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 BalancedIncompleteBlockDesign(v, k, use_LJCR=False):
r"""
Returns 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)
- ``use_LJCR`` (boolean) -- whether to query the La Jolla Covering
Repository for the design when Sage does not know how to build it (see
:meth:`~sage.combinat.designs.covering_design.best_known_covering_design_www`). This
requires internet.
.. SEEALSO::
* :meth:`steiner_triple_system`
* :meth:`v_4_1_BIBD`
TODO:
* Implement `(v,5,1)`-BIBD using `this text <http://www.argilo.net/files/bibd.pdf>`_.
* Implement other constructions from the Handbook of Combinatorial
Designs.
EXAMPLES::
sage: designs.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(21,5, use_LJCR=True) # optional - internet
sage: B # optional - internet
Incidence structure with 21 points and 21 blocks
sage: B.blocks() # optional - internet
[[0, 1, 2, 3, 20], [0, 4, 8, 12, 16], [0, 5, 10, 15, 19],
[0, 6, 11, 13, 17], [0, 7, 9, 14, 18], [1, 4, 11, 14, 19],
[1, 5, 9, 13, 16], [1, 6, 8, 15, 18], [1, 7, 10, 12, 17],
[2, 4, 9, 15, 17], [2, 5, 11, 12, 18], [2, 6, 10, 14, 16],
[2, 7, 8, 13, 19], [3, 4, 10, 13, 18], [3, 5, 8, 14, 17],
[3, 6, 9, 12, 19], [3, 7, 11, 15, 16], [4, 5, 6, 7, 20],
[8, 9, 10, 11, 20], [12, 13, 14, 15, 20], [16, 17, 18, 19, 20]]
sage: designs.BalancedIncompleteBlockDesign(20,5, use_LJCR=True) # optional - internet
Traceback (most recent call last):
...
ValueError: No such design exists !
TESTS:
A BIBD from a Finite Projective Plane::
sage: _ = designs.BalancedIncompleteBlockDesign(21,5)
"""
if ((binomial(v, 2) % binomial(k, 2) != 0) or (v - 1) % (k - 1) != 0):
raise ValueError("No such design exists !")
if k == 2:
from itertools import combinations
return BlockDesign(v, combinations(range(v), 2), test=False)
if k == 3:
return steiner_triple_system(v)
if k == 4:
return BlockDesign(v, v_4_1_BIBD(v), test=False)
if v == (k - 1)**2 + k and is_prime_power(k - 1):
from block_design import ProjectivePlaneDesign
return ProjectivePlaneDesign(k - 1)
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:
raise ValueError("No such design exists !")
B = B.incidence_structure()
if len(B.blcks) == expected_n_of_blocks:
return B
raise ValueError("I don't know how to build this design.")
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 BalancedIncompleteBlockDesign(v,k,existence=False,use_LJCR=False):
r"""
Returns 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, returns:
- ``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
:meth:`~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.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(21,5, use_LJCR=True) # optional - internet
sage: B # optional - internet
Incidence structure with 21 points and 21 blocks
sage: B.blocks() # optional - internet
[[0, 1, 2, 3, 20], [0, 4, 8, 12, 16], [0, 5, 10, 15, 19],
[0, 6, 11, 13, 17], [0, 7, 9, 14, 18], [1, 4, 11, 14, 19],
[1, 5, 9, 13, 16], [1, 6, 8, 15, 18], [1, 7, 10, 12, 17],
[2, 4, 9, 15, 17], [2, 5, 11, 12, 18], [2, 6, 10, 14, 16],
[2, 7, 8, 13, 19], [3, 4, 10, 13, 18], [3, 5, 8, 14, 17],
[3, 6, 9, 12, 19], [3, 7, 11, 15, 16], [4, 5, 6, 7, 20],
[8, 9, 10, 11, 20], [12, 13, 14, 15, 20], [16, 17, 18, 19, 20]]
sage: designs.BalancedIncompleteBlockDesign(20,5, use_LJCR=True) # optional - internet
Traceback (most recent call last):
...
ValueError: No such design exists !
sage: designs.BalancedIncompleteBlockDesign(16,6)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build this design.
