def v_5_1_BIBD(v, check=True):
r"""
Return a `(v,5,1)`-BIBD.
This method follows the constuction from [ClaytonSmith]_.
INPUT:
- ``v`` (integer)
.. SEEALSO::
* :func:`balanced_incomplete_block_design`
EXAMPLES::
sage: from sage.combinat.designs.bibd import v_5_1_BIBD
sage: i = 0
sage: while i<200:
....: i += 20
....: _ = v_5_1_BIBD(i+1)
....: _ = v_5_1_BIBD(i+5)
TESTS:
Check that the needed difference families are there::
sage: for v in [21,41,61,81,141,161,281]:
....: assert designs.difference_family(v,5,existence=True)
....: _ = designs.difference_family(v,5)
"""
v = int(v)
assert (v > 1)
assert (v%20 == 5 or v%20 == 1) # note: equivalent to (v-1)%4 == 0 and (v*(v-1))%20 == 0
# Lemma 27
if v%5 == 0 and (v//5)%4 == 1 and is_prime_power(v//5):
bibd = BIBD_5q_5_for_q_prime_power(v//5)
# Lemma 28
elif v in [21,41,61,81,141,161,281]:
from difference_family import difference_family
G,D = difference_family(v,5)
bibd = BIBD_from_difference_family(G, D, check=False)
# Lemma 29
elif v == 165:
bibd = BIBD_from_PBD(v_5_1_BIBD(41,check=False),165,5,check=False)
elif v == 181:
bibd = BIBD_from_PBD(v_5_1_BIBD(45,check=False),181,5,check=False)
elif v in (201,285,301,401,421,425):
# Call directly the BIBD_from_TD function
# note: there are (201,5,1) and (421,5)-difference families that can be
# obtained from the general constructor
bibd = BIBD_from_TD(v,5)
# Theorem 31.2
elif (v-1)//4 in [80, 81, 85, 86, 90, 91, 95, 96, 110, 111, 115, 116, 120, 121, 250, 251, 255, 256, 260, 261, 265, 266, 270, 271]:
r = (v-1)//4
if r <= 96:
k,t,u = 5, 16, r-80
elif r <= 121:
k,t,u = 10, 11, r-110
else:
k,t,u = 10, 25, r-250
bibd = BIBD_from_PBD(PBD_from_TD(k,t,u),v,5,check=False)
else:
r,s,t,u = _get_r_s_t_u(v)
bibd = BIBD_from_PBD(PBD_from_TD(5,t,u),v,5,check=False)
if check:
assert is_pairwise_balanced_design(bibd,v,[5])
return bibd
def balanced_incomplete_block_design(v, k, existence=False, use_LJCR=False):
r"""
Return a BIBD of parameters `v,k`.
A Balanced Incomplete Block Design of parameters `v,k` is a collection
`\mathcal C` of `k`-subsets of `V=\{0,\dots,v-1\}` such that for any two
distinct elements `x,y\in V` there is a unique element `S\in \mathcal C`
such that `x,y\in S`.
More general definitions sometimes involve a `\lambda` parameter, and we
assume here that `\lambda=1`.
For more information on BIBD, see the
:wikipedia:`corresponding Wikipedia entry <Block_design#Definition_of_a_BIBD_.28or_2-design.29>`.
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.
- ``use_LJCR`` (boolean) -- whether to query the La Jolla Covering
Repository for the design when Sage does not know how to build it (see
:func:`~sage.combinat.designs.covering_design.best_known_covering_design_www`). This
requires internet.
.. SEEALSO::
* :func:`steiner_triple_system`
* :func:`v_4_1_BIBD`
* :func:`v_5_1_BIBD`
TODO:
* Implement other constructions from the Handbook of Combinatorial
Designs.
