class CoveringDesign(SageObject):
"""
Covering design.
INPUT:
data -- parameters (v,k,t)
size
incidence structure (points and blocks) -- default points are $[0,...v-1]$
lower bound for such a design
database information: creator, method, timestamp
"""
def __init__(self,
v=0,
k=0,
t=0,
size=0,
points=[],
blocks=[],
low_bd=0,
method='',
creator='',
timestamp=''):
"""
EXAMPLES:
sage: C=CoveringDesign(5,4,3,4,range(5),[[0,1,2,3],[0,1,2,4],[0,1,3,4],[0,2,3,4]],4, 'Lexicographic Covering')
sage: print C
C(5,4,3) = 4
Method: Lexicographic Covering
0 1 2 3
0 1 2 4
0 1 3 4
0 2 3 4
"""
self.__v = v
self.__k = k
self.__t = t
self.__size = size
if low_bd > 0:
self.__low_bd = low_bd
else:
self.__low_bd = schonheim(v, k, t)
self.__method = method
self.__creator = creator
self.__timestamp = timestamp
self.__incidence_structure = IncidenceStructure(points, blocks)
def __repr__(self):
"""
A print method, giving the parameters and any other information about the covering (but not the blocks).
EXAMPLES:
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C
(7,3,2)-covering design of size 7
Lower bound: 7
Method: Projective Plane
"""
repr = '(%d,%d,%d)-covering design of size %d\n' % (
self.__v, self.__k, self.__t, self.__size)
repr += 'Lower bound: %d\n' % (self.__low_bd)
if self.__creator != '':
repr += 'Created by: %s\n' % (self.__creator)
if self.__method != '':
repr += 'Method: %s\n' % (self.__method)
if self.__timestamp != '':
repr += 'Submitted on: %s\n' % (self.__timestamp)
return repr
def __str__(self):
"""
A print method, displaying a covering design's parameters and blocks.
OUTPUT:
covering design parameters and blocks, in a readable form
EXAMPLES:
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: print C
C(7,3,2) = 7
Method: Projective Plane
0 1 2
0 3 4
0 5 6
1 3 5
1 4 6
2 3 6
2 4 5
"""
if self.__size == self.__low_bd: # check if the covering is known to be optimal
repr = 'C(%d,%d,%d) = %d\n' % (self.__v, self.__k, self.__t,
self.__size)
else:
repr = '%d <= C(%d,%d,%d) <= %d\n' % (
self.__low_bd, self.__v, self.__k, self.__t, self.__size)
if self.__creator != '':
repr += 'Created by: %s\n' % (self.__creator)
if self.__method != '':
repr += 'Method: %s\n' % (self.__method)
if self.__timestamp != '':
repr += 'Submitted on: %s\n' % (self.__timestamp)
for i in range(self.__size):
for j in range(self.__k):
repr = repr + str(
self.__incidence_structure.blocks()[i][j]) + ' '
repr += '\n'
return repr
def is_covering(self):
"""
Checks that all t-sets are in fact covered.
INPUT:
putative covering design
OUTPUT:
True if all t-sets are in at least one block
NOTES:
This is very slow and wasteful of memory. A faster Cython
version will be added soon.
EXAMPLES:
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.is_covering()
True
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 6]],0, 'not a covering') # last block altered
sage: C.is_covering()
False
"""
v = self.__v
k = self.__k
t = self.__t
Svt = Combinations(range(v), t)
Skt = Combinations(range(k), t)
tset = {} # tables of t-sets: False = uncovered, True = covered
for i in Svt:
tset[tuple(i)] = False
# mark all t-sets covered by each block
for a in self.__incidence_structure.blocks():
for z in Skt:
y = [a[x] for x in z]
tset[tuple(y)] = True
for i in Svt:
if tset[tuple(i)] == False: # uncovered
return False
return True # everything was covered
def v(self):
"""
Return v, the number of points in the covering design
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.v()
7
"""
return self.__v
def k(self):
"""
Return k, the size of blocks of the covering design
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.k()
3
"""
return self.__k
def t(self):
"""
Return t, the size of sets which must be covered by the blocks of the covering design
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.t()
2
"""
return self.__t
def size(self):
"""
Return the number of blocks in the covering design
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.size()
7
"""
return self.__size
def low_bd(self):
"""
Return a lower bound for the number of blocks a covering design with these parameters could have.
