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 orthogonal_array(k,n,t=2,check=True,existence=False):
r"""
Return an orthogonal array of parameters `k,n,t`.
An orthogonal array of parameters `k,n,t` is a matrix with `k` columns
filled with integers from `[n]` in such a way that for any `t` columns, each
of the `n^t` possible rows occurs exactly once. In
particular, the matrix has `n^t` rows.
More general definitions sometimes involve a `\lambda` parameter, and we
assume here that `\lambda=1`.
For more information on orthogonal arrays, see
:wikipedia:`Orthogonal_array`.
INPUT:
- ``k`` -- (integer) number of columns. If ``k=None`` it is set to the
largest value available.
- ``n`` -- (integer) number of symbols
- ``t`` -- (integer; default: 2) -- strength of the array
- ``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.
- ``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.
.. NOTE::
When ``k=None`` and ``existence=True`` the function returns an
integer, i.e. the largest `k` such that we can build a `TD(k,n)`.
OUTPUT:
The kind of output depends on the input:
- if ``existence=False`` (the default) then the output is a list of lists
that represent an orthogonal array with parameters ``k`` and ``n``
- if ``existence=True`` and ``k`` is an integer, then the function returns a
troolean: either ``True``, ``Unknown`` or ``False``
- if ``existence=True`` and ``k=None`` then the output is the largest value
of ``k`` for which Sage knows how to compute a `TD(k,n)`.
.. NOTE::
This method implements theorems from [Stinson2004]_. See the code's
documentation for details.
.. SEEALSO::
When `t=2` an orthogonal array is also a transversal design (see
:func:`transversal_design`) and a family of mutually orthogonal latin
squares (see
:func:`~sage.combinat.designs.latin_squares.mutually_orthogonal_latin_squares`).
EXAMPLES::
sage: designs.orthogonal_array(3,2)
[[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
sage: designs.orthogonal_array(5,5)
[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3],
[0, 3, 1, 4, 2], [0, 4, 3, 2, 1], [1, 0, 4, 3, 2],
[1, 1, 1, 1, 1], [1, 2, 3, 4, 0], [1, 3, 0, 2, 4],
[1, 4, 2, 0, 3], [2, 0, 3, 1, 4], [2, 1, 0, 4, 3],
[2, 2, 2, 2, 2], [2, 3, 4, 0, 1], [2, 4, 1, 3, 0],
[3, 0, 2, 4, 1], [3, 1, 4, 2, 0], [3, 2, 1, 0, 4],
[3, 3, 3, 3, 3], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3],
[4, 1, 3, 0, 2], [4, 2, 0, 3, 1], [4, 3, 2, 1, 0],
[4, 4, 4, 4, 4]]
What is the largest value of `k` for which Sage knows how to compute a
`OA(k,14,2)`?::
sage: designs.orthogonal_array(None,14,existence=True)
6
If you ask for an orthogonal array that does not exist, then the function
either raise an ``EmptySetError`` (if it knows that such an orthogonal array
does not exist) or a ``NotImplementedError``::
sage: designs.orthogonal_array(4,2)
Traceback (most recent call last):
...
EmptySetError: No Orthogonal Array exists when k>=n+t except when n<=1
sage: designs.orthogonal_array(12,20)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build an OA(12,20)!
Note that these errors correspond respectively to the answers ``False`` and
``Unknown`` when the parameter ``existence`` is set to ``True``::
sage: designs.orthogonal_array(4,2,existence=True)
False
sage: designs.orthogonal_array(12,20,existence=True)
Unknown
TESTS:
The special cases `n=0,1`::
sage: designs.orthogonal_array(3,0)
[]
sage: designs.orthogonal_array(3,1)
[[0, 0, 0]]
sage: designs.orthogonal_array(None,0,existence=True)
+Infinity
sage: designs.orthogonal_array(None,1,existence=True)
+Infinity
sage: designs.orthogonal_array(None,1)
Traceback (most recent call last):
...
ValueError: there is no upper bound on k when 0<=n<=1
sage: designs.orthogonal_array(None,0)
Traceback (most recent call last):
...
ValueError: there is no upper bound on k when 0<=n<=1
sage: designs.orthogonal_array(16,0)
[]
sage: designs.orthogonal_array(16,1)
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]
when `t>2` and `k=None`::
sage: t = 3
sage: designs.orthogonal_array(None,5,t=t,existence=True) == t
True
sage: _ = designs.orthogonal_array(t,5,t)
"""
from latin_squares import mutually_orthogonal_latin_squares
from database import OA_constructions, MOLS_constructions
from block_design import projective_plane, projective_plane_to_OA
from orthogonal_arrays_recursive import find_recursive_construction
assert n>=0
# If k is set to None we find the largest value available
if k is None:
if n == 0 or n == 1:
if existence:
from sage.rings.infinity import Infinity
return Infinity
raise ValueError("there is no upper bound on k when 0<=n<=1")
elif t == 2 and projective_plane(n,existence=True):
k = n+1
else:
for k in range(t-1,n+2):
if not orthogonal_array(k+1,n,t=t,existence=True):
break
if existence:
return k
if k < t:
raise ValueError("undefined for k<t")
if existence and _OA_cache_get(k,n) is not None and t == 2:
return _OA_cache_get(k,n)
may_be_available = _OA_cache_construction_available(k,n) is not False
if n <= 1:
if existence:
return True
OA = [[0]*k]*n
elif k >= n+t:
# When t=2 then k<n+t as it is equivalent to the existence of n-1 MOLS.
# When t>2 the submatrix defined by the rows whose first t-2 elements
# are 0s yields a OA with t=2 and k-(t-2) columns. Thus k-(t-2) < n+2,
# i.e. k<n+t.
if existence:
return False
raise EmptySetError("No Orthogonal Array exists when k>=n+t except when n<=1")
elif k <= t:
if existence:
return True
from itertools import product
return map(list, product(range(n), repeat=k))
elif t != 2:
if existence:
return Unknown
raise NotImplementedError("Only trivial orthogonal arrays are implemented for t>=2")
elif k <= 3:
if existence:
return True
return [[i,j,(i+j)%n] for i in xrange(n) for j in xrange(n)]
# projective spaces are equivalent to OA(n+1,n,2)
elif (projective_plane(n, existence=True) or
(k == n+1 and projective_plane(n, existence=True) is False)):
_OA_cache_set(n+1,n,projective_plane(n, existence=True))
if k == n+1:
if existence:
return projective_plane(n, existence=True)
p = projective_plane(n, check=False)
OA = projective_plane_to_OA(p, check=False)
else:
if existence:
return True
p = projective_plane(n, check=False)
OA = [l[:k] for l in projective_plane_to_OA(p, check=False)]
# Constructions from the database
elif may_be_available and n in OA_constructions and k <= OA_constructions[n][0]:
_OA_cache_set(OA_constructions[n][0],n,True)
if existence:
return True
_, construction = OA_constructions[n]
OA = OA_from_wider_OA(construction(),k)
# Constructions from the database II
elif may_be_available and k <= 6 and n == 12:
_OA_cache_set(6,12,True)
if existence:
return True
else:
from database import TD_6_12
TD = TD_6_12()
OA = [[x%n for x in R] for R in TD]
# Constructions from the database III
# Section 6.5.1 from [Stinson2004]
elif may_be_available and n in MOLS_constructions and k-2 <= MOLS_constructions[n][0]:
_OA_cache_set(MOLS_constructions[n][0]+2,n,True)
if existence:
return True
else:
construction = MOLS_constructions[n][1]
mols = construction()
OA = [[i,j]+[m[i,j] for m in mols]
for i in range(n) for j in range(n)]
OA = OA_from_wider_OA(OA,k)
elif may_be_available and find_recursive_construction(k,n):
_OA_cache_set(k,n,True)
if existence:
return True
f,args = find_recursive_construction(k,n)
OA = f(*args)
else:
_OA_cache_set(k,n,Unknown)
if existence:
return Unknown
raise NotImplementedError("I don't know how to build an OA({},{})!".format(k,n))
if check:
assert is_orthogonal_array(OA,k,n,t)
return OA
def orthogonal_array(k,
n,
t=2,
check=True,
existence=False,
who_asked=tuple()):
r"""
Return an orthogonal array of parameters `k,n,t`.
An orthogonal array of parameters `k,n,t` is a matrix with `k` columns
filled with integers from `[n]` in such a way that for any `t` columns, each
of the `n^t` possible rows occurs exactly once. In
particular, the matrix has `n^t` rows.
More general definitions sometimes involve a `\lambda` parameter, and we
assume here that `\lambda=1`.
For more information on orthogonal arrays, see
:wikipedia:`Orthogonal_array`.
INPUT:
- ``k`` -- (integer) number of columns. If ``k=None`` it is set to the
largest value available.
- ``n`` -- (integer) number of symbols
- ``t`` -- (integer; default: 2) -- strength of the array
- ``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.
- ``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.
.. NOTE::
When ``k=None`` and ``existence=True`` the function returns an
integer, i.e. the largest `k` such that we can build a `TD(k,n)`.
- ``who_asked`` (internal use only) -- because of the equivalence between
OA/TD/MOLS, each of the three constructors calls the others. We must keep
track of who calls who in order to avoid infinite loops. ``who_asked`` is
the tuple of the other functions that were called before this one on the
same input `k,n`.
OUTPUT:
The kind of output depends on the input:
- if ``existence=False`` (the default) then the output is a list of lists
that represent an orthogonal array with parameters ``k`` and ``n``
- if ``existence=True`` and ``k`` is an integer, then the function returns a
troolean: either ``True``, ``Unknown`` or ``False``
- if ``existence=True`` and ``k=None`` then the output is the largest value
of ``k`` for which Sage knows how to compute a `TD(k,n)`.
.. NOTE::
This method implements theorems from [Stinson2004]_. See the code's
documentation for details.
.. SEEALSO::
When `t=2` an orthogonal array is also a transversal design (see
:func:`transversal_design`) and a family of mutually orthogonal latin
squares (see
:func:`~sage.combinat.designs.latin_squares.mutually_orthogonal_latin_squares`).
EXAMPLES::
sage: designs.orthogonal_array(3,2)
[[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
sage: designs.orthogonal_array(5,5)
[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3],
[0, 3, 1, 4, 2], [0, 4, 3, 2, 1], [1, 0, 4, 3, 2],
[1, 1, 1, 1, 1], [1, 2, 3, 4, 0], [1, 3, 0, 2, 4],
[1, 4, 2, 0, 3], [2, 0, 3, 1, 4], [2, 1, 0, 4, 3],
[2, 2, 2, 2, 2], [2, 3, 4, 0, 1], [2, 4, 1, 3, 0],
[3, 0, 2, 4, 1], [3, 1, 4, 2, 0], [3, 2, 1, 0, 4],
[3, 3, 3, 3, 3], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3],
[4, 1, 3, 0, 2], [4, 2, 0, 3, 1], [4, 3, 2, 1, 0],
[4, 4, 4, 4, 4]]
What is the largest value of `k` for which Sage knows how to compute a
`OA(k,14,2)`?::
sage: designs.orthogonal_array(None,14,existence=True)
6
If you ask for an orthogonal array that does not exist, then the function
either raise an `EmptySetError` (if it knows that such an orthogonal array
does not exist) or a `NotImplementedError`::
sage: designs.orthogonal_array(4,2)
Traceback (most recent call last):
...
