This module gathers some construction related to orthogonal arrays (or
transversal designs). One can build an (or check that it can be built)
from the Sage console with designs.orthogonal_arrays.build:
sage: OA = designs.orthogonal_arrays.build(4,8)
See also the modules orthogonal_arrays_build_recursive or orthogonal_arrays_find_recursive for recursive constructions.
This module defines the following functions:
orthogonal_array() | Return an orthogonal array of parameters ![]() |
transversal_design() | Return a transversal design of parameters ![]() |
incomplete_orthogonal_array() | Return an ![]() |
is_transversal_design() | Check that a given set of blocks B is a transversal design. |
is_orthogonal_array() | Check that the integer matrix ![]() ![]() |
wilson_construction() | Return a ![]() ![]() |
TD_product() | Return the product of two transversal designs. |
OA_find_disjoint_blocks() | Return ![]() ![]() |
OA_relabel() | Return a relabelled version of the OA. |
OA_from_quasi_difference_matrix() | Return an Orthogonal Array from a Quasi-Difference matrix |
OA_from_Vmt() | Return an Orthogonal Array from a ![]() |
OA_from_PBD() | Return an ![]() |
OA_n_times_2_pow_c_from_matrix() | Return an ![]() ![]() |
OA_from_wider_OA() | Return the first ![]() ![]() |
QDM_from_Vmt() | Return a QDM a ![]() |
REFERENCES:
[CD96] | Making the MOLS table Charles Colbourn and Jeffrey Dinitz Computational and constructive design theory vol 368,pages 67-134 1996 |
Functions related to orthogonal arrays.
An orthogonal array of parameters is a matrix with
columns
filled with integers from
in such a way that for any
columns, each
of the
possible rows occurs exactly once. In particular, the matrix
has
rows.
For more information on orthogonal arrays, see Wikipedia article Orthogonal_array.
From here you have access to:
EXAMPLES:
sage: designs.orthogonal_arrays.build(3,2)
[[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
sage: designs.orthogonal_arrays.build(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 for which Sage knows how to compute a
?:
sage: designs.orthogonal_arrays.largest_available_k(14)
6
If you ask for an orthogonal array that does not exist, then you will either obtain an EmptySetError (if it knows that such an orthogonal array does not exist) or a NotImplementedError:
sage: designs.orthogonal_arrays.build(4,2)
Traceback (most recent call last):
...
EmptySetError: There exists no OA(4,2) as k(=4)>n+t-1=3
sage: designs.orthogonal_arrays.build(12,20)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build an OA(12,20)!
Return an of strength
An orthogonal array of parameters is a matrix with
columns filled with integers from
in such a way that for any
columns, each of the
possible rows occurs exactly
once. In particular, the matrix has
rows.
More general definitions sometimes involve a parameter, and we
assume here that
.
For more information on orthogonal arrays, see Wikipedia article Orthogonal_array.
INPUT:
EXAMPLES:
sage: designs.orthogonal_arrays.build(3,3,resolvable=True) # indirect doctest
[[0, 0, 0],
[1, 2, 1],
[2, 1, 2],
[0, 2, 2],
[1, 1, 0],
[2, 0, 1],
[0, 1, 1],
[1, 0, 2],
[2, 2, 0]]
sage: OA_7_50 = designs.orthogonal_arrays.build(7,50) # indirect doctest
Return the existence status of an
INPUT:
Warning
The function does not only return booleans, but True, False, or Unknown.
See also
EXAMPLE:
sage: designs.orthogonal_arrays.exists(3,6) # indirect doctest
True
sage: designs.orthogonal_arrays.exists(4,6) # indirect doctest
Unknown
sage: designs.orthogonal_arrays.exists(7,6) # indirect doctest
False
Return a string describing how to builds an
INPUT:
EXAMPLE:
sage: designs.orthogonal_arrays.explain_construction(9,565)
"Wilson's construction n=23.24+13 with master design OA(9+1,23)"
sage: designs.orthogonal_arrays.explain_construction(10,154)
'the database contains a (137,10;1,0;17)-quasi difference matrix'
Return whether Sage can build an .
INPUT:
See also
EXAMPLE:
sage: designs.orthogonal_arrays.is_available(3,6) # indirect doctest
True
sage: designs.orthogonal_arrays.is_available(4,6) # indirect doctest
False
Return the largest such that Sage can build an
.