Typically this is the Schonheim bound, but for some parameters better bounds have been shown.
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.low_bd()
7
"""
return self.__low_bd
def method(self):
"""
Return the method used to create the covering design
This field is optional, and is used in a database to give information about how coverings were constructed
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.method()
'Projective Plane'
"""
return self.__method
def creator(self):
"""
Return the creator of the covering design
This field is optional, and is used in a database to give attribution for the covering design
It can refer to the person who submitted it, or who originally gave a construction
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane','Gino Fano')
sage: C.creator()
'Gino Fano'
"""
return self.__creator
def timestamp(self):
"""
Return the time that the covering was submitted to the database
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane','Gino Fano','1892-01-01 00:00:00')
sage: C.timestamp() #Fano had an article in 1892, I don't know the date it appeared
'1892-01-01 00:00:00'
"""
return self.__timestamp
def incidence_structure(self):
"""
Return the incidence structure of a covering design, without all the extra parameters.
EXAMPLES:
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: D = C.incidence_structure()
sage: D.points()
[0, 1, 2, 3, 4, 5, 6]
sage: D.blocks()
[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]
"""
return self.__incidence_structure
class CoveringDesign(SageObject):
"""
Covering design.
INPUT:
- ``v,k,t`` -- integer parameters of the covering design.
- ``size`` (integer)
- ``points`` -- list of points (default points are `[0,...v-1]`).
- ``blocks``
- ``low_bd`` (integer) -- lower bound for such a design
- ``method, creator, timestamp`` -- database information.
"""
def __init__(self, v=0, k=0, t=0, size=0, points=[], blocks=[], low_bd=0, method='', creator ='',timestamp=''):
"""
EXAMPLES::
sage: C=CoveringDesign(5,4,3,4,range(5),[[0,1,2,3],[0,1,2,4],[0,1,3,4],[0,2,3,4]],4, 'Lexicographic Covering')
sage: print(C)
C(5,4,3) = 4
Method: Lexicographic Covering
0 1 2 3
0 1 2 4
0 1 3 4
0 2 3 4
"""
self.__v = v
self.__k = k
self.__t = t
self.__size = size
if low_bd > 0:
self.__low_bd = low_bd
else:
self.__low_bd = schonheim(v,k,t)
self.__method = method
self.__creator = creator
self.__timestamp = timestamp
self.__incidence_structure = IncidenceStructure(points, blocks)
def __repr__(self):
"""
A print method, giving the parameters and any other
information about the covering (but not the blocks).
EXAMPLES::
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C
(7,3,2)-covering design of size 7
Lower bound: 7
Method: Projective Plane
"""
repr = '(%d,%d,%d)-covering design of size %d\n' % (self.__v,
self.__k,
self.__t,
self.__size)
repr += 'Lower bound: %d\n' % (self.__low_bd)
if self.__creator != '':
repr += 'Created by: %s\n' % (self.__creator)
if self.__method != '':
repr += 'Method: %s\n' % (self.__method)
if self.__timestamp != '':
repr += 'Submitted on: %s\n' % (self.__timestamp)
return repr
def __str__(self):
"""
A print method, displaying a covering design's parameters and blocks.
OUTPUT:
covering design parameters and blocks, in a readable form
EXAMPLES::
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: print(C)
C(7,3,2) = 7
Method: Projective Plane
0 1 2
0 3 4
0 5 6
1 3 5
1 4 6
2 3 6
2 4 5
"""
if self.__size == self.__low_bd: # check if the covering is known to be optimal
repr = 'C(%d,%d,%d) = %d\n'%(self.__v,self.__k,self.__t,self.__size)
else:
repr = '%d <= C(%d,%d,%d) <= %d\n'%(self.__low_bd,self.__v,self.__k,self.__t,self.__size);
if self.__creator != '':
repr += 'Created by: %s\n'%(self.__creator)
if self.__method != '':
repr += 'Method: %s\n'%(self.__method)
if self.__timestamp != '':
repr += 'Submitted on: %s\n'%(self.__timestamp)
for i in range(self.__size):
for j in range(self.__k):
repr = repr + str(self.__incidence_structure.blocks()[i][j]) + ' '
repr += '\n'
return repr
def is_covering(self):
"""
Checks that all `t`-sets are in fact covered by the blocks of
``self``
.. NOTE::
This is very slow and wasteful of memory.
