KTS = BalancedIncompleteBlockDesign(
v, [tr for cl in classes for tr in cl], k=3, lambd=1, copy=False)
KTS._classes = classes
return KTS
# Construction 1.2 from [Stinson91] (originally Theorem 5 from [RCW71])
#
# For all prime powers q=1 mod 6, there exists a KTS(3q)
elif (v // 3) % 6 == 1 and is_prime_power(v // 3):
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
q = v // 3
K = GF(q, 'x')
a = K.primitive_element()
t = (q - 1) // 6
A0 = [(0, 0), (0, 1), (0, 2)]
B = [[(a**i, j), (a**(i + 2 * t), j), (a**(i + 4 * t), j)]
for j in range(3) for i in range(t)]
A = [[(a**i, 0), (a**(i + 2 * t), 1), (a**(i + 4 * t), 2)]
for i in range(6 * t)]
# Action of K on the points
action = lambda v, x: (v + x[0], x[1])
# relabel to integer
relabel = {(p, j): i + j * q
for i, p in enumerate(K) for j in range(3)}
B0 = [A0] + B + A[t:2 * t] + A[3 * t:4 * t] + A[5 * t:6 * t]
# Classes
classes = [[[relabel[action(p, x)] for x in tr] for tr in B0]
for p in K]
for i in list(range(t)) + list(range(2 * t, 3 * t)) + list(
range(4 * t, 5 * t)):
classes.append([[relabel[action(p, x)] for x in A[i]] for p in K])
KTS = BalancedIncompleteBlockDesign(
v, [tr for cl in classes for tr in cl], k=3, lambd=1, copy=False)
KTS._classes = classes
return KTS
else:
# This is Lemma IX.6.4 from [BJL99].
#
# This construction takes a (v,{4,7})-PBD. All points are doubled (x has
# a copy x'), and an infinite point \infty is added.
#
# On all blocks of 2*4 points we "paste" a KTS(2*4+1) using the infinite
# point, in such a way that all {x,x',infty} are set of the design. We
# do the same for blocks with 2*7 points using a KTS(2*7+1).
#
# Note that the triples of points equal to {x,x',\infty} will be added
# several times.
#
# As all those subdesigns are resolvable, each class of the KTS(n) is
# obtained by considering a set {x,x',\infty} and all sets of all
# parallel classes of the subdesign which contain this set.
# We create the small KTS(n') we need, and relabel them such that
# 01(n'-1),23(n'-1),... are blocks of the design.
gdd4 = kirkman_triple_system(9)
gdd7 = kirkman_triple_system(15)
X = [B for B in gdd4 if 8 in B]
for b in X:
b.remove(8)
X = sum(X, []) + [8]
gdd4.relabel({v: i for i, v in enumerate(X)})
gdd4 = gdd4.is_resolvable(True)[1] # the relabeled classes
X = [B for B in gdd7 if 14 in B]
for b in X:
b.remove(14)
X = sum(X, []) + [14]
gdd7.relabel({v: i for i, v in enumerate(X)})