INPUT:
EXAMPLE:
sage: designs.orthogonal_arrays.largest_available_k(0)
+Infinity
sage: designs.orthogonal_arrays.largest_available_k(1)
+Infinity
sage: designs.orthogonal_arrays.largest_available_k(10)
4
sage: designs.orthogonal_arrays.largest_available_k(27)
28
sage: designs.orthogonal_arrays.largest_available_k(100)
10
sage: designs.orthogonal_arrays.largest_available_k(-1)
Traceback (most recent call last):
...
ValueError: n(=-1) was expected to be >=0
Return disjoint blocks contained in a given
.
blocks of an
are said to be disjoint if they all have
different values for a every given index, i.e. if they correspond to
disjoint blocks in the
assciated with the
.
INPUT:
See also
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import OA_find_disjoint_blocks
sage: k=3;n=4;x=3
sage: Bs = OA_find_disjoint_blocks(designs.orthogonal_arrays.build(k,n),k,n,x)
sage: assert len(Bs) == x
sage: for i in range(k):
....: assert len(set([B[i] for B in Bs])) == x
sage: OA_find_disjoint_blocks(designs.orthogonal_arrays.build(k,n),k,n,5)
Traceback (most recent call last):
...
ValueError: There does not exist 5 disjoint blocks in this OA(3,4)
Return an from a PBD
Construction
Let be a
-PBD. If there exists for every
a
(i.e. if there exist
idempotent MOLS), then
one can obtain a
by concatenating:
Note
This function raises an exception when Sage is unable to build the necessary designs.
INPUT:
EXAMPLES:
We start from the example VI.1.2 from the [DesignHandbook] to build an
:
sage: from sage.combinat.designs.orthogonal_arrays import OA_from_PBD
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: pbd = [[0,1,2,3],[0,4,5,6],[0,7,8,9],[1,4,7],[1,5,8],
....: [1,6,9],[2,4,9],[2,5,7],[2,6,8],[3,4,8],[3,5,9],[3,6,7]]
sage: oa = OA_from_PBD(3,10,pbd)
sage: is_orthogonal_array(oa, 3, 10)
True
But we cannot build an for this PBD (although there
exists an
:
sage: OA_from_PBD(4,10,pbd)
Traceback (most recent call last):
...
EmptySetError: There is no OA(n+1,n) - 3.OA(n+1,1) as all blocks do intersect in a projective plane.
Or an (as the PBD has 10 points):
sage: _ = OA_from_PBD(3,6,pbd)
Traceback (most recent call last):
...
RuntimeError: PBD is not a valid Pairwise Balanced Design on [0,...,5]
Return an Orthogonal Array from a
INPUT:
EXAMPLES:
sage: _ = designs.orthogonal_arrays.build(6,46) # indirect doctest
Return an Orthogonal Array from a Quasi-Difference matrix
Difference Matrices
Let be a group of order
. A difference matrix
is a
matrix with entries from
such that for any
the set
is equal to
.
By concatenating the matrices
(where
), one obtains a
matrix of size
which is also an
.
Quasi-difference Matrices
A quasi-difference matrix is a difference matrix with missing entries. The
construction above can be applied again in this case, where the missing
entries in each column of are replaced by unique values on which
has
a trivial action.
This produces an incomplete orthogonal array with a “hole” (i.e. missing
rows) of size ‘u’ (i.e. the number of missing values per column of ). If
there exists an
, then adding the rows of this
to the
incomplete orthogonal array should lead to an OA...
Formal definition (from the Handbook of Combinatorial Designs [DesignHandbook])
Let be an abelian group of order
. A
-quasi-difference matrix (QDM) is a matrix
with
rows and
columns, with each entry either
empty or containing an element of
. Each column contains exactly
entries, and each row contains at most one empty entry. Furthermore, for
each
the multiset
contains every nonzero element of exactly
times, and contains
0 exactly
times.
Construction
If a -QDM exists and
, then an
exists. Start with a
-QDM
over the group
. Append
rows of zeroes. Then select
elements
not in
, and replace the empty
entries, each by one of these infinite symbols, so that
appears
exactly once in each column. Develop the resulting matrix over the group
(leaving infinite symbols fixed), to obtain a
matrix
. Then
is an orthogonal array with
columns and index
, having
symbols and one hole of size
.
Adding to an
with elements
yields
the
.
For more information, see the Handbook of Combinatorial Designs [DesignHandbook] or http://web.cs.du.edu/~petr/milehigh/2013/Colbourn.pdf.