EmptySetError: No Orthogonal Array exists when k>=n+t except when n=1
sage: designs.orthogonal_array(12,20)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build this orthogonal array!
Note that these errors correspond respectively to the answers ``False`` and
``Unknown`` when the parameter ``existence`` is set to ``True``::
sage: designs.orthogonal_array(4,2,existence=True)
False
sage: designs.orthogonal_array(12,20,existence=True)
Unknown
TESTS:
The special case `n=1`::
sage: designs.orthogonal_array(3,1)
[[0, 0, 0]]
sage: designs.orthogonal_array(None,1,existence=True)
+Infinity
sage: designs.orthogonal_array(None,1)
Traceback (most recent call last):
...
ValueError: there are no bound on k when n=1.
sage: designs.orthogonal_array(None,14,existence=True)
6
sage: designs.orthogonal_array(16,1)
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]
"""
from sage.rings.finite_rings.constructor import FiniteField
from latin_squares import mutually_orthogonal_latin_squares
from database import OA_constructions
from block_design import projective_plane, projective_plane_to_OA
# If k is set to None we find the largest value available
if k is None:
if n == 1:
if existence:
from sage.rings.infinity import Infinity
return Infinity
raise ValueError("there are no bound on k when n=1.")
for k in range(1, n + 2):
if not orthogonal_array(k + 1, n, existence=True):
break
if existence:
return k
if k < 2:
raise ValueError("undefined for k less than 2")
if n == 1:
OA = [[0] * k]
elif k >= n + t:
# When t=2 then k<n+t as it is equivalent to the existence of n-1 MOLS.
# When t>2 the submatrix defined by the rows whose first t-2 elements
# are 0s yields a OA with t=2 and k-(t-2) columns. Thus k-(t-2) < n+2,
# i.e. k<n+t.
if existence:
return False
raise EmptySetError(
"No Orthogonal Array exists when k>=n+t except when n=1")
elif k == t:
if existence:
return True
from itertools import product
OA = map(list, product(range(n), repeat=k))
elif n in OA_constructions and k <= OA_constructions[n][0]:
if existence:
return True
_, construction = OA_constructions[n]
OA = OA_from_wider_OA(construction(), k)
# projective spaces are equivalent to OA(n+1,n,2)
elif (t == 2
and (projective_plane(n, existence=True) or
(k == n + 1 and projective_plane(n, existence=True) is False))):
if k == n + 1:
if existence:
return projective_plane(n, existence=True)
p = projective_plane(n, check=False)
OA = projective_plane_to_OA(p)
else:
if existence:
return True
p = projective_plane(n, check=False)
OA = [l[:k] for l in projective_plane_to_OA(p)]
# Constructions from the database
elif n in OA_constructions and k <= OA_constructions[n][0]:
if existence:
return True
_, construction = OA_constructions[n]
OA = OA_from_wider_OA(construction(), k)
elif (t == 2 and transversal_design not in who_asked
and transversal_design(
k, n, existence=True, who_asked=who_asked +
(orthogonal_array, )) is not Unknown):
# forward existence
if transversal_design(k,
n,
existence=True,
who_asked=who_asked + (orthogonal_array, )):
if existence:
return True
else:
TD = transversal_design(k,
n,
check=False,
who_asked=who_asked +
(orthogonal_array, ))
OA = [[x % n for x in R] for R in TD]
# forward non-existence
else:
if existence:
return False
raise EmptySetError("There exists no OA" + str((k, n)) + "!")
# Section 6.5.1 from [Stinson2004]
elif (t == 2 and mutually_orthogonal_latin_squares not in who_asked
and mutually_orthogonal_latin_squares(
n,
k - 2,
existence=True,
who_asked=who_asked + (orthogonal_array, )) is not Unknown):
# forward existence
if mutually_orthogonal_latin_squares(n,
k - 2,
existence=True,
who_asked=who_asked +
(orthogonal_array, )):
if existence:
return True
else:
mols = mutually_orthogonal_latin_squares(n,
k - 2,
who_asked=who_asked +
(orthogonal_array, ))
OA = [[i, j] + [m[i, j] for m in mols] for i in range(n)
for j in range(n)]
# forward non-existence
else:
if existence:
return False
raise EmptySetError("There exists no OA" + str((k, n)) + "!")
else:
if existence:
return Unknown
raise NotImplementedError(
"I don't know how to build this orthogonal array!")
if check:
assert is_orthogonal_array(OA, k, n, t)
return OA
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 orthogonal_array(k, n, t=2, check=True, existence=False):
r"""
Return an orthogonal array of parameters `k,n,t`.
An orthogonal array of parameters `k,n,t` is a matrix with `k` columns
filled with integers from `[n]` in such a way that for any `t` columns, each
of the `n^t` possible rows occurs exactly once. In
particular, the matrix has `n^t` rows.
More general definitions sometimes involve a `\lambda` parameter, and we
assume here that `\lambda=1`.
For more information on orthogonal arrays, see
:wikipedia:`Orthogonal_array`.