gdd7 = gdd7.is_resolvable(True)[1] # the relabeled classes
# The first parallel class contains 01(n'-1), the second contains
# 23(n'-1), etc..
# Then remove the blocks containing (n'-1)
for B in gdd4:
for i, b in enumerate(B):
if 8 in b:
j = min(b)
del B[i]
B.insert(0, j)
break
gdd4.sort()
for B in gdd4:
B.pop(0)
for B in gdd7:
for i, b in enumerate(B):
if 14 in b:
j = min(b)
del B[i]
B.insert(0, j)
break
gdd7.sort()
for B in gdd7:
B.pop(0)
# Pasting the KTS(n') without {x,x',\infty} blocks
classes = [[] for i in range((v - 1) // 2)]
gdd = {4: gdd4, 7: gdd7}
for B in PBD_4_7((v - 1) // 2, check=False):
for i, classs in enumerate(gdd[len(B)]):
classes[B[i]].extend([[2 * B[x // 2] + x % 2 for x in BB]
for BB in classs])
# The {x,x',\infty} blocks
for i, classs in enumerate(classes):
classs.append([2 * i, 2 * i + 1, v - 1])
KTS = BalancedIncompleteBlockDesign(
v,
blocks=[tr for cl in classes for tr in cl],
k=3,
lambd=1,
check=True,
copy=False)
KTS._classes = classes
assert KTS.is_resolvable()
return KTS
def GDD_4_2(q, existence=False, check=True):
r"""
Return a `(2q,\{4\},\{2\})`-GDD for `q` a prime power with `q\equiv 1\pmod{6}`.
This method implements Lemma VII.5.17 from [BJL99] (p.495).
INPUT:
- ``q`` (integer)
- ``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.
- ``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.
EXAMPLE::
sage: from sage.combinat.designs.group_divisible_designs import GDD_4_2
sage: GDD_4_2(7,existence=True)
True
sage: GDD_4_2(7)
Group Divisible Design on 14 points of type 2^7
sage: GDD_4_2(8,existence=True)
Unknown
sage: GDD_4_2(8)
Traceback (most recent call last):
...
NotImplementedError
"""
if q <= 1 or q % 6 != 1 or not is_prime_power(q):
if existence:
return Unknown
raise NotImplementedError
if existence:
return True
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
G = GF(q, "x")
w = G.primitive_element()
e = w ** ((q - 1) // 3)
# A first parallel class is defined. G acts on it, which yields all others.
first_class = [[(0, 0), (1, w ** i), (1, e * w ** i), (1, e * e * w ** i)] for i in range((q - 1) // 6)]
label = {p: i for i, p in enumerate(G)}
classes = [[[2 * label[x[1] + g] + (x[0] + j) % 2 for x in S] for S in first_class] for g in G for j in range(2)]
return GroupDivisibleDesign(
2 * q,
groups=[[i, i + 1] for i in range(0, 2 * q, 2)],
blocks=sum(classes, []),
K=[4],
G=[2],
check=check,
copy=False,
)
def v_4_1_rbibd(v,existence=False):
r"""
Return a `(v,4,1)`-RBIBD.
INPUT:
- `n` (integer)
- ``existence`` (boolean; ``False`` by default) -- whether to build the
design or only answer whether it exists.
.. SEEALSO::
- :meth:`IncidenceStructure.is_resolvable`
- :func:`resolvable_balanced_incomplete_block_design`
.. NOTE::
A resolvable `(v,4,1)`-BIBD exists whenever `1\equiv 4\pmod(12)`. This
function, however, only implements a construction of `(v,4,1)`-BIBD such
that `v=3q+1\equiv 1\pmod{3}` where `q` is a prime power (see VII.7.5.a
from [BJL99]_).
EXAMPLE::
sage: rBIBD = designs.resolvable_balanced_incomplete_block_design(28,4)
sage: rBIBD.is_resolvable()
True
sage: rBIBD.is_t_design(return_parameters=True)
(True, (2, 28, 4, 1))
TESTS::
sage: for q in prime_powers(2,30):
....: if (3*q+1)%12 == 4:
....: _ = designs.resolvable_balanced_incomplete_block_design(3*q+1,4) # indirect doctest
"""
# Volume 1, VII.7.5.a from [BJL99]_
if v%3 != 1 or not is_prime_power((v-1)//3):
if existence:
return Unknown
raise NotImplementedError("I don't know how to build a ({},{},1)-RBIBD!".format(v,4))
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
q = (v-1)//3
nn = (q-1)//4
G = GF(q,'x')
w = G.primitive_element()
e = w**(nn)
assert e**2 == -1
first_class = [[(w**i,j),(-w**i,j),(e*w**i,j+1),(-e*w**i,j+1)]
for i in range(nn) for j in range(3)]
first_class.append([(0,0),(0,1),(0,2),'inf'])
label = {p:i for i,p in enumerate(G)}
classes = [[[v-1 if x=='inf' else (x[1]%3)*q+label[x[0]+g] for x in S]
for S in first_class]
for g in G]
BIBD = BalancedIncompleteBlockDesign(v,
blocks = sum(classes,[]),
k=4,
check=True,
copy=False)
BIBD._classes = classes
assert BIBD.is_resolvable()
return BIBD
def kirkman_triple_system(v,existence=False):
r"""
Return a Kirkman Triple System on `v` points.