EXAMPLES::
sage: designs.balanced_incomplete_block_design(7, 3).blocks()
[[0, 1, 3], [0, 2, 4], [0, 5, 6], [1, 2, 6], [1, 4, 5], [2, 3, 5], [3, 4, 6]]
sage: B = designs.balanced_incomplete_block_design(66, 6, use_LJCR=True) # optional - internet
sage: B # optional - internet
Incidence structure with 66 points and 143 blocks
sage: B.blocks() # optional - internet
[[0, 1, 2, 3, 4, 65], [0, 5, 24, 25, 39, 57], [0, 6, 27, 38, 44, 55], ...
sage: designs.balanced_incomplete_block_design(66, 6, use_LJCR=True) # optional - internet
Incidence structure with 66 points and 143 blocks
sage: designs.balanced_incomplete_block_design(216, 6)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build a (216,6,1)-BIBD!
TESTS::
sage: designs.balanced_incomplete_block_design(85,5,existence=True)
True
sage: _ = designs.balanced_incomplete_block_design(85,5)
A BIBD from a Finite Projective Plane::
sage: _ = designs.balanced_incomplete_block_design(21,5)
Some trivial BIBD::
sage: designs.balanced_incomplete_block_design(10,10)
(10,10,1)-Balanced Incomplete Block Design
sage: designs.balanced_incomplete_block_design(1,10)
(1,0,1)-Balanced Incomplete Block Design
Existence of BIBD with `k=3,4,5`::
sage: [v for v in xrange(50) if designs.balanced_incomplete_block_design(v,3,existence=True)]
[1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49]
sage: [v for v in xrange(100) if designs.balanced_incomplete_block_design(v,4,existence=True)]
[1, 4, 13, 16, 25, 28, 37, 40, 49, 52, 61, 64, 73, 76, 85, 88, 97]
sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,5,existence=True)]
[1, 5, 21, 25, 41, 45, 61, 65, 81, 85, 101, 105, 121, 125, 141, 145]
For `k > 5` there are currently very few constructions::
sage: [v for v in xrange(300) if designs.balanced_incomplete_block_design(v,6,existence=True) is True]
[1, 6, 31, 66, 76, 91, 96, 106, 111, 121, 126, 136, 141, 151, 156, 171, 181, 186, 196, 201, 211, 241, 271]
sage: [v for v in xrange(300) if designs.balanced_incomplete_block_design(v,6,existence=True) is Unknown]
[51, 61, 81, 166, 216, 226, 231, 246, 256, 261, 276, 286, 291]
Here are some constructions with `k \geq 7` and `v` a prime power::
sage: designs.balanced_incomplete_block_design(169,7)
(169,7,1)-Balanced Incomplete Block Design
sage: designs.balanced_incomplete_block_design(617,8)
(617,8,1)-Balanced Incomplete Block Design
sage: designs.balanced_incomplete_block_design(433,9)
(433,9,1)-Balanced Incomplete Block Design
sage: designs.balanced_incomplete_block_design(1171,10)
(1171,10,1)-Balanced Incomplete Block Design
And we know some inexistence results::
sage: designs.balanced_incomplete_block_design(21,6,existence=True)
False
"""
lmbd = 1
# Trivial BIBD
if v == 1:
if existence:
return True
return BalancedIncompleteBlockDesign(v, [], check=False)
if k == v:
if existence:
return True
return BalancedIncompleteBlockDesign(v, [range(v)], check=False, copy=False)