INPUT:
EXAMPLES:
sage: _ = designs.orthogonal_arrays.build(6,20) # indirect doctest
Return the first columns of
.
If has
columns, this function returns
immediately.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import OA_from_wider_OA
sage: OA_from_wider_OA(designs.orthogonal_arrays.build(6,20,2),1)[:5]
[(19,), (19,), (19,), (19,), (19,)]
sage: _ = designs.orthogonal_arrays.build(5,46) # indirect doctest
Return an from a constrained
-difference
matrix.
This construction appears in [AbelCheng1994] and [AbelThesis].
Let be an additive Abelian group. We denote by
a
-hyperplane
in
.
Let be a
array with entries in
and
be a vector with
entries in
. Let
and
be
respectively the part of the array that belong to
and
.
The input and
must satisfy the following conditions. For any
and
:
Under these conditions, it is easy to check that the array whose rows
of length
indexed by
given by
where
is a
-difference matrix.
INPUT:
Note
By convention, a multiplicative generator of
is fixed
(inside the function). The hyperplane
is the one spanned by
. The
part of the input matrix
and
vector
are given in the following form: the integer
corresponds
to the element
and None corresponds to
.
See also
Several examples use this construction:
EXAMPLE:
sage: from sage.combinat.designs.orthogonal_arrays import OA_n_times_2_pow_c_from_matrix
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: A = [
....: [(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None)],
....: [(0,None),(1,None), (2,2), (3,2), (4,2),(2,None),(3,None),(4,None), (0,2), (1,2)],
....: [(0,None), (2,5), (4,5), (1,2), (3,6), (3,4), (0,0), (2,1), (4,1), (1,6)],
....: [(0,None), (3,4), (1,4), (4,0), (2,5),(3,None), (1,0), (4,1), (2,2), (0,3)],
....: ]
sage: Y = [None, 0, 1, 6]
sage: OA = OA_n_times_2_pow_c_from_matrix(5,3,GF(5),A,Y)
sage: is_orthogonal_array(OA,5,40,2)
True
sage: A[0][0] = (1,None)
sage: OA_n_times_2_pow_c_from_matrix(5,3,GF(5),A,Y)
Traceback (most recent call last):
...
ValueError: the first part of the matrix A must be a
(G,k-1,2)-difference matrix
sage: A[0][0] = (0,0)
sage: OA_n_times_2_pow_c_from_matrix(5,3,GF(5),A,Y)
Traceback (most recent call last):
...
ValueError: B_2,0 - B_0,0 = B_2,6 - B_0,6 but the associated part of the
matrix C does not satisfies the required condition
REFERENCES:
[AbelThesis] | On the Existence of Balanced Incomplete Block Designs and Transversal Designs, Julian R. Abel, PhD Thesis, University of New South Wales, 1995 |
[AbelCheng1994] | R.J.R. Abel and Y.W. Cheng, Some new MOLS of order 2np for p a prime power, The Australasian Journal of Combinatorics, vol 10 (1994) |
Return a relabelled version of the OA.
INPUT:
OA – an OA, or rather a list of blocks of length , each
of which contains integers from
to
.
k,n (integers)
blocks (list of blocks) – relabels the integers of the OA
from into
in such a way that the
blocks from block are respectively relabeled as
[n-i,...,n-i], ..., [n-1,...,n-1]. Thus, the blocks from
this list are expected to have disjoint values for each
coordinate.
If set to the empty list (default) no such relabelling is performed.
matrix – a matrix of dimensions such that if the i th
coordinate of a block is
, this
will be relabelled with
matrix[i][x]. This is not necessarily an integer between
and
, and it is not necessarily an integer either. This is
performed after the previous relabelling.
If set to None (default) no such relabelling is performed.
Note
A None coordinate in one block remains a None coordinate in the final block.