A Kirkman Triple System `KTS(v)` is a resolvable Steiner Triple System. It
exists if and only if `v\equiv 3\pmod{6}`.
INPUT:
- `n` (integer)
- ``existence`` (boolean; ``False`` by default) -- whether to build the
`KTS(n)` or only answer whether it exists.
.. SEEALSO::
:meth:`IncidenceStructure.is_resolvable`
EXAMPLES:
A solution to Kirkmman's original problem::
sage: kts = designs.kirkman_triple_system(15)
sage: classes = kts.is_resolvable(1)[1]
sage: names = '0123456789abcde'
sage: to_name = lambda (r,s,t): ' '+names[r]+names[s]+names[t]+' '
sage: rows = [' '.join(('Day {}'.format(i) for i in range(1,8)))]
sage: rows.extend(' '.join(map(to_name,row)) for row in zip(*classes))
sage: print '\n'.join(rows)
Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7
07e 18e 29e 3ae 4be 5ce 6de
139 24a 35b 46c 05d 167 028
26b 03c 14d 257 368 049 15a
458 569 06a 01b 12c 23d 347
acd 7bd 78c 89d 79a 8ab 9bc
TESTS::
sage: for i in range(3,300,6):
....: _ = designs.kirkman_triple_system(i)
"""
if v%6 != 3:
if existence:
return False
raise ValueError("There is no KTS({}) as v!=3 mod(6)".format(v))
if existence:
return False
elif v == 3:
return BalancedIncompleteBlockDesign(3,[[0,1,2]],k=3,lambd=1)
elif v == 9:
classes = [[[0, 1, 5], [2, 6, 7], [3, 4, 8]],
[[1, 6, 8], [3, 5, 7], [0, 2, 4]],
[[1, 4, 7], [0, 3, 6], [2, 5, 8]],
[[4, 5, 6], [0, 7, 8], [1, 2, 3]]]
KTS = BalancedIncompleteBlockDesign(v,[tr for cl in classes for tr in cl],k=3,lambd=1,copy=False)
KTS._classes = classes
return KTS
# Construction 1.1 from [Stinson91] (originally Theorem 6 from [RCW71])
#
# For all prime powers q=1 mod 6, there exists a KTS(2q+1)
elif ((v-1)//2)%6 == 1 and is_prime_power((v-1)//2):
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
q = (v-1)//2
K = GF(q,'x')
a = K.primitive_element()
t = (q-1)/6
# m is the solution of a^m=(a^t+1)/2
from sage.groups.generic import discrete_log
m = discrete_log((a**t+1)/2, a)
assert 2*a**m == a**t+1
# First parallel class
first_class = [[(0,1),(0,2),'inf']]
b0 = K.one(); b1 = a**t; b2 = a**m
first_class.extend([(b0*a**i,1),(b1*a**i,1),(b2*a**i,2)]
for i in range(t)+range(2*t,3*t)+range(4*t,5*t))
b0 = a**(m+t); b1=a**(m+3*t); b2=a**(m+5*t)
first_class.extend([[(b0*a**i,2),(b1*a**i,2),(b2*a**i,2)]
for i in range(t)])
# Action of K on the points
action = lambda v,x : (v+x[0],x[1]) if len(x) == 2 else x
# relabel to integer
relabel = {(p,x): i+(x-1)*q
for i,p in enumerate(K)