# Non-existence of BIBD
if (v < k or
k < 2 or
(v-1) % (k-1) != 0 or
(v*(v-1)) % (k*(k-1)) != 0 or
# From the Handbook of combinatorial designs:
#
# With lambda>1 other exceptions are
# (15,5,2),(21,6,2),(22,7,2),(22,8,4).
(k==6 and v in [36,46]) or
(k==7 and v == 43) or
# Fisher's inequality
(v*(v-1))/(k*(k-1)) < v):
if existence:
return False
raise EmptySetError("There exists no ({},{},{})-BIBD".format(v,k,lmbd))
if k == 2:
if existence:
return True
from itertools import combinations
return BalancedIncompleteBlockDesign(v, combinations(range(v),2), check=False, copy=True)
if k == 3:
if existence:
return v%6 == 1 or v%6 == 3
return steiner_triple_system(v)
if k == 4:
if existence:
return v%12 == 1 or v%12 == 4
return BalancedIncompleteBlockDesign(v, v_4_1_BIBD(v), copy=False)
if k == 5:
if existence:
return v%20 == 1 or v%20 == 5
return BalancedIncompleteBlockDesign(v, v_5_1_BIBD(v), copy=False)
from difference_family import difference_family
from database import BIBD_constructions
if (v,k,1) in BIBD_constructions:
if existence:
return True
return BlockDesign(v,BIBD_constructions[(v,k,1)](), copy=False)
if BIBD_from_arc_in_desarguesian_projective_plane(v,k,existence=True):
if existence:
return True
B = BIBD_from_arc_in_desarguesian_projective_plane(v,k)
return BalancedIncompleteBlockDesign(v, B, copy=False)
if BIBD_from_TD(v,k,existence=True):
if existence:
return True
return BalancedIncompleteBlockDesign(v, BIBD_from_TD(v,k), copy=False)
if v == (k-1)**2+k and is_prime_power(k-1):
if existence:
return True
from block_design import projective_plane
return BalancedIncompleteBlockDesign(v, projective_plane(k-1),copy=False)
if difference_family(v,k,existence=True):
if existence:
return True
G,D = difference_family(v,k)
return BalancedIncompleteBlockDesign(v, BIBD_from_difference_family(G,D,check=False), copy=False)
if use_LJCR:
from covering_design import best_known_covering_design_www
B = best_known_covering_design_www(v,k,2)
# Is it a BIBD or just a good covering ?
expected_n_of_blocks = binomial(v,2)/binomial(k,2)
if B.low_bd() > expected_n_of_blocks:
if existence:
return False
raise EmptySetError("There exists no ({},{},{})-BIBD".format(v,k,lmbd))
B = B.incidence_structure()
if B.num_blocks() == expected_n_of_blocks:
if existence:
return True
else:
return B
if existence:
return Unknown
else:
raise NotImplementedError("I don't know how to build a ({},{},1)-BIBD!".format(v,k))
def v_4_1_BIBD(v, check=True):
r"""
Return a `(v,4,1)`-BIBD.
A `(v,4,1)`-BIBD is an edge-decomposition of the complete graph `K_v` into
copies of `K_4`. For more information, see
:func:`balanced_incomplete_block_design`. It exists if and only if `v\equiv 1,4
\pmod {12}`.
See page 167 of [Stinson2004]_ for the construction details.
.. SEEALSO::
* :func:`balanced_incomplete_block_design`
INPUT:
- ``v`` (integer) -- number of points.
- ``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.
EXAMPLES::
sage: from sage.combinat.designs.bibd import v_4_1_BIBD # long time
sage: for n in range(13,100): # long time
....: if n%12 in [1,4]: # long time
....: _ = v_4_1_BIBD(n, check = True) # long time
TESTS:
Check that the `(25,4)` and `(37,4)`-difference family are available::
sage: assert designs.difference_family(25,4,existence=True)
sage: _ = designs.difference_family(25,4)
sage: assert designs.difference_family(37,4,existence=True)
sage: _ = designs.difference_family(37,4)
Check some larger `(v,4,1)`-BIBD (see :trac:`17557`)::
sage: for v in range(400): # long time
....: if v%12 in [1,4]: # long time
....: _ = designs.balanced_incomplete_block_design(v,4) # long time
"""
k = 4
if v == 0:
return []
if v <= 12 or v%12 not in [1,4]:
raise EmptySetError("A K_4-decomposition of K_v exists iif v=2,4 mod 12, v>12 or v==0")