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import OA_relabel
sage: OA = designs.orthogonal_arrays.build(3,2)
sage: OA_relabel(OA,3,2,matrix=[["A","B"],["C","D"],["E","F"]])
[['A', 'C', 'E'], ['A', 'D', 'F'], ['B', 'C', 'F'], ['B', 'D', 'E']]
sage: TD = OA_relabel(OA,3,2,matrix=[[0,1],[2,3],[4,5]]); TD
[[0, 2, 4], [0, 3, 5], [1, 2, 5], [1, 3, 4]]
sage: from sage.combinat.designs.orthogonal_arrays import is_transversal_design
sage: is_transversal_design(TD,3,2)
True
Making sure that [2,2,2,2] is a block of . We do this
by relabelling block [0,0,0,0] which belongs to the design:
sage: designs.orthogonal_arrays.build(4,3)
[[0, 0, 0, 0], [0, 1, 2, 1], [0, 2, 1, 2], [1, 0, 2, 2], [1, 1, 1, 0], [1, 2, 0, 1], [2, 0, 1, 1], [2, 1, 0, 2], [2, 2, 2, 0]]
sage: OA_relabel(designs.orthogonal_arrays.build(4,3),4,3,blocks=[[0,0,0,0]])
[[2, 2, 2, 2], [2, 0, 1, 0], [2, 1, 0, 1], [0, 2, 1, 1], [0, 0, 0, 2], [0, 1, 2, 0], [1, 2, 0, 0], [1, 0, 2, 1], [1, 1, 1, 2]]
TESTS:
sage: OA_relabel(designs.orthogonal_arrays.build(3,2),3,2,blocks=[[0,1],[0,1]])
Traceback (most recent call last):
...
RuntimeError: Two block have the same coordinate for one of the k dimensions
Return a QDM from a
Definition
Let be a prime power and let
for
integers. Let
be a primitive element of
. A
vector is a vector
for which, for each
, the differences
represent the cyclotomic classes of
(compute subscripts
modulo
). In other words, for fixed
, is
and
then
Construction of a quasi-difference matrix from a `V(m,t)` vector
Starting with a vector
, form a single row of
length
whose first entry is empty, and whose remaining entries are
. Form
rows by multiplying this row by the
th
roots, i.e. the powers of
. From each of these
rows, form
rows by taking the
cyclic shifts of the row. The result is a
.
For more information, refer to the Handbook of Combinatorial Designs [DesignHandbook].
INPUT:
See also
EXAMPLES:
sage: _ = designs.orthogonal_arrays.build(6,46) # indirect doctest
Return the product of two transversal designs.
From a transversal design of parameters
and a transversal
design
of parameters
, this function returns a transversal
design of parameters
where
.
Formally, if the groups of are
and the groups of
are
, the groups of the product design are
and its blocks are the
where
is a
block of
and
is a block of
.
INPUT:
Note
This function uses transversal designs with
both as input and
ouptut.
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import TD_product
sage: TD1 = designs.transversal_design(6,7)
sage: TD2 = designs.transversal_design(6,12)
sage: TD6_84 = TD_product(6,TD1,7,TD2,12)
Bases: sage.combinat.designs.incidence_structures.GroupDivisibleDesign
Class for Transversal Designs
INPUT:
EXAMPLES:
sage: designs.transversal_design(None,5)
Transversal Design TD(6,5)
sage: designs.transversal_design(None,30)
Transversal Design TD(6,30)
sage: designs.transversal_design(None,36)
Transversal Design TD(10,36)
Return an .
An is an orthogonal array from
which have been removed disjoint
. If there exist
they can be used to fill the holes and give rise
to an
.
A very useful particular case (see e.g. the Wilson construction in
wilson_construction()) is when all . In that case the
incomplete design is a
. Such design is equivalent to
transversal design
from which has been removed
disjoint
blocks.
INPUT:
k,n (integers)
holes_sizes (list of integers) – respective sizes of the holes to be found.
Note
Right now the feature is only available when all holes have size 1,
i.e. .
resolvable (boolean) – set to True if you want the design to be
resolvable. The classes of the resolvable design are obtained as the first
blocks, then the next
blocks, etc ... Set to False 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 sage.misc.unknown).
- False – meaning that the design does not exist.
Note
By convention, the ground set is always and the
holes are
,
, etc.
See also
EXAMPLES:
sage: IOA = designs.incomplete_orthogonal_array(3,3,[1,1,1])
sage: IOA
[[0, 1, 2], [0, 2, 1], [1, 0, 2], [1, 2, 0], [2, 0, 1], [2, 1, 0]]
sage: missing_blocks = [[0,0,0],[1,1,1],[2,2,2]]
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: is_orthogonal_array(IOA + missing_blocks,3,3,2)
True
TESTS:
Affine planes and projective planes:
sage: for q in xrange(2,100):
....: if is_prime_power(q):
....: assert designs.incomplete_orthogonal_array(q,q,[1]*q,existence=True)
....: assert not designs.incomplete_orthogonal_array(q+1,q,[1]*2,existence=True)
Further tests:
sage: designs.incomplete_orthogonal_array(8,4,[1,1,1],existence=True)
False
sage: designs.incomplete_orthogonal_array(5,10,[1,1,1],existence=True)
Unknown
sage: designs.incomplete_orthogonal_array(5,10,[1,1,1])
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build an OA(5,10)!
sage: designs.incomplete_orthogonal_array(4,3,[1,1])
Traceback (most recent call last):
...