for x in [1,2]}
relabel['inf'] = 2*q
classes = [[[relabel[action(p,x)] for x in tr] for tr in first_class]
for p in K]
KTS = BalancedIncompleteBlockDesign(v,[tr for cl in classes for tr in cl],k=3,lambd=1,copy=False)
KTS._classes = classes
return KTS
# Construction 1.2 from [Stinson91] (originally Theorem 5 from [RCW71])
#
# For all prime powers q=1 mod 6, there exists a KTS(3q)
elif (v//3)%6 == 1 and is_prime_power(v//3):
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
q = v//3
K = GF(q,'x')
a = K.primitive_element()
t = (q-1)/6
A0 = [(0,0),(0,1),(0,2)]
B = [[(a**i,j),(a**(i+2*t),j),(a**(i+4*t),j)] for j in range(3)
for i in range(t)]
A = [[(a**i,0),(a**(i+2*t),1),(a**(i+4*t),2)] for i in range(6*t)]
# Action of K on the points
action = lambda v,x: (v+x[0],x[1])
# relabel to integer
relabel = {(p,j): i+j*q
for i,p in enumerate(K)
for j in range(3)}
B0 = [A0] + B + A[t:2*t] + A[3*t:4*t] + A[5*t:6*t]
# Classes
classes = [[[relabel[action(p,x)] for x in tr] for tr in B0]
for p in K]
for i in range(t)+range(2*t,3*t)+range(4*t,5*t):
classes.append([[relabel[action(p,x)] for x in A[i]] for p in K])
KTS = BalancedIncompleteBlockDesign(v,[tr for cl in classes for tr in cl],k=3,lambd=1,copy=False)
KTS._classes = classes
return KTS
else:
# This is Lemma IX.6.4 from [BJL99].
#
# This construction takes a (v,{4,7})-PBD. All points are doubled (x has
# a copy x'), and an infinite point \infty is added.
#
# On all blocks of 2*4 points we "paste" a KTS(2*4+1) using the infinite
# point, in such a way that all {x,x',infty} are set of the design. We
# do the same for blocks with 2*7 points using a KTS(2*7+1).
#
# Note that the triples of points equal to {x,x',\infty} will be added
# several times.
#
# As all those subdesigns are resolvable, each class of the KTS(n) is
# obtained by considering a set {x,x',\infty} and all sets of all
# parallel classes of the subdesign which contain this set.
# We create the small KTS(n') we need, and relabel them such that
# 01(n'-1),23(n'-1),... are blocks of the design.
gdd4 = kirkman_triple_system(9)
gdd7 = kirkman_triple_system(15)
X = [B for B in gdd4 if 8 in B]
for b in X:
b.remove(8)
X = sum(X, []) + [8]
gdd4.relabel({v:i for i,v in enumerate(X)})
gdd4 = gdd4.is_resolvable(True)[1] # the relabeled classes
X = [B for B in gdd7 if 14 in B]
for b in X:
b.remove(14)
X = sum(X, []) + [14]
gdd7.relabel({v:i for i,v in enumerate(X)})