# Step 1. Base cases.
if v == 13:
# note: this construction can also be obtained from difference_family
from block_design import projective_plane
return projective_plane(3)._blocks
if v == 16:
from block_design import AffineGeometryDesign
from sage.rings.finite_rings.constructor import FiniteField
return AffineGeometryDesign(2,1,FiniteField(4,'x'))._blocks
if v == 25 or v == 37:
from difference_family import difference_family
G,D = difference_family(v,4)
return BIBD_from_difference_family(G,D,check=False)
if v == 28:
return [[0, 1, 23, 26], [0, 2, 10, 11], [0, 3, 16, 18], [0, 4, 15, 20],
[0, 5, 8, 9], [0, 6, 22, 25], [0, 7, 14, 21], [0, 12, 17, 27],
[0, 13, 19, 24], [1, 2, 24, 27], [1, 3, 11, 12], [1, 4, 17, 19],
[1, 5, 14, 16], [1, 6, 9, 10], [1, 7, 20, 25], [1, 8, 15, 22],
[1, 13, 18, 21], [2, 3, 21, 25], [2, 4, 12, 13], [2, 5, 18, 20],
[2, 6, 15, 17], [2, 7, 19, 22], [2, 8, 14, 26], [2, 9, 16, 23],
[3, 4, 22, 26], [3, 5, 7, 13], [3, 6, 14, 19], [3, 8, 20, 23],
[3, 9, 15, 27], [3, 10, 17, 24], [4, 5, 23, 27], [4, 6, 7, 8],
[4, 9, 14, 24], [4, 10, 16, 21], [4, 11, 18, 25], [5, 6, 21, 24],
[5, 10, 15, 25], [5, 11, 17, 22], [5, 12, 19, 26], [6, 11, 16, 26],
[6, 12, 18, 23], [6, 13, 20, 27], [7, 9, 17, 18], [7, 10, 26, 27],
[7, 11, 23, 24], [7, 12, 15, 16], [8, 10, 18, 19], [8, 11, 21, 27],
[8, 12, 24, 25], [8, 13, 16, 17], [9, 11, 19, 20], [9, 12, 21, 22],
[9, 13, 25, 26], [10, 12, 14, 20], [10, 13, 22, 23], [11, 13, 14, 15],
[14, 17, 23, 25], [14, 18, 22, 27], [15, 18, 24, 26], [15, 19, 21, 23],
[16, 19, 25, 27], [16, 20, 22, 24], [17, 20, 21, 26]]
# Step 2 : this is function PBD_4_5_8_9_12
PBD = PBD_4_5_8_9_12((v-1)/(k-1),check=False)
# Step 3 : Theorem 7.20
bibd = BIBD_from_PBD(PBD,v,k,check=False)
if check:
assert is_pairwise_balanced_design(bibd,v,[k])
return bibd
def balanced_incomplete_block_design(v, k, existence=False, use_LJCR=False):
r"""
Return a BIBD of parameters `v,k`.
A Balanced Incomplete Block Design of parameters `v,k` is a collection
`\mathcal C` of `k`-subsets of `V=\{0,\dots,v-1\}` such that for any two
distinct elements `x,y\in V` there is a unique element `S\in \mathcal C`
such that `x,y\in S`.
More general definitions sometimes involve a `\lambda` parameter, and we
assume here that `\lambda=1`.
For more information on BIBD, see the
:wikipedia:`corresponding Wikipedia entry <Block_design#Definition_of_a_BIBD_.28or_2-design.29>`.
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.
- ``use_LJCR`` (boolean) -- whether to query the La Jolla Covering
Repository for the design when Sage does not know how to build it (see
:func:`~sage.combinat.designs.covering_design.best_known_covering_design_www`). This
requires internet.
.. SEEALSO::
* :func:`steiner_triple_system`
* :func:`v_4_1_BIBD`
* :func:`v_5_1_BIBD`
TODO:
* Implement other constructions from the Handbook of Combinatorial
Designs.
EXAMPLES::
sage: designs.balanced_incomplete_block_design(7, 3).blocks()
[[0, 1, 3], [0, 2, 4], [0, 5, 6], [1, 2, 6], [1, 4, 5], [2, 3, 5], [3, 4, 6]]
sage: B = designs.balanced_incomplete_block_design(66, 6, use_LJCR=True) # optional - internet
sage: B # optional - internet
Incidence structure with 66 points and 143 blocks
sage: B.blocks() # optional - internet
[[0, 1, 2, 3, 4, 65], [0, 5, 24, 25, 39, 57], [0, 6, 27, 38, 44, 55], ...
sage: designs.balanced_incomplete_block_design(66, 6, use_LJCR=True) # optional - internet
Incidence structure with 66 points and 143 blocks
sage: designs.balanced_incomplete_block_design(141, 6)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build a (141,6,1)-BIBD!