EXAMPLES::
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.is_covering()
True
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 6]],0, 'not a covering') # last block altered
sage: C.is_covering()
False
"""
v = self.__v
k = self.__k
t = self.__t
Svt = Combinations(range(v),t)
Skt = Combinations(range(k),t)
tset = {} # tables of t-sets: False = uncovered, True = covered
for i in Svt:
tset[tuple(i)] = False
# mark all t-sets covered by each block
for a in self.__incidence_structure.blocks():
for z in Skt:
y = [a[x] for x in z]
tset[tuple(y)] = True
for i in Svt:
if not tset[tuple(i)]: # uncovered
return False
return True # everything was covered
def v(self):
"""
Return `v`, the number of points in the covering design.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.v()
7
"""
return self.__v
def k(self):
"""
Return `k`, the size of blocks of the covering design
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.k()
3
"""
return self.__k
def t(self):
"""
Return `t`, the size of sets which must be covered by the
blocks of the covering design
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.t()
2
"""
return self.__t
def size(self):
"""
Return the number of blocks in the covering design
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.size()
7
"""
return self.__size
def low_bd(self):
"""
Return a lower bound for the number of blocks a covering
design with these parameters could have.
Typically this is the Schonheim bound, but for some parameters
better bounds have been shown.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.low_bd()
7
"""
return self.__low_bd
def method(self):
"""
Return the method used to create the covering design
This field is optional, and is used in a database to give information about how coverings were constructed
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: C.method()
'Projective Plane'
"""
return self.__method
def creator(self):
"""
Return the creator of the covering design
This field is optional, and is used in a database to give
attribution for the covering design It can refer to the person
who submitted it, or who originally gave a construction
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane','Gino Fano')
sage: C.creator()
'Gino Fano'
"""
return self.__creator
def timestamp(self):
"""
Return the time that the covering was submitted to the database
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane','Gino Fano','1892-01-01 00:00:00')
sage: C.timestamp() #Fano had an article in 1892, I don't know the date it appeared
'1892-01-01 00:00:00'
"""
return self.__timestamp
def incidence_structure(self):
"""
Return the incidence structure of a covering design, without all the extra parameters.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C=CoveringDesign(7,3,2,7,range(7),[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]],0, 'Projective Plane')
sage: D = C.incidence_structure()
sage: D.ground_set()
[0, 1, 2, 3, 4, 5, 6]
sage: D.blocks()
[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]
"""
return self.__incidence_structure
class CoveringDesign(SageObject):
"""
Covering design.
INPUT:
- ``v``, ``k``, ``t`` -- integer parameters of the covering design
- ``size`` (integer)
- ``points`` -- list of points (default points are `[0, ..., v-1]`)
- ``blocks``
- ``low_bd`` (integer) -- lower bound for such a design
- ``method``, ``creator``, ``timestamp`` -- database information
"""
def __init__(self,
v=0,
k=0,
t=0,
size=0,
points=None,
blocks=None,
low_bd=0,
method='',
creator='',
timestamp=''):
"""
EXAMPLES::
sage: C = CoveringDesign(5, 4, 3, 4, range(5), [[0, 1, 2, 3],
....: [0, 1, 2, 4], [0, 1, 3, 4], [0, 2, 3, 4]], 4,
....: 'Lexicographic Covering')
sage: print(C)
C(5, 4, 3) = 4
Method: Lexicographic Covering
0 1 2 3
0 1 2 4
0 1 3 4
0 2 3 4
"""
self.__v = v
self.__k = k
self.__t = t
self.__size = size
if low_bd > 0:
self.__low_bd = low_bd
else:
self.__low_bd = schonheim(v, k, t)
self.__method = method
self.__creator = creator
self.__timestamp = timestamp
if points is None:
points = []
if blocks is None:
blocks = []
self.__incidence_structure = IncidenceStructure(points, blocks)
def __repr__(self):
"""
Return the representation of this covering design.
This has the parameters but not the blocks.