gdd7 = gdd7.is_resolvable(True)[1] # the relabeled classes
# The first parallel class contains 01(n'-1), the second contains
# 23(n'-1), etc..
# Then remove the blocks containing (n'-1)
for B in gdd4:
for i,b in enumerate(B):
if 8 in b: j = min(b); del B[i]; B.insert(0,j); break
gdd4.sort()
for B in gdd4:
B.pop(0)
for B in gdd7:
for i,b in enumerate(B):
if 14 in b: j = min(b); del B[i]; B.insert(0,j); break
gdd7.sort()
for B in gdd7:
B.pop(0)
# Pasting the KTS(n') without {x,x',\infty} blocks
classes = [[] for i in range((v-1)/2)]
gdd = {4:gdd4, 7: gdd7}
for B in PBD_4_7((v-1)//2,check=False):
for i,classs in enumerate(gdd[len(B)]):
classes[B[i]].extend([[2*B[x//2]+x%2 for x in BB] for BB in classs])
# The {x,x',\infty} blocks
for i,classs in enumerate(classes):
classs.append([2*i,2*i+1,v-1])
KTS = BalancedIncompleteBlockDesign(v,
blocks = [tr for cl in classes for tr in cl],
k=3,
lambd=1,
check=True,
copy =False)
KTS._classes = classes
assert KTS.is_resolvable()
return KTS
def GDD_4_2(q,existence=False,check=True):
r"""
Return a `(2q,\{4\},\{2\})`-GDD for `q` a prime power with `q\equiv 1\pmod{6}`.
This method implements Lemma VII.5.17 from [BJL99] (p.495).
INPUT:
- ``q`` (integer)
- ``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.
- ``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.
EXAMPLE::
sage: from sage.combinat.designs.group_divisible_designs import GDD_4_2
sage: GDD_4_2(7,existence=True)
True
sage: GDD_4_2(7)
Group Divisible Design on 14 points of type 2^7
sage: GDD_4_2(8,existence=True)
Unknown
sage: GDD_4_2(8)
Traceback (most recent call last):
...
NotImplementedError
"""
if q <=1 or q%6 != 1 or not is_prime_power(q):
if existence:
return Unknown
raise NotImplementedError
if existence:
return True
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
G = GF(q,'x')
w = G.primitive_element()
e = w**((q - 1) // 3)
# A first parallel class is defined. G acts on it, which yields all others.
first_class = [[(0,0),(1,w**i),(1,e*w**i),(1,e*e*w**i)]
for i in range((q - 1) // 6)]
label = {p:i for i,p in enumerate(G)}
classes = [[[2*label[x[1]+g]+(x[0]+j)%2 for x in S]
for S in first_class]
for g in G for j in range(2)]
return GroupDivisibleDesign(2*q,
groups = [[i,i+1] for i in range(0,2*q,2)],
blocks = sum(classes,[]),
K = [4],
G = [2],
check = check,
copy = False)
def __init__(self,
q,
level,
info_magma=None,
grouptype=None,
magma=None,
compute_presentation=True):
from sage.modular.arithgroup.congroup_gamma import Gamma_constructor
assert grouptype in ['SL2', 'PSL2']
self._grouptype = grouptype
self._compute_presentation = compute_presentation
self.magma = magma
self.F = QQ
self.q = ZZ(q)
self.discriminant = ZZ(1)
self.level = ZZ(level / self.q)
if self.level != 1 and compute_presentation:
raise NotImplementedError
self._Gamma = Gamma_constructor(self.q)
self._Gamma_farey = self._Gamma.farey_symbol()
self.F_units = []
self._prec_inf = -1
self.B = MatrixSpace(QQ, 2, 2)
# Here we initialize the non-split Cartan, properly
self.GFq = FiniteField(self.q)
if not self.GFq(-1).is_square():
self.eps = ZZ(-1)
else:
self.eps = ZZ(2)
while self.GFq(self.eps).is_square():
self.eps += 1
epsinv = (self.GFq(self.eps)**-1).lift()