TESTS::
sage: designs.BalancedIncompleteBlockDesign(85,5,existence=True)
True
sage: _ = designs.BalancedIncompleteBlockDesign(85,5)
A BIBD from a Finite Projective Plane::
sage: _ = designs.BalancedIncompleteBlockDesign(21,5)
Some trivial BIBD::
sage: designs.BalancedIncompleteBlockDesign(10,10)
Incidence structure with 10 points and 1 blocks
sage: designs.BalancedIncompleteBlockDesign(1,10)
Incidence structure with 1 points and 0 blocks
Existence of BIBD with `k=3,4,5`::
sage: [v for v in xrange(50) if designs.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(v,6,existence=True) is True]
[1, 6, 31]
sage: [v for v in xrange(150) if designs.BalancedIncompleteBlockDesign(v,6,existence=True) is Unknown]
[16, 21, 36, 46, 51, 61, 66, 76, 81, 91, 96, 106, 111, 121, 126, 136, 141]
"""
if v == 1:
if existence:
return True
return BlockDesign(v, [], test=False)
if k == v:
if existence:
return True
return BlockDesign(v, [range(v)], test=False)
if v < k or k < 2 or (v-1) % (k-1) != 0 or (v*(v-1)) % (k*(k-1)) != 0:
if existence:
return False
raise EmptySetError("No such design exists !")
if k == 2:
if existence:
return True
from itertools import combinations
return BlockDesign(v, combinations(range(v),2), test = False)
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 BlockDesign(v, v_4_1_BIBD(v), test = False)
if k == 5:
if existence:
return v%20 == 1 or v%20 == 5
return BlockDesign(v, v_5_1_BIBD(v), test = False)
if BIBD_from_TD(v,k,existence=True):
if existence:
return True
return BlockDesign(v, BIBD_from_TD(v,k))
if v == (k-1)**2+k and is_prime_power(k-1):
if existence:
return True
from block_design import projective_plane
return projective_plane(k-1)
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("No such design exists !")
B = B.incidence_structure()
if len(B.blcks) == 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 this design.")
def BalancedIncompleteBlockDesign(v, k, existence=False, use_LJCR=False):
r"""
Returns 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, returns:
- ``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
:meth:`~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.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(21,5, use_LJCR=True) # optional - internet
sage: B # optional - internet
Incidence structure with 21 points and 21 blocks
sage: B.blocks() # optional - internet
[[0, 1, 2, 3, 20], [0, 4, 8, 12, 16], [0, 5, 10, 15, 19],
[0, 6, 11, 13, 17], [0, 7, 9, 14, 18], [1, 4, 11, 14, 19],
[1, 5, 9, 13, 16], [1, 6, 8, 15, 18], [1, 7, 10, 12, 17],
[2, 4, 9, 15, 17], [2, 5, 11, 12, 18], [2, 6, 10, 14, 16],
[2, 7, 8, 13, 19], [3, 4, 10, 13, 18], [3, 5, 8, 14, 17],
[3, 6, 9, 12, 19], [3, 7, 11, 15, 16], [4, 5, 6, 7, 20],
[8, 9, 10, 11, 20], [12, 13, 14, 15, 20], [16, 17, 18, 19, 20]]
sage: designs.BalancedIncompleteBlockDesign(20,5, use_LJCR=True) # optional - internet
Traceback (most recent call last):
...
ValueError: No such design exists !
sage: designs.BalancedIncompleteBlockDesign(16,6)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build this design.
TESTS::
sage: designs.BalancedIncompleteBlockDesign(85,5,existence=True)
True
sage: _ = designs.BalancedIncompleteBlockDesign(85,5)
A BIBD from a Finite Projective Plane::
sage: _ = designs.BalancedIncompleteBlockDesign(21,5)
Some trivial BIBD::
sage: designs.BalancedIncompleteBlockDesign(10,10)
Incidence structure with 10 points and 1 blocks
sage: designs.BalancedIncompleteBlockDesign(1,10)
Incidence structure with 1 points and 0 blocks
Existence of BIBD with `k=3,4,5`::
sage: [v for v in xrange(50) if designs.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(v,6,existence=True) is True]
[1, 6, 31]
sage: [v for v in xrange(150) if designs.BalancedIncompleteBlockDesign(v,6,existence=True) is Unknown]
[16, 21, 36, 46, 51, 61, 66, 76, 81, 91, 96, 106, 111, 121, 126, 136, 141]
"""
if v == 1:
if existence:
return True
return BlockDesign(v, [], test=False)
if k == v:
if existence:
return True
return BlockDesign(v, [range(v)], test=False)
if v < k or k < 2 or (v - 1) % (k - 1) != 0 or (v *
(v - 1)) % (k *
(k - 1)) != 0:
if existence:
return False
raise EmptySetError("No such design exists !")
