def resolvable_balanced_incomplete_block_design(v,k,existence=False):
r"""
Return a resolvable BIBD of parameters `v,k`.
A BIBD is said to be *resolvable* if its blocks can be partitionned into
parallel classes, i.e. partitions of the ground set.
INPUT:
- ``v,k`` (integers)
- ``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.
.. SEEALSO::
- :meth:`IncidenceStructure.is_resolvable`
- :func:`~sage.combinat.designs.bibd.balanced_incomplete_block_design`
EXAMPLE::
sage: KTS15 = designs.resolvable_balanced_incomplete_block_design(15,3); KTS15
(15,3,1)-Balanced Incomplete Block Design
sage: KTS15.is_resolvable()
True
TESTS::
sage: for v in range(40):
....: for k in range(v):
....: if designs.resolvable_balanced_incomplete_block_design(v,k,existence=True):
....: _ = designs.resolvable_balanced_incomplete_block_design(v,k)
"""
# Trivial cases
if v==1 or k==v:
return balanced_incomplete_block_design(v,k,existence=existence)
# Non-existence of resolvable BIBD
if (v < k or
k < 2 or
v%k != 0 or
(v-1) % (k-1) != 0 or
(v*(v-1)) % (k*(k-1)) != 0 or
# From the Handbook of combinatorial designs:
#
# With lambda>1 the other exceptions is
# (15,5,2)
(k==6 and v == 36) or
# Fisher's inequality
(v*(v-1))/(k*(k-1)) < v):
if existence:
return False
raise EmptySetError("There exists no ({},{},{})-RBIBD".format(v,k,1))
if k==2:
if existence:
return True
classes = [[[(c+i)%(v-1),(c+v-i)%(v-1)] for i in range(1,v//2)]
for c in range(v-1)]
for i,classs in enumerate(classes):
classs.append([v-1,i])
B = BalancedIncompleteBlockDesign(v,
sum(classes,[]),
k = k,
check=True,
copy=False)
B._classes = classes
return B
elif k==3:
return kirkman_triple_system(v,existence=existence)
elif k==4:
return v_4_1_rbibd(v,existence=existence)
else:
if existence:
return Unknown
raise NotImplementedError("I don't know how to build a ({},{},1)-RBIBD!".format(v,3))
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.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 join(rows,'\n')
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.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.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 resolvable_balanced_incomplete_block_design(v, k, existence=False):
r"""
Return a resolvable BIBD of parameters `v,k`.
A BIBD is said to be *resolvable* if its blocks can be partitionned into
parallel classes, i.e. partitions of the ground set.
INPUT:
- ``v,k`` (integers)
- ``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.
.. SEEALSO::
- :meth:`IncidenceStructure.is_resolvable`
- :func:`~sage.combinat.designs.bibd.balanced_incomplete_block_design`
EXAMPLE::
sage: KTS15 = designs.resolvable_balanced_incomplete_block_design(15,3); KTS15
(15,3,1)-Balanced Incomplete Block Design
sage: KTS15.is_resolvable()
True
TESTS::
sage: for v in range(40):
....: for k in range(v):
....: if designs.resolvable_balanced_incomplete_block_design(v,k,existence=True):
....: _ = designs.resolvable_balanced_incomplete_block_design(v,k)
"""
# Trivial cases
if v == 1 or k == v:
return balanced_incomplete_block_design(v, k, existence=existence)
# Non-existence of resolvable BIBD
if (v < k or k < 2 or v % k != 0 or (v - 1) % (k - 1) != 0
or (v * (v - 1)) % (k * (k - 1)) != 0 or
# From the Handbook of combinatorial designs:
#
# With lambda>1 the other exceptions is
# (15,5,2)
(k == 6 and v == 36) or
# Fisher's inequality
(v * (v - 1)) / (k * (k - 1)) < v):
if existence:
return False
raise EmptySetError("There exists no ({},{},{})-RBIBD".format(v, k, 1))
if k == 2:
if existence:
return True
classes = [[[(c + i) % (v - 1), (c + v - i) % (v - 1)]
for i in range(1, v // 2)] for c in range(v - 1)]
for i, classs in enumerate(classes):
classs.append([v - 1, i])
B = BalancedIncompleteBlockDesign(v,
sum(classes, []),
k=k,
check=True,
copy=False)
B._classes = classes
return B
elif k == 3:
return kirkman_triple_system(v, existence=existence)
elif k == 4:
return v_4_1_rbibd(v, existence=existence)
else:
if existence:
return Unknown
raise NotImplementedError(
"I don't know how to build a ({},{},1)-RBIBD!".format(v, 3))
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