TESTS::
sage: designs.balanced_incomplete_block_design(85,5,existence=True)
True
sage: _ = designs.balanced_incomplete_block_design(85,5)
A BIBD from a Finite Projective Plane::
sage: _ = designs.balanced_incomplete_block_design(21,5)
Some trivial BIBD::
sage: designs.balanced_incomplete_block_design(10,10)
(10,10,1)-Balanced Incomplete Block Design
sage: designs.balanced_incomplete_block_design(1,10)
(1,0,1)-Balanced Incomplete Block Design
Existence of BIBD with `k=3,4,5`::
sage: [v for v in xrange(50) if designs.balanced_incomplete_block_design(v,3,existence=True)]
[1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49]
sage: [v for v in xrange(100) if designs.balanced_incomplete_block_design(v,4,existence=True)]
[1, 4, 13, 16, 25, 28, 37, 40, 49, 52, 61, 64, 73, 76, 85, 88, 97]
sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,5,existence=True)]
[1, 5, 21, 25, 41, 45, 61, 65, 81, 85, 101, 105, 121, 125, 141, 145]
For `k > 5` there are currently very few constructions::
sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,6,existence=True) is True]
[1, 6, 31, 91, 121]
sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,6,existence=True) is Unknown]
[51, 61, 66, 76, 81, 96, 106, 111, 126, 136, 141]
Here are some constructions with `k \geq 7` and `v` a prime power::
sage: designs.balanced_incomplete_block_design(169,7)
(169,7,1)-Balanced Incomplete Block Design
sage: designs.balanced_incomplete_block_design(617,8)
(617,8,1)-Balanced Incomplete Block Design
sage: designs.balanced_incomplete_block_design(433,9)
(433,9,1)-Balanced Incomplete Block Design
sage: designs.balanced_incomplete_block_design(1171,10)
(1171,10,1)-Balanced Incomplete Block Design
And we know some inexistence results::
sage: designs.balanced_incomplete_block_design(21,6,existence=True)
False
"""
lmbd = 1
# Trivial BIBD
if v == 1:
if existence:
return True
return BalancedIncompleteBlockDesign(v, [], check=False)
if k == v:
if existence:
return True
return BalancedIncompleteBlockDesign(v, [range(v)],
check=False,
copy=False)
# Non-existence of BIBD
if (v < k or k < 2 or (v - 1) % (k - 1) != 0
or (v * (v - 1)) % (k * (k - 1)) != 0 or
# From the Handbook of combinatorial designs:
#
# With lambda>1 other exceptions are
# (15,5,2),(21,6,2),(22,7,2),(22,8,4).
(k == 6 and v in [36, 46]) or (k == 7 and v == 43) or
# Fisher's inequality
(v * (v - 1)) / (k * (k - 1)) < v):
if existence:
return False
raise EmptySetError("There exists no ({},{},{})-BIBD".format(
v, k, lmbd))
if k == 2:
if existence:
return True
from itertools import combinations
return BalancedIncompleteBlockDesign(v,
combinations(range(v), 2),
check=False,
copy=True)
if k == 3:
if existence:
return v % 6 == 1 or v % 6 == 3
return steiner_triple_system(v)
if k == 4:
if existence:
return v % 12 == 1 or v % 12 == 4
return BalancedIncompleteBlockDesign(v, v_4_1_BIBD(v), copy=False)
if k == 5:
if existence:
return v % 20 == 1 or v % 20 == 5
return BalancedIncompleteBlockDesign(v, v_5_1_BIBD(v), copy=False)
from difference_family import difference_family
from database import BIBD_constructions
if (v, k, 1) in BIBD_constructions:
if existence:
return True
return BlockDesign(v, BIBD_constructions[(v, k, 1)](), copy=False)
if BIBD_from_TD(v, k, existence=True):
if existence:
return True
return BalancedIncompleteBlockDesign(v, BIBD_from_TD(v, k), copy=False)
if v == (k - 1)**2 + k and is_prime_power(k - 1):
if existence:
return True
from block_design import projective_plane
return BalancedIncompleteBlockDesign(v,
projective_plane(k - 1),
copy=False)
if difference_family(v, k, existence=True):
if existence:
return True
G, D = difference_family(v, k)
return BalancedIncompleteBlockDesign(v,
BIBD_from_difference_family(
G, D, check=False),
copy=False)
if use_LJCR:
from covering_design import best_known_covering_design_www
B = best_known_covering_design_www(v, k, 2)