INPUT:
- ``k`` -- (integer) number of columns. If ``k=None`` it is set to the
largest value available.
- ``n`` -- (integer) number of symbols
- ``t`` -- (integer; default: 2) -- strength of the array
- ``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.
- ``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.
.. NOTE::
When ``k=None`` and ``existence=True`` the function returns an
integer, i.e. the largest `k` such that we can build a `TD(k,n)`.
OUTPUT:
The kind of output depends on the input:
- if ``existence=False`` (the default) then the output is a list of lists
that represent an orthogonal array with parameters ``k`` and ``n``
- if ``existence=True`` and ``k`` is an integer, then the function returns a
troolean: either ``True``, ``Unknown`` or ``False``
- if ``existence=True`` and ``k=None`` then the output is the largest value
of ``k`` for which Sage knows how to compute a `TD(k,n)`.
.. NOTE::
This method implements theorems from [Stinson2004]_. See the code's
documentation for details.
.. SEEALSO::
When `t=2` an orthogonal array is also a transversal design (see
:func:`transversal_design`) and a family of mutually orthogonal latin
squares (see
:func:`~sage.combinat.designs.latin_squares.mutually_orthogonal_latin_squares`).
EXAMPLES::
sage: designs.orthogonal_array(3,2)
[[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
sage: designs.orthogonal_array(5,5)
[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3],
[0, 3, 1, 4, 2], [0, 4, 3, 2, 1], [1, 0, 4, 3, 2],
[1, 1, 1, 1, 1], [1, 2, 3, 4, 0], [1, 3, 0, 2, 4],
[1, 4, 2, 0, 3], [2, 0, 3, 1, 4], [2, 1, 0, 4, 3],
[2, 2, 2, 2, 2], [2, 3, 4, 0, 1], [2, 4, 1, 3, 0],
[3, 0, 2, 4, 1], [3, 1, 4, 2, 0], [3, 2, 1, 0, 4],
[3, 3, 3, 3, 3], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3],
[4, 1, 3, 0, 2], [4, 2, 0, 3, 1], [4, 3, 2, 1, 0],
[4, 4, 4, 4, 4]]
What is the largest value of `k` for which Sage knows how to compute a
`OA(k,14,2)`?::
sage: designs.orthogonal_array(None,14,existence=True)
6
If you ask for an orthogonal array that does not exist, then the function
either raise an ``EmptySetError`` (if it knows that such an orthogonal array
does not exist) or a ``NotImplementedError``::
sage: designs.orthogonal_array(4,2)
Traceback (most recent call last):
...
EmptySetError: No Orthogonal Array exists when k>=n+t except when n<=1
sage: designs.orthogonal_array(12,20)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build an OA(12,20)!
Note that these errors correspond respectively to the answers ``False`` and
``Unknown`` when the parameter ``existence`` is set to ``True``::
sage: designs.orthogonal_array(4,2,existence=True)
False
sage: designs.orthogonal_array(12,20,existence=True)
Unknown
TESTS:
The special cases `n=0,1`::
sage: designs.orthogonal_array(3,0)
[]
sage: designs.orthogonal_array(3,1)
[[0, 0, 0]]
sage: designs.orthogonal_array(None,0,existence=True)
+Infinity
sage: designs.orthogonal_array(None,1,existence=True)
+Infinity
sage: designs.orthogonal_array(None,1)
Traceback (most recent call last):
...
ValueError: there is no upper bound on k when 0<=n<=1
sage: designs.orthogonal_array(None,0)
Traceback (most recent call last):
...
ValueError: there is no upper bound on k when 0<=n<=1
sage: designs.orthogonal_array(16,0)
[]
sage: designs.orthogonal_array(16,1)
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]
when `t>2` and `k=None`::
sage: t = 3
sage: designs.orthogonal_array(None,5,t=t,existence=True) == t
True
sage: _ = designs.orthogonal_array(t,5,t)
"""
from latin_squares import mutually_orthogonal_latin_squares
from database import OA_constructions, MOLS_constructions
from block_design import projective_plane, projective_plane_to_OA
from orthogonal_arrays_recursive import find_recursive_construction
assert n >= 0
# If k is set to None we find the largest value available
if k is None:
if n == 0 or n == 1:
if existence:
from sage.rings.infinity import Infinity
return Infinity
raise ValueError("there is no upper bound on k when 0<=n<=1")
elif t == 2 and projective_plane(n, existence=True):
k = n + 1
else:
for k in range(t - 1, n + 2):
if not orthogonal_array(k + 1, n, t=t, existence=True):
break
if existence:
return k
if k < t:
raise ValueError("undefined for k<t")
if existence and _OA_cache_get(k, n) is not None and t == 2:
return _OA_cache_get(k, n)
may_be_available = _OA_cache_construction_available(k, n) is not False
if n <= 1:
if existence:
return True
OA = [[0] * k] * n
elif k >= n + t:
# When t=2 then k<n+t as it is equivalent to the existence of n-1 MOLS.
# When t>2 the submatrix defined by the rows whose first t-2 elements
# are 0s yields a OA with t=2 and k-(t-2) columns. Thus k-(t-2) < n+2,
# i.e. k<n+t.
if existence:
return False
raise EmptySetError(
"No Orthogonal Array exists when k>=n+t except when n<=1")
elif k <= t:
if existence:
return True
from itertools import product
return map(list, product(range(n), repeat=k))
elif t != 2:
if existence:
return Unknown
raise NotImplementedError(
"Only trivial orthogonal arrays are implemented for t>=2")
elif k <= 3:
if existence:
return True
return [[i, j, (i + j) % n] for i in xrange(n) for j in xrange(n)]
# projective spaces are equivalent to OA(n+1,n,2)
elif (projective_plane(n, existence=True)
or (k == n + 1 and projective_plane(n, existence=True) is False)):
_OA_cache_set(n + 1, n, projective_plane(n, existence=True))
if k == n + 1:
if existence:
return projective_plane(n, existence=True)
p = projective_plane(n, check=False)
OA = projective_plane_to_OA(p, check=False)
else:
if existence:
return True
p = projective_plane(n, check=False)
OA = [l[:k] for l in projective_plane_to_OA(p, check=False)]
# Constructions from the database
elif may_be_available and n in OA_constructions and k <= OA_constructions[
n][0]:
_OA_cache_set(OA_constructions[n][0], n, True)
if existence:
return True
_, construction = OA_constructions[n]
OA = OA_from_wider_OA(construction(), k)
# Constructions from the database II
elif may_be_available and k <= 6 and n == 12:
_OA_cache_set(6, 12, True)
if existence:
return True
else:
from database import TD_6_12
TD = TD_6_12()
OA = [[x % n for x in R] for R in TD]
# Constructions from the database III
# Section 6.5.1 from [Stinson2004]
elif may_be_available and n in MOLS_constructions and k - 2 <= MOLS_constructions[
n][0]:
_OA_cache_set(MOLS_constructions[n][0] + 2, n, True)
if existence:
return True
else:
construction = MOLS_constructions[n][1]
mols = construction()
OA = [[i, j] + [m[i, j] for m in mols] for i in range(n)
for j in range(n)]
OA = OA_from_wider_OA(OA, k)
elif may_be_available and find_recursive_construction(k, n):
_OA_cache_set(k, n, True)
if existence:
return True
f, args = find_recursive_construction(k, n)
OA = f(*args)
else:
_OA_cache_set(k, n, Unknown)
if existence:
return Unknown
raise NotImplementedError(
"I don't know how to build an OA({},{})!".format(k, n))
if check:
assert is_orthogonal_array(OA, k, n, t)
return OA
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 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_4_1_BIBD(v, check=True):
r"""
Returns 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
:meth:`BalancedIncompleteBlockDesign`. It exists if and only if `v\equiv 1,4
\pmod {12}`.
See page 167 of [Stinson2004]_ for the construction details.
.. SEEALSO::
* :meth:`BalancedIncompleteBlockDesign`
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
"""
from sage.rings.finite_rings.constructor import FiniteField
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:
from block_design import projective_plane
return projective_plane(3).blocks()
if v == 16:
from block_design import AffineGeometryDesign
return AffineGeometryDesign(2,1,FiniteField(4,'x')).blocks()
if v == 25:
return [[0, 1, 17, 22], [0, 2, 11, 21], [0, 3, 15, 18], [0, 4, 7, 13],
[0, 5, 12, 14], [0, 6, 19, 23], [0, 8, 16, 24], [0, 9, 10, 20],
[1, 2, 3, 4], [1, 5, 6, 7], [1, 8, 12, 15], [1, 9, 13, 16],
[1, 10, 11, 14], [1, 18, 20, 23], [1, 19, 21, 24], [2, 5, 15, 24],
[2, 6, 9, 17], [2, 7, 14, 18], [2, 8, 22, 23], [2, 10, 12, 13],
[2, 16, 19, 20], [3, 5, 16, 22], [3, 6, 11, 20], [3, 7, 12, 19],
[3, 8, 9, 14], [3, 10, 17, 24], [3, 13, 21, 23], [4, 5, 10, 23],
[4, 6, 8, 21], [4, 9, 18, 24], [4, 11, 15, 16], [4, 12, 17, 20],
[4, 14, 19, 22], [5, 8, 13, 20], [5, 9, 11, 19], [5, 17, 18, 21],
[6, 10, 15, 22], [6, 12, 16, 18], [6, 13, 14, 24], [7, 8, 11, 17],
[7, 9, 15, 23], [7, 10, 16, 21], [7, 20, 22, 24], [8, 10, 18, 19],
[9, 12, 21, 22], [11, 12, 23, 24], [11, 13, 18, 22], [13, 15, 17, 19],
[14, 15, 20, 21], [14, 16, 17, 23]]
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]]
if v == 37:
return [[0, 1, 3, 24], [0, 2, 23, 36], [0, 4, 26, 32], [0, 5, 9, 31],
[0, 6, 11, 15], [0, 7, 17, 25], [0, 8, 20, 27], [0, 10, 18, 30],