EXAMPLES::
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C
(7, 3, 2)-covering design of size 7
Lower bound: 7
Method: Projective Plane
"""
repr = ('(%d, %d, %d)-covering design of size %d\n' %
(self.__v, self.__k, self.__t, self.__size))
repr += 'Lower bound: %d\n' % (self.__low_bd)
if self.__creator:
repr += 'Created by: %s\n' % (self.__creator)
if self.__method:
repr += 'Method: %s\n' % (self.__method)
if self.__timestamp:
repr += 'Submitted on: %s\n' % (self.__timestamp)
return repr[:-1]
def __str__(self):
"""
Return the string for this covering design.
This has the parameters and the blocks in readable form.
EXAMPLES::
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: print(C)
C(7, 3, 2) = 7
Method: Projective Plane
0 1 2
0 3 4
0 5 6
1 3 5
1 4 6
2 3 6
2 4 5
"""
if self.__size == self.__low_bd: # check if covering is optimal
repr = ('C(%d, %d, %d) = %d\n' %
(self.__v, self.__k, self.__t, self.__size))
else:
repr = ('%d <= C(%d, %d, %d) <= %d\n' %
(self.__low_bd, self.__v, self.__k, self.__t, self.__size))
if self.__creator:
repr += 'Created by: %s\n' % (self.__creator)
if self.__method:
repr += 'Method: %s\n' % (self.__method)
if self.__timestamp:
repr += 'Submitted on: %s\n' % (self.__timestamp)
return repr + '\n'.join(
' '.join(str(k) for k in block)
for block in self.__incidence_structure.blocks())
def is_covering(self):
"""
Check all `t`-sets are in fact covered by the blocks of ``self``.
.. NOTE::
This is very slow and wasteful of memory.
EXAMPLES::
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.is_covering()
True
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6],
....: [2, 4, 6]], 0, 'not a covering') # last block altered
sage: C.is_covering()
False
"""
v = self.__v
k = self.__k
t = self.__t
Svt = Combinations(range(v), t)
Skt = Combinations(range(k), t)
tset = {} # tables of t-sets: False = uncovered, True = covered
for i in Svt:
tset[tuple(i)] = False
# mark all t-sets covered by each block
for a in self.__incidence_structure.blocks():
for z in Skt:
y = (a[x] for x in z)
tset[tuple(y)] = True
for i in Svt:
if not tset[tuple(i)]: # uncovered
return False
return True # everything was covered
def v(self):
"""
Return `v`, the number of points in the covering design.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.v()
7
"""
return self.__v
def k(self):
"""
Return `k`, the size of blocks of the covering design
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.k()
3
"""
return self.__k
def t(self):
"""
Return `t`, the size of sets which must be covered by the
blocks of the covering design
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.t()
2
"""
return self.__t
def size(self):
"""
Return the number of blocks in the covering design
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.size()
7
"""
return self.__size
def low_bd(self):
"""
Return a lower bound for the number of blocks a covering
design with these parameters could have.
Typically this is the Schonheim bound, but for some parameters
better bounds have been shown.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.low_bd()
7
"""
return self.__low_bd
def method(self):
"""
Return the method used to create the covering design.
This field is optional, and is used in a database to give information
about how coverings were constructed.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.method()
'Projective Plane'
"""
return self.__method
def creator(self):
"""
Return the creator of the covering design
This field is optional, and is used in a database to give
attribution for the covering design It can refer to the person
who submitted it, or who originally gave a construction
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6],
....: [2, 4, 5]],0, 'Projective Plane', 'Gino Fano')
sage: C.creator()
'Gino Fano'
"""
return self.__creator
def timestamp(self):
"""
Return the time that the covering was submitted to the database
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]],0, 'Projective Plane',
....: 'Gino Fano', '1892-01-01 00:00:00')
sage: C.timestamp() # No exact date known; in Fano's 1892 article
'1892-01-01 00:00:00'
"""
return self.__timestamp
def incidence_structure(self):
"""
Return the incidence structure of this design, without extra parameters.
EXAMPLES::
sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....: [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....: [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: D = C.incidence_structure()
sage: D.ground_set()
[0, 1, 2, 3, 4, 5, 6]
sage: D.blocks()
[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5],
[1, 4, 6], [2, 3, 6], [2, 4, 5]]
"""
return self.__incidence_structure