EmptySetError: There is no OA(n+1,n) - 2.OA(n+1,1) as all blocks do
intersect in a projective plane.
sage: n=10
sage: k=designs.orthogonal_arrays.largest_available_k(n)
sage: designs.incomplete_orthogonal_array(k,n,[1,1,1],existence=True)
True
sage: _ = designs.incomplete_orthogonal_array(k,n,[1,1,1])
sage: _ = designs.incomplete_orthogonal_array(k,n,[1])
A resolvable . We check that extending each class and
adding the
blocks turns it into an
.:
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: k,n=5,7
sage: OA = designs.incomplete_orthogonal_array(k,n,[1]*n,resolvable=True)
sage: classes = [OA[i*n:(i+1)*n] for i in range(n-1)]
sage: for classs in classes: # The design is resolvable !
....: assert(len(set(col))==n for col in zip(*classs))
sage: OA.extend([[i]*(k) for i in range(n)])
sage: for i,R in enumerate(OA):
....: R.append(i//n)
sage: is_orthogonal_array(OA,k+1,n)
True
Non-existent resolvable incomplete OA:
sage: designs.incomplete_orthogonal_array(9,13,[1]*10,resolvable=True,existence=True)
False
sage: designs.incomplete_orthogonal_array(9,13,[1]*10,resolvable=True)
Traceback (most recent call last):
...
EmptySetError: There is no resolvable incomplete OA(9,13) whose holes' sizes sum to 10!=n(=13)
Error message for big holes:
sage: designs.incomplete_orthogonal_array(6,4*9,[9,9,8])
Traceback (most recent call last):
...
NotImplementedError: I was not able to build this OA(6,36)-OA(6,8)-2.OA(6,9)
REFERENCES:
[BvR82] | More mutually orthogonal Latin squares, Andries Brouwer and John van Rees Discrete Mathematics vol.39, num.3, pages 263-281 1982 |
Check that a given set of blocks B is a transversal design.
See transversal_design() for a definition.
INPUT:
Note
The tranversal design must have as a ground set,
partitioned as
sets of size
:
.
EXAMPLES:
sage: TD = designs.transversal_design(5, 5, check=True) # indirect doctest
sage: from sage.combinat.designs.orthogonal_arrays import is_transversal_design
sage: is_transversal_design(TD, 5, 5)
True
sage: is_transversal_design(TD, 4, 4)
False
Return the largest such that Sage can build an
.
INPUT:
EXAMPLE:
sage: designs.orthogonal_arrays.largest_available_k(0)
+Infinity
sage: designs.orthogonal_arrays.largest_available_k(1)
+Infinity
sage: designs.orthogonal_arrays.largest_available_k(10)
4
sage: designs.orthogonal_arrays.largest_available_k(27)
28
sage: designs.orthogonal_arrays.largest_available_k(100)
10
sage: designs.orthogonal_arrays.largest_available_k(-1)
Traceback (most recent call last):
...
ValueError: n(=-1) was expected to be >=0
Return an orthogonal array of parameters .
An orthogonal array of parameters is a matrix with
columns
filled with integers from
in such a way that for any
columns, each
of the
possible rows occurs exactly once. In
particular, the matrix has
rows.
More general definitions sometimes involve a parameter, and we
assume here that
.
An orthogonal array is said to be resolvable if it corresponds to a resolvable transversal design (see sage.combinat.designs.incidence_structures.IncidenceStructure.is_resolvable()).
For more information on orthogonal arrays, see Wikipedia article 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
resolvable (boolean) – set to True if you want the design to be
resolvable. The classes of the resolvable design are obtained as the
first
blocks, then the next
blocks, etc ... Set to False by
default.
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 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 such that we can build a
.
explain_construction (boolean) – return a string describing the construction.
OUTPUT:
The kind of output depends on the input:
Note
This method implements theorems from [Stinson2004]. See the code’s documentation for details.