[0, 12, 19, 29], [0, 13, 14, 16], [0, 21, 34, 35], [0, 22, 28, 33],
[1, 2, 4, 25], [1, 5, 27, 33], [1, 6, 10, 32], [1, 7, 12, 16],
[1, 8, 18, 26], [1, 9, 21, 28], [1, 11, 19, 31], [1, 13, 20, 30],
[1, 14, 15, 17], [1, 22, 35, 36], [1, 23, 29, 34], [2, 3, 5, 26],
[2, 6, 28, 34], [2, 7, 11, 33], [2, 8, 13, 17], [2, 9, 19, 27],
[2, 10, 22, 29], [2, 12, 20, 32], [2, 14, 21, 31], [2, 15, 16, 18],
[2, 24, 30, 35], [3, 4, 6, 27], [3, 7, 29, 35], [3, 8, 12, 34],
[3, 9, 14, 18], [3, 10, 20, 28], [3, 11, 23, 30], [3, 13, 21, 33],
[3, 15, 22, 32], [3, 16, 17, 19], [3, 25, 31, 36], [4, 5, 7, 28],
[4, 8, 30, 36], [4, 9, 13, 35], [4, 10, 15, 19], [4, 11, 21, 29],
[4, 12, 24, 31], [4, 14, 22, 34], [4, 16, 23, 33], [4, 17, 18, 20],
[5, 6, 8, 29], [5, 10, 14, 36], [5, 11, 16, 20], [5, 12, 22, 30],
[5, 13, 25, 32], [5, 15, 23, 35], [5, 17, 24, 34], [5, 18, 19, 21],
[6, 7, 9, 30], [6, 12, 17, 21], [6, 13, 23, 31], [6, 14, 26, 33],
[6, 16, 24, 36], [6, 18, 25, 35], [6, 19, 20, 22], [7, 8, 10, 31],
[7, 13, 18, 22], [7, 14, 24, 32], [7, 15, 27, 34], [7, 19, 26, 36],
[7, 20, 21, 23], [8, 9, 11, 32], [8, 14, 19, 23], [8, 15, 25, 33],
[8, 16, 28, 35], [8, 21, 22, 24], [9, 10, 12, 33], [9, 15, 20, 24],
[9, 16, 26, 34], [9, 17, 29, 36], [9, 22, 23, 25], [10, 11, 13, 34],
[10, 16, 21, 25], [10, 17, 27, 35], [10, 23, 24, 26], [11, 12, 14, 35],
[11, 17, 22, 26], [11, 18, 28, 36], [11, 24, 25, 27], [12, 13, 15, 36],
[12, 18, 23, 27], [12, 25, 26, 28], [13, 19, 24, 28], [13, 26, 27, 29],
[14, 20, 25, 29], [14, 27, 28, 30], [15, 21, 26, 30], [15, 28, 29, 31],
[16, 22, 27, 31], [16, 29, 30, 32], [17, 23, 28, 32], [17, 30, 31, 33],
[18, 24, 29, 33], [18, 31, 32, 34], [19, 25, 30, 34], [19, 32, 33, 35],
[20, 26, 31, 35], [20, 33, 34, 36], [21, 27, 32, 36]]
# 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:
_check_pbd(bibd,v,[k])
return bibd
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 v_4_1_BIBD(v, check=True):
r"""
Returns 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
:meth:`BalancedIncompleteBlockDesign`. It exists if and only if `v\equiv 1,4
\pmod {12}`.
See page 167 of [Stinson2004]_ for the construction details.
.. SEEALSO::
* :meth:`BalancedIncompleteBlockDesign`
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
"""
from sage.rings.finite_rings.constructor import FiniteField
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:
from block_design import projective_plane
return projective_plane(3).blocks()
if v == 16:
from block_design import AffineGeometryDesign
return AffineGeometryDesign(2, 1, FiniteField(4, 'x')).blocks()
if v == 25:
return [[0, 1, 17, 22], [0, 2, 11, 21], [0, 3, 15, 18], [0, 4, 7, 13],
[0, 5, 12, 14], [0, 6, 19, 23], [0, 8, 16, 24], [0, 9, 10, 20],
[1, 2, 3, 4], [1, 5, 6, 7], [1, 8, 12, 15], [1, 9, 13, 16],
[1, 10, 11, 14], [1, 18, 20, 23], [1, 19, 21, 24],
[2, 5, 15, 24], [2, 6, 9, 17], [2, 7, 14, 18], [2, 8, 22, 23],
[2, 10, 12, 13], [2, 16, 19, 20], [3, 5, 16, 22],
[3, 6, 11, 20], [3, 7, 12, 19], [3, 8, 9, 14], [3, 10, 17, 24],
[3, 13, 21, 23], [4, 5, 10, 23], [4, 6, 8, 21], [4, 9, 18, 24],
[4, 11, 15, 16], [4, 12, 17, 20], [4, 14, 19, 22],
[5, 8, 13, 20], [5, 9, 11, 19], [5, 17, 18,
21], [6, 10, 15, 22],
[6, 12, 16, 18], [6, 13, 14, 24], [7, 8, 11,
17], [7, 9, 15, 23],
[7, 10, 16, 21], [7, 20, 22, 24], [8, 10, 18, 19],
[9, 12, 21, 22], [11, 12, 23, 24], [11, 13, 18, 22],
[13, 15, 17, 19], [14, 15, 20, 21], [14, 16, 17, 23]]
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]]
if v == 37:
return [[0, 1, 3, 24], [0, 2, 23, 36], [0, 4, 26, 32], [0, 5, 9, 31],
[0, 6, 11, 15], [0, 7, 17, 25], [0, 8, 20,
27], [0, 10, 18, 30],
[0, 12, 19, 29], [0, 13, 14, 16], [0, 21, 34, 35],
[0, 22, 28, 33], [1, 2, 4, 25], [1, 5, 27, 33], [1, 6, 10, 32],
[1, 7, 12, 16], [1, 8, 18, 26], [1, 9, 21,
28], [1, 11, 19, 31],
[1, 13, 20, 30], [1, 14, 15, 17], [1, 22, 35, 36],
[1, 23, 29, 34], [2, 3, 5, 26], [2, 6, 28, 34], [2, 7, 11, 33],
[2, 8, 13, 17], [2, 9, 19, 27], [2, 10, 22, 29],
[2, 12, 20, 32], [2, 14, 21, 31], [2, 15, 16, 18],
[2, 24, 30, 35], [3, 4, 6, 27], [3, 7, 29, 35], [3, 8, 12, 34],
[3, 9, 14, 18], [3, 10, 20, 28], [3, 11, 23, 30],
[3, 13, 21, 33], [3, 15, 22, 32], [3, 16, 17, 19],
[3, 25, 31, 36], [4, 5, 7, 28], [4, 8, 30, 36], [4, 9, 13, 35],