N = self.level
q = self.q
self.Obasis = [
matrix(ZZ, 2, 2, v)
for v in [[1, 0, 0, 1], [0, q, 0, 0], [0, N *
epsinv, N, 0], [0, 0, 0, q]]
]
x = QQ['x'].gen()
K = FiniteField(self.q**2, 'z', modulus=x * x - self.eps)
x = K.primitive_element()
x1 = x
while x1.multiplicative_order() != self.q + 1 or x1.norm() != 1:
x1 *= x
a, b = x1.polynomial().list() # represents a+b*sqrt(eps)
a = a.lift()
b = b.lift()
self.extra_matrix = self.B(
lift(matrix(ZZ, 2, 2, [a, b, b * self.eps, a]), self.q))
self.extra_matrix_inverse = ~self.extra_matrix
if compute_presentation:
self.Ugens = []
self._gens = []
temp_relation_words = []
I = SL2Z([1, 0, 0, 1])
E = SL2Z([-1, 0, 0, -1])
minus_one = []
for i, g in enumerate(self._Gamma_farey.generators()):
newg = self.B([g.a(), g.b(), g.c(), g.d()])
if newg == I:
continue
self.Ugens.append(newg)
self._gens.append(
self.element_class(self,
quaternion_rep=newg,
word_rep=[i + 1],
check=False))
if g.matrix()**2 == I.matrix():
temp_relation_words.append([i + 1, i + 1])
if minus_one is not None:
temp_relation_words.append([-i - 1] + minus_one)
else:
minus_one = [i + 1]
elif g.matrix()**2 == E.matrix():
temp_relation_words.append([i + 1, i + 1, i + 1, i + 1])
if minus_one is not None:
temp_relation_words.append([-i - 1, -i - 1] +
minus_one)
else:
minus_one = [i + 1, i + 1]
elif g.matrix()**3 == I.matrix():
temp_relation_words.append([i + 1, i + 1, i + 1])
elif g.matrix()**3 == E.matrix():
temp_relation_words.append(
[i + 1, i + 1, i + 1, i + 1, i + 1, i + 1])
if minus_one is not None:
temp_relation_words.append([-i - 1, -i - 1, -i - 1] +
minus_one)
else:
minus_one = [i + 1, i + 1, i + 1]
else:
assert g.matrix()**24 != I.matrix()
# The extra matrix is added
i = len(self.Ugens)
self.extra_matrix_index = i
self.Ugens.append(self.extra_matrix)
self._gens.append(
self.element_class(self,
quaternion_rep=self.Ugens[i],
word_rep=[i + 1],
check=False))
# The new relations are also added
w = self._get_word_rep_initial(self.extra_matrix**(self.q + 1))
temp_relation_words.append(w + ([-i - 1] * (self.q + 1)))
for j, g in enumerate(self.Ugens[:-1]):
g1 = self.extra_matrix_inverse * g * self.extra_matrix
w = self._get_word_rep_initial(g1)
new_rel = w + [-i - 1, -j - 1, i + 1]
temp_relation_words.append(new_rel)
self.F_unit_offset = len(self.Ugens)
if minus_one is not None:
self.minus_one_long = syllables_to_tietze(minus_one)
self._relation_words = []
for rel in temp_relation_words:
sign = multiply_out(rel, self.Ugens, self.B(1))
if sign == self.B(1) or 'P' in grouptype:
self._relation_words.append(rel)
else:
assert sign == self.B(-1)
newrel = rel + self.minus_one
sign = multiply_out(newrel, self.Ugens, self.B(1))
assert sign == self.B(1)
self._relation_words.append(newrel)
ArithGroup_generic.__init__(self)
Parent.__init__(self)
def __init__(self,q,level,info_magma = None,grouptype = None,magma = None, compute_presentation = True):
from sage.modular.arithgroup.congroup_gamma import Gamma_constructor
assert grouptype in ['SL2','PSL2']
self._grouptype = grouptype
self._compute_presentation = compute_presentation
self.magma = magma
self.F = QQ
self.q = ZZ(q)
self.discriminant = ZZ(1)
self.level = ZZ(level/self.q)