if k == 2:
if existence:
return True
from itertools import combinations
return BlockDesign(v, combinations(range(v), 2), test=False)
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 BlockDesign(v, v_4_1_BIBD(v), test=False)
if k == 5:
if existence:
return v % 20 == 1 or v % 20 == 5
return BlockDesign(v, v_5_1_BIBD(v), test=False)
if BIBD_from_TD(v, k, existence=True):
if existence:
return True
return BlockDesign(v, BIBD_from_TD(v, k))
if v == (k - 1)**2 + k and is_prime_power(k - 1):
if existence:
return True
from block_design import projective_plane
return projective_plane(k - 1)
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("No such design exists !")
B = B.incidence_structure()
if len(B.blcks) == 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 this design.")
def BalancedIncompleteBlockDesign(v,k,use_LJCR=False):
r"""
Returns 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)
- ``use_LJCR`` (boolean) -- whether to query the La Jolla Covering
Repository for the design when Sage does not know how to build it (see
:meth:`~sage.combinat.designs.covering_design.best_known_covering_design_www`). This
requires internet.
.. SEEALSO::
* :meth:`steiner_triple_system`
* :meth:`v_4_1_BIBD`
TODO:
* Implement `(v,5,1)`-BIBD using `this text <http://www.argilo.net/files/bibd.pdf>`_.
* Implement other constructions from the Handbook of Combinatorial
Designs.
EXAMPLES::
sage: designs.BalancedIncompleteBlockDesign(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.BalancedIncompleteBlockDesign(21,5, use_LJCR=True) # optional - internet
sage: B # optional - internet
Incidence structure with 21 points and 21 blocks
sage: B.blocks() # optional - internet
[[0, 1, 2, 3, 20], [0, 4, 8, 12, 16], [0, 5, 10, 15, 19],
[0, 6, 11, 13, 17], [0, 7, 9, 14, 18], [1, 4, 11, 14, 19],
[1, 5, 9, 13, 16], [1, 6, 8, 15, 18], [1, 7, 10, 12, 17],
[2, 4, 9, 15, 17], [2, 5, 11, 12, 18], [2, 6, 10, 14, 16],
[2, 7, 8, 13, 19], [3, 4, 10, 13, 18], [3, 5, 8, 14, 17],
[3, 6, 9, 12, 19], [3, 7, 11, 15, 16], [4, 5, 6, 7, 20],
[8, 9, 10, 11, 20], [12, 13, 14, 15, 20], [16, 17, 18, 19, 20]]
sage: designs.BalancedIncompleteBlockDesign(20,5, use_LJCR=True) # optional - internet
Traceback (most recent call last):
...
ValueError: No such design exists !
TESTS:
A BIBD from a Finite Projective Plane::
sage: _ = designs.BalancedIncompleteBlockDesign(21,5)
"""
if ((binomial(v,2)%binomial(k,2) != 0) or
(v-1)%(k-1) != 0):
raise ValueError("No such design exists !")
if k == 2:
from itertools import combinations
return BlockDesign(v, combinations(range(v),2), test = False)
if k == 3:
return steiner_triple_system(v)
if k == 4:
return BlockDesign(v, v_4_1_BIBD(v), test = False)
if v == (k-1)**2+k and is_prime_power(k-1):
from block_design import ProjectivePlaneDesign
return ProjectivePlaneDesign(k-1)
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:
raise ValueError("No such design exists !")
B = B.incidence_structure()
if len(B.blcks) == expected_n_of_blocks:
return B
raise ValueError("I don't know how to build this design.")