[4, 10, 15, 19], [4, 11, 21, 29], [4, 12, 24, 31],
[4, 14, 22, 34], [4, 16, 23, 33], [4, 17, 18,
20], [5, 6, 8, 29],
[5, 10, 14, 36], [5, 11, 16, 20], [5, 12, 22, 30],
[5, 13, 25, 32], [5, 15, 23, 35], [5, 17, 24, 34],
[5, 18, 19, 21], [6, 7, 9, 30], [6, 12, 17,
21], [6, 13, 23, 31],
[6, 14, 26, 33], [6, 16, 24, 36], [6, 18, 25, 35],
[6, 19, 20, 22], [7, 8, 10, 31], [7, 13, 18, 22],
[7, 14, 24, 32], [7, 15, 27, 34], [7, 19, 26, 36],
[7, 20, 21, 23], [8, 9, 11, 32], [8, 14, 19, 23],
[8, 15, 25, 33], [8, 16, 28, 35], [8, 21, 22, 24],
[9, 10, 12, 33], [9, 15, 20, 24], [9, 16, 26, 34],
[9, 17, 29, 36], [9, 22, 23, 25], [10, 11, 13, 34],
[10, 16, 21, 25], [10, 17, 27, 35], [10, 23, 24, 26],
[11, 12, 14, 35], [11, 17, 22, 26], [11, 18, 28, 36],
[11, 24, 25, 27], [12, 13, 15, 36], [12, 18, 23, 27],
[12, 25, 26, 28], [13, 19, 24, 28], [13, 26, 27, 29],
[14, 20, 25, 29], [14, 27, 28, 30], [15, 21, 26, 30],
[15, 28, 29, 31], [16, 22, 27, 31], [16, 29, 30, 32],
[17, 23, 28, 32], [17, 30, 31, 33], [18, 24, 29, 33],
[18, 31, 32, 34], [19, 25, 30, 34], [19, 32, 33, 35],
[20, 26, 31, 35], [20, 33, 34, 36], [21, 27, 32, 36]]
# 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:
_check_pbd(bibd, v, [k])
return bibd
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::
* :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::
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)
A trivial BIBD::
sage: designs.BalancedIncompleteBlockDesign(10,10)
Incidence structure with 10 points and 1 blocks
"""
if ((binomial(v, 2) % binomial(k, 2) != 0) or (v - 1) % (k - 1) != 0):
if existence:
return False
raise ValueError("No such design exists !")
if k == v:
if existence:
return True
return BlockDesign(v, [range(v)], test=False)
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 bool((n % 6) in [1, 3])
return steiner_triple_system(v)
if k == 4:
if existence:
return bool((n % 12) in [1, 4])
return BlockDesign(v, v_4_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 ValueError("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:
from sage.misc.unknown import Unknown
return Unknown
else:
raise ValueError("I don't know how to build this design.")
def orthogonal_array(k,n,t=2,check=True,existence=False,who_asked=tuple()):
r"""
Return an orthogonal array of parameters `k,n,t`.
An orthogonal array of parameters `k,n,t` is a matrix with `k` columns
filled with integers from `[n]` in such a way that for any `t` columns, each
of the `n^t` possible rows occurs exactly once. In
particular, the matrix has `n^t` rows.
More general definitions sometimes involve a `\lambda` parameter, and we
assume here that `\lambda=1`.
For more information on orthogonal arrays, see
:wikipedia:`Orthogonal_array`.
INPUT:
- ``k`` -- (integer) number of columns. If ``k=None`` it is set to the
largest value available.
- ``n`` -- (integer) number of symbols
- ``t`` -- (integer; default: 2) -- strength of the array
- ``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.
- ``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.
.. NOTE::
When ``k=None`` and ``existence=True`` the function returns an
integer, i.e. the largest `k` such that we can build a `TD(k,n)`.
- ``who_asked`` (internal use only) -- because of the equivalence between
OA/TD/MOLS, each of the three constructors calls the others. We must keep
track of who calls who in order to avoid infinite loops. ``who_asked`` is
the tuple of the other functions that were called before this one on the
same input `k,n`.
OUTPUT:
The kind of output depends on the input:
- if ``existence=False`` (the default) then the output is a list of lists
that represent an orthogonal array with parameters ``k`` and ``n``
- if ``existence=True`` and ``k`` is an integer, then the function returns a
troolean: either ``True``, ``Unknown`` or ``False``
- if ``existence=True`` and ``k=None`` then the output is the largest value
of ``k`` for which Sage knows how to compute a `TD(k,n)`.
.. NOTE::
This method implements theorems from [Stinson2004]_. See the code's
documentation for details.
.. SEEALSO::
When `t=2` an orthogonal array is also a transversal design (see
:func:`transversal_design`) and a family of mutually orthogonal latin
squares (see
:func:`~sage.combinat.designs.latin_squares.mutually_orthogonal_latin_squares`).