See also
When an orthogonal array is also a transversal design (see
transversal_design()) and a family of mutually orthogonal latin
squares (see
mutually_orthogonal_latin_squares()).
TESTS:
The special cases :
sage: designs.orthogonal_arrays.build(3,0)
[]
sage: designs.orthogonal_arrays.build(3,1)
[[0, 0, 0]]
sage: designs.orthogonal_arrays.largest_available_k(0)
+Infinity
sage: designs.orthogonal_arrays.largest_available_k(1)
+Infinity
sage: designs.orthogonal_arrays.build(16,0)
[]
sage: designs.orthogonal_arrays.build(16,1)
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]
when and
:
sage: t = 3
sage: designs.orthogonal_arrays.largest_available_k(5,t=t) == t
True
sage: _ = designs.orthogonal_arrays.build(t,5,t)
Return a transversal design of parameters .
A transversal design of parameters is a collection
of
subsets of
(where the groups
are
disjoint and have cardinality
) such that:
More general definitions sometimes involve a parameter, and we
assume here that
.
For more information on transversal designs, see http://mathworld.wolfram.com/TransversalDesign.html.
INPUT:
– integers. If k is None it is set to the largest value
available.
resolvable (boolean) – set to True if you want the design to be
resolvable (see
sage.combinat.designs.incidence_structures.IncidenceStructure.is_resolvable()). The
classes of the resolvable design are obtained as the first
blocks,
then the next
blocks, etc ... Set to False by default.
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 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 such that we can build a
.
OUTPUT:
The kind of output depends on the input:
See also
orthogonal_array() – a tranversal design is equivalent to an
orthogonal array
.
EXAMPLES:
sage: TD = designs.transversal_design(5,5); TD
Transversal Design TD(5,5)
sage: TD.blocks()
[[0, 5, 10, 15, 20], [0, 6, 12, 18, 24], [0, 7, 14, 16, 23],
[0, 8, 11, 19, 22], [0, 9, 13, 17, 21], [1, 5, 14, 18, 22],
[1, 6, 11, 16, 21], [1, 7, 13, 19, 20], [1, 8, 10, 17, 24],
[1, 9, 12, 15, 23], [2, 5, 13, 16, 24], [2, 6, 10, 19, 23],
[2, 7, 12, 17, 22], [2, 8, 14, 15, 21], [2, 9, 11, 18, 20],
[3, 5, 12, 19, 21], [3, 6, 14, 17, 20], [3, 7, 11, 15, 24],
[3, 8, 13, 18, 23], [3, 9, 10, 16, 22], [4, 5, 11, 17, 23],
[4, 6, 13, 15, 22], [4, 7, 10, 18, 21], [4, 8, 12, 16, 20],
[4, 9, 14, 19, 24]]
Some examples of the maximal number of transversal Sage is able to build:
sage: TD_4_10 = designs.transversal_design(4,10)
sage: designs.transversal_design(5,10,existence=True)
Unknown
For prime powers, there is an explicit construction which gives a
:
sage: designs.transversal_design(4, 3, existence=True)
True
sage: designs.transversal_design(674, 673, existence=True)
True
For other values of n it depends:
sage: designs.transversal_design(7, 6, existence=True)
False
sage: designs.transversal_design(4, 6, existence=True)
Unknown
sage: designs.transversal_design(3, 6, existence=True)
True
sage: designs.transversal_design(11, 10, existence=True)
False
sage: designs.transversal_design(4, 10, existence=True)
True
sage: designs.transversal_design(5, 10, existence=True)
Unknown
sage: designs.transversal_design(7, 20, existence=True)
Unknown
sage: designs.transversal_design(6, 12, existence=True)
True
sage: designs.transversal_design(7, 12, existence=True)
True
sage: designs.transversal_design(8, 12, existence=True)
Unknown
sage: designs.transversal_design(6, 20, existence = True)
True
sage: designs.transversal_design(7, 20, existence = True)
Unknown
If you ask for a transversal design that Sage is not able to build then an EmptySetError or a NotImplementedError is raised:
sage: designs.transversal_design(47, 100)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build a TD(47,100)!
sage: designs.transversal_design(55, 54)
Traceback (most recent call last):
...
EmptySetError: There exists no TD(55,54)!