# Is it a BIBD or just a good covering ?
expected_n_of_blocks = binomial(v, 2) / binomial(k, 2)
if B.low_bd() > expected_n_of_blocks:
if existence:
return False
raise EmptySetError("There exists no ({},{},{})-BIBD".format(
v, k, lmbd))
B = B.incidence_structure()
if B.num_blocks() == expected_n_of_blocks:
if existence:
return True
else:
return B
if existence:
return Unknown
else:
raise NotImplementedError(
"I don't know how to build a ({},{},1)-BIBD!".format(v, k))
def v_4_1_BIBD(v, check=True):
r"""
Return a `(v,4,1)`-BIBD.
A `(v,4,1)`-BIBD is an edge-decomposition of the complete graph `K_v` into
copies of `K_4`. For more information, see
:func:`balanced_incomplete_block_design`. It exists if and only if `v\equiv 1,4
\pmod {12}`.
See page 167 of [Stinson2004]_ for the construction details.
.. SEEALSO::
* :func:`balanced_incomplete_block_design`
INPUT:
- ``v`` (integer) -- number of points.
- ``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.
EXAMPLES::
sage: from sage.combinat.designs.bibd import v_4_1_BIBD # long time
sage: for n in range(13,100): # long time
....: if n%12 in [1,4]: # long time
....: _ = v_4_1_BIBD(n, check = True) # long time
TESTS:
Check that the `(25,4)` and `(37,4)`-difference family are available::
sage: assert designs.difference_family(25,4,existence=True)
sage: _ = designs.difference_family(25,4)
sage: assert designs.difference_family(37,4,existence=True)
sage: _ = designs.difference_family(37,4)
Check some larger `(v,4,1)`-BIBD (see :trac:`17557`)::
sage: for v in range(400): # long time
....: if v%12 in [1,4]: # long time
....: _ = designs.balanced_incomplete_block_design(v,4) # long time
"""
k = 4
if v == 0:
return []
if v <= 12 or v % 12 not in [1, 4]:
raise EmptySetError(
"A K_4-decomposition of K_v exists iif v=2,4 mod 12, v>12 or v==0")
# Step 1. Base cases.
if v == 13:
# note: this construction can also be obtained from difference_family
from block_design import projective_plane
return projective_plane(3)._blocks
if v == 16:
from block_design import AffineGeometryDesign
from sage.rings.finite_rings.constructor import FiniteField
return AffineGeometryDesign(2, 1, FiniteField(4, 'x'))._blocks
if v == 25 or v == 37:
from difference_family import difference_family
G, D = difference_family(v, 4)
return BIBD_from_difference_family(G, D, check=False)
if v == 28:
return [[0, 1, 23, 26], [0, 2, 10, 11], [0, 3, 16, 18], [0, 4, 15, 20],
[0, 5, 8, 9], [0, 6, 22, 25], [0, 7, 14, 21], [0, 12, 17, 27],
[0, 13, 19, 24],
[1, 2, 24, 27], [1, 3, 11, 12], [1, 4, 17, 19], [1, 5, 14, 16],
[1, 6, 9, 10], [1, 7, 20, 25], [1, 8, 15, 22], [1, 13, 18, 21],
[2, 3, 21, 25], [2, 4, 12, 13], [2, 5, 18, 20], [2, 6, 15, 17],
[2, 7, 19, 22], [2, 8, 14, 26], [2, 9, 16, 23], [3, 4, 22, 26],