if self.level != 1 and compute_presentation:
raise NotImplementedError
self._Gamma = Gamma_constructor(self.q)
self._Gamma_farey = self._Gamma.farey_symbol()
self.F_units = []
self._prec_inf = -1
self.B = MatrixSpace(QQ,2,2)
self._O_discriminant = ZZ.ideal(self.level * self.q)
# Here we initialize the non-split Cartan, properly
self.GFq = FiniteField(self.q)
if not self.GFq(-1).is_square():
self.eps = ZZ(-1)
else:
self.eps = ZZ(2)
while self.GFq(self.eps).is_square():
self.eps += 1
epsinv = (self.GFq(self.eps)**-1).lift()
N = self.level
q = self.q
self.Obasis = [matrix(ZZ,2,2,v) for v in [[1,0,0,1], [0,q,0,0], [0,N*epsinv,N,0], [0,0,0,q]]]
x = QQ['x'].gen()
K = FiniteField(self.q**2,'z',modulus = x*x - self.eps)
x = K.primitive_element()
x1 = x
while x1.multiplicative_order() != self.q+1 or x1.norm() != 1:
x1 *= x
a, b = x1.polynomial().list() # represents a+b*sqrt(eps)
a = a.lift()
b = b.lift()
self.extra_matrix = self.B(lift(matrix(ZZ,2,2,[a,b,b*self.eps,a]),self.q))
self.extra_matrix_inverse = ~self.extra_matrix
if compute_presentation:
self.Ugens = []
self._gens = []
temp_relation_words = []
I = SL2Z([1,0,0,1])
E = SL2Z([-1,0,0,-1])
minus_one = []
for i,g in enumerate(self._Gamma_farey.generators()):
newg = self.B([g.a(),g.b(),g.c(),g.d()])
if newg == I:
continue
self.Ugens.append(newg)
self._gens.append(self.element_class(self,quaternion_rep = newg, word_rep = [i+1],check = False))
if g.matrix()**2 == I.matrix():
temp_relation_words.append([i+1, i+1])
if minus_one is not None:
temp_relation_words.append([-i-1]+minus_one)
else:
minus_one = [i+1]
elif g.matrix()**2 == E.matrix():
temp_relation_words.append([i+1,i+1,i+1,i+1])
if minus_one is not None:
temp_relation_words.append([-i-1,-i-1]+minus_one)
else:
minus_one = [i+1, i+1]
elif g.matrix()**3 == I.matrix():
temp_relation_words.append([i+1, i+1, i+1])
elif g.matrix()**3 == E.matrix():
temp_relation_words.append([i+1, i+1, i+1, i+1, i+1, i+1])
if minus_one is not None:
temp_relation_words.append([-i-1, -i-1, -i-1]+minus_one)
else:
minus_one = [i+1, i+1, i+1]
else:
assert g.matrix()**24 != I.matrix()
# The extra matrix is added
i = len(self.Ugens)
self.extra_matrix_index = i
self.Ugens.append(self.extra_matrix)
self._gens.append(self.element_class(self,quaternion_rep = self.Ugens[i], word_rep = [i+1],check = False))
# The new relations are also added
w = self._get_word_rep_initial(self.extra_matrix**(self.q+1))
temp_relation_words.append( w + ([-i-1] * (self.q+1)))
for j,g in enumerate(self.Ugens[:-1]):
g1 = self.extra_matrix_inverse * g * self.extra_matrix
w = self._get_word_rep_initial(g1)
new_rel = w + [-i-1, -j-1, i+1]
temp_relation_words.append(new_rel)
self.F_unit_offset = len(self.Ugens)
if minus_one is not None:
self.minus_one_long = syllables_to_tietze(minus_one)
self._relation_words = []
for rel in temp_relation_words:
sign = multiply_out(rel, self.Ugens, self.B(1))
if sign == self.B(1) or 'P' in grouptype:
self._relation_words.append(rel)
else:
assert sign == self.B(-1)
newrel = rel + self.minus_one
sign = multiply_out(newrel, self.Ugens, self.B(1))
assert sign == self.B(1)
self._relation_words.append(newrel)
ArithGroup_generic.__init__(self)
Parent.__init__(self)