EXAMPLES::
sage: designs.orthogonal_array(3,2)
[[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
sage: designs.orthogonal_array(5,5)
[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3],
[0, 3, 1, 4, 2], [0, 4, 3, 2, 1], [1, 0, 4, 3, 2],
[1, 1, 1, 1, 1], [1, 2, 3, 4, 0], [1, 3, 0, 2, 4],
[1, 4, 2, 0, 3], [2, 0, 3, 1, 4], [2, 1, 0, 4, 3],
[2, 2, 2, 2, 2], [2, 3, 4, 0, 1], [2, 4, 1, 3, 0],
[3, 0, 2, 4, 1], [3, 1, 4, 2, 0], [3, 2, 1, 0, 4],
[3, 3, 3, 3, 3], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3],
[4, 1, 3, 0, 2], [4, 2, 0, 3, 1], [4, 3, 2, 1, 0],
[4, 4, 4, 4, 4]]
What is the largest value of `k` for which Sage knows how to compute a
`OA(k,14,2)`?::
sage: designs.orthogonal_array(None,14,existence=True)
6
If you ask for an orthogonal array that does not exist, then the function
either raise an `EmptySetError` (if it knows that such an orthogonal array
does not exist) or a `NotImplementedError`::
sage: designs.orthogonal_array(4,2)
Traceback (most recent call last):
...
EmptySetError: No Orthogonal Array exists when k>=n+t except when n=1
sage: designs.orthogonal_array(12,20)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build this orthogonal array!
Note that these errors correspond respectively to the answers ``False`` and
``Unknown`` when the parameter ``existence`` is set to ``True``::
sage: designs.orthogonal_array(4,2,existence=True)
False
sage: designs.orthogonal_array(12,20,existence=True)
Unknown
TESTS:
The special case `n=1`::
sage: designs.orthogonal_array(3,1)
[[0, 0, 0]]
sage: designs.orthogonal_array(None,1,existence=True)
+Infinity
sage: designs.orthogonal_array(None,1)
Traceback (most recent call last):
...
ValueError: there are no bound on k when n=1.
sage: designs.orthogonal_array(None,14,existence=True)
6
sage: designs.orthogonal_array(16,1)
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]
"""
from sage.rings.finite_rings.constructor import FiniteField
from latin_squares import mutually_orthogonal_latin_squares
from database import OA_constructions
from block_design import projective_plane, projective_plane_to_OA
# If k is set to None we find the largest value available
if k is None:
if n == 1:
if existence:
from sage.rings.infinity import Infinity
return Infinity
raise ValueError("there are no bound on k when n=1.")
for k in range(1,n+2):
if not orthogonal_array(k+1,n,existence=True):
break
if existence:
return k
if k < 2:
raise ValueError("undefined for k less than 2")
if n == 1:
OA = [[0]*k]
elif k >= n+t:
# When t=2 then k<n+t as it is equivalent to the existence of n-1 MOLS.
# When t>2 the submatrix defined by the rows whose first t-2 elements
# are 0s yields a OA with t=2 and k-(t-2) columns. Thus k-(t-2) < n+2,
# i.e. k<n+t.
if existence:
return False
raise EmptySetError("No Orthogonal Array exists when k>=n+t except when n=1")
elif k == t:
if existence:
return True
from itertools import product
OA = map(list, product(range(n), repeat=k))
elif n in OA_constructions and k <= OA_constructions[n][0]:
if existence:
return True
_, construction = OA_constructions[n]
OA = OA_from_wider_OA(construction(),k)
# projective spaces are equivalent to OA(n+1,n,2)
elif (t == 2 and
(projective_plane(n, existence=True) or
(k == n+1 and projective_plane(n, existence=True) is False))):
if k == n+1:
if existence:
return projective_plane(n, existence=True)
p = projective_plane(n, check=False)
OA = projective_plane_to_OA(p)
else:
if existence:
return True
p = projective_plane(n, check=False)
OA = [l[:k] for l in projective_plane_to_OA(p)]
# Constructions from the database
elif n in OA_constructions and k <= OA_constructions[n][0]:
if existence:
return True
_, construction = OA_constructions[n]
OA = OA_from_wider_OA(construction(),k)
elif (t == 2 and transversal_design not in who_asked and
transversal_design(k,n,existence=True,who_asked=who_asked+(orthogonal_array,)) is not Unknown):
# forward existence
if transversal_design(k,n,existence=True,who_asked=who_asked+(orthogonal_array,)):
if existence:
return True
else:
TD = transversal_design(k,n,check=False,who_asked=who_asked+(orthogonal_array,))
OA = [[x%n for x in R] for R in TD]
# forward non-existence
else:
if existence:
return False
raise EmptySetError("There exists no OA"+str((k,n))+"!")
# Section 6.5.1 from [Stinson2004]
elif (t == 2 and mutually_orthogonal_latin_squares not in who_asked and
mutually_orthogonal_latin_squares(n,k-2, existence=True,who_asked=who_asked+(orthogonal_array,)) is not Unknown):
# forward existence
if mutually_orthogonal_latin_squares(n,k-2, existence=True,who_asked=who_asked+(orthogonal_array,)):
if existence:
return True
else:
mols = mutually_orthogonal_latin_squares(n,k-2,who_asked=who_asked+(orthogonal_array,))
OA = [[i,j]+[m[i,j] for m in mols]
for i in range(n) for j in range(n)]
# forward non-existence
else:
if existence:
return False
raise EmptySetError("There exists no OA"+str((k,n))+"!")
else:
if existence:
return Unknown
raise NotImplementedError("I don't know how to build this orthogonal array!")
if check:
assert is_orthogonal_array(OA,k,n,t)
return OA