[3, 5, 7, 13], [3, 6, 14, 19], [3, 8, 20, 23], [3, 9, 15, 27],
[3, 10, 17, 24], [4, 5, 23, 27], [4, 6, 7, 8], [4, 9, 14, 24],
[4, 10, 16, 21], [4, 11, 18, 25], [5, 6, 21, 24],
[5, 10, 15, 25], [5, 11, 17, 22], [5, 12, 19, 26],
[6, 11, 16, 26], [6, 12, 18, 23], [6, 13, 20, 27],
[7, 9, 17, 18], [7, 10, 26, 27], [7, 11, 23, 24],
[7, 12, 15, 16], [8, 10, 18, 19], [8, 11, 21, 27],
[8, 12, 24, 25], [8, 13, 16, 17], [9, 11, 19, 20],
[9, 12, 21, 22], [9, 13, 25, 26], [10, 12, 14, 20],
[10, 13, 22, 23], [11, 13, 14, 15], [14, 17, 23, 25],
[14, 18, 22, 27], [15, 18, 24, 26], [15, 19, 21, 23],
[16, 19, 25, 27], [16, 20, 22, 24], [17, 20, 21, 26]]
# Step 2 : this is function PBD_4_5_8_9_12
PBD = PBD_4_5_8_9_12((v - 1) / (k - 1), check=False)
# Step 3 : Theorem 7.20
bibd = BIBD_from_PBD(PBD, v, k, check=False)
if check:
assert is_pairwise_balanced_design(bibd, v, [k])
return bibd
def v_5_1_BIBD(v, check=True):
r"""
Return a `(v,5,1)`-BIBD.
This method follows the constuction from [ClaytonSmith]_.
INPUT:
- ``v`` (integer)
.. SEEALSO::
* :func:`balanced_incomplete_block_design`
EXAMPLES::
sage: from sage.combinat.designs.bibd import v_5_1_BIBD
sage: i = 0
sage: while i<200:
....: i += 20
....: _ = v_5_1_BIBD(i+1)
....: _ = v_5_1_BIBD(i+5)
TESTS:
Check that the needed difference families are there::
sage: for v in [21,41,61,81,141,161,281]:
....: assert designs.difference_family(v,5,existence=True)
....: _ = designs.difference_family(v,5)
"""
v = int(v)
assert (v > 1)
assert (v % 20 == 5 or v % 20 == 1
) # note: equivalent to (v-1)%4 == 0 and (v*(v-1))%20 == 0
# Lemma 27
if v % 5 == 0 and (v // 5) % 4 == 1 and is_prime_power(v // 5):
bibd = BIBD_5q_5_for_q_prime_power(v // 5)
# Lemma 28
elif v in [21, 41, 61, 81, 141, 161, 281]:
from difference_family import difference_family
G, D = difference_family(v, 5)
bibd = BIBD_from_difference_family(G, D, check=False)
# Lemma 29
elif v == 165:
bibd = BIBD_from_PBD(v_5_1_BIBD(41, check=False), 165, 5, check=False)
elif v == 181:
bibd = BIBD_from_PBD(v_5_1_BIBD(45, check=False), 181, 5, check=False)
elif v in (201, 285, 301, 401, 421, 425):
# Call directly the BIBD_from_TD function
# note: there are (201,5,1) and (421,5)-difference families that can be
# obtained from the general constructor
bibd = BIBD_from_TD(v, 5)
# Theorem 31.2
elif (v - 1) // 4 in [
80, 81, 85, 86, 90, 91, 95, 96, 110, 111, 115, 116, 120, 121, 250,
251, 255, 256, 260, 261, 265, 266, 270, 271
]:
r = (v - 1) // 4
if r <= 96:
k, t, u = 5, 16, r - 80
elif r <= 121:
k, t, u = 10, 11, r - 110
else:
k, t, u = 10, 25, r - 250
bibd = BIBD_from_PBD(PBD_from_TD(k, t, u), v, 5, check=False)
else:
r, s, t, u = _get_r_s_t_u(v)
bibd = BIBD_from_PBD(PBD_from_TD(5, t, u), v, 5, check=False)
if check:
assert is_pairwise_balanced_design(bibd, v, [5])
return bibd