Those two errors correspond respectively to the cases where Sage answer Unknown or False when the parameter existence is set to True:
sage: designs.transversal_design(47, 100, existence=True)
Unknown
sage: designs.transversal_design(55, 54, existence=True)
False
If for a given you want to know the largest
for which Sage is able
to build a
just call the function with
set to None and
existence set to True as follows:
sage: designs.transversal_design(None, 6, existence=True)
3
sage: designs.transversal_design(None, 20, existence=True)
6
sage: designs.transversal_design(None, 30, existence=True)
6
sage: designs.transversal_design(None, 120, existence=True)
9
TESTS:
The case when :
sage: designs.transversal_design(5,1).blocks()
[[0, 1, 2, 3, 4]]
Obtained through Wilson’s decomposition:
sage: _ = designs.transversal_design(4,38)
Obtained through product decomposition:
sage: _ = designs.transversal_design(6,60)
sage: _ = designs.transversal_design(5,60) # checks some tricky divisibility error
For small values of the parameter n we check the coherence of the function transversal_design():
sage: for n in xrange(2,25): # long time -- 15 secs
....: i = 2
....: while designs.transversal_design(i, n, existence=True) is True:
....: i += 1
....: _ = designs.transversal_design(i-1, n)
....: assert designs.transversal_design(None, n, existence=True) == i - 1
....: j = i
....: while designs.transversal_design(j, n, existence=True) is Unknown:
....: try:
....: _ = designs.transversal_design(j, n)
....: raise AssertionError("no NotImplementedError")
....: except NotImplementedError:
....: pass
....: j += 1
....: k = j
....: while k < n+4:
....: assert designs.transversal_design(k, n, existence=True) is False
....: try:
....: _ = designs.transversal_design(k, n)
....: raise AssertionError("no EmptySetError")
....: except EmptySetError:
....: pass
....: k += 1
....: print "%2d: (%2d, %2d)"%(n,i,j)
2: ( 4, 4)
3: ( 5, 5)
4: ( 6, 6)
5: ( 7, 7)
6: ( 4, 7)
7: ( 9, 9)
8: (10, 10)
9: (11, 11)
10: ( 5, 11)
11: (13, 13)
12: ( 8, 14)
13: (15, 15)
14: ( 7, 15)
15: ( 7, 17)
16: (18, 18)
17: (19, 19)
18: ( 8, 20)
19: (21, 21)
20: ( 7, 22)
21: ( 8, 22)
22: ( 6, 23)
23: (25, 25)
24: (10, 26)
The special case :
sage: designs.transversal_design(3, 1).blocks()
[[0, 1, 2]]
sage: designs.transversal_design(None, 1, existence=True)
+Infinity
sage: designs.transversal_design(None, 1)
Traceback (most recent call last):
...
ValueError: there is no upper bound on k when 0<=n<=1
Resolvable TD:
sage: k,n = 5,15
sage: TD = designs.transversal_design(k,n,resolvable=True)
sage: TD.is_resolvable()
True
sage: r = designs.transversal_design(None,n,resolvable=True,existence=True)
sage: non_r = designs.transversal_design(None,n,existence=True)
sage: r + 1 == non_r
True
Returns a from a truncated
by Wilson’s
construction.
Simple form:
Let be a truncated
with
truncated columns of sizes
, whose blocks have sizes in
. If there
exist:
Then there exists an . The construction is a
generalization of Lemma 3.16 in [HananiBIBD].
Brouwer-Van Rees form:
Let be a truncated
with
truncated columns of sizes
. Let the set
of the
points of column
be
partitionned into
. Let
be integers
such that:
Then there exists an . This construction appears
in [BvR82].
INPUT:
OA – an incomplete orthogonal array with columns. The elements
of a column of size
must belong to
. The missing entries
of a block are represented by None values. If OA=None, it is
defined as a truncated orthogonal arrays with
columns.
k,r,m (integers)
u (list) – two cases depending on the form to use:
- Simple form: a list of length
such that column k+i has size u[i]. The untruncated points of column k+i are assumed to be [0,...,u[i]-1].
- Brouwer-Van Rees form: a list of length
such that u[i] is the list of pairs
. The untruncated points of column k+i are assumed to be
where
. Besides, the first
points represent
, the next
points represent
, etc...
explain_construction (boolean) – return a string describing the construction.
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.
REFERENCE:
[HananiBIBD] | Balanced incomplete block designs and related designs, Haim Hanani, Discrete Mathematics 11.3 (1975) pages 255-369. |
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import wilson_construction
sage: from sage.combinat.designs.orthogonal_arrays import OA_relabel
sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_wilson_decomposition_with_one_truncated_group
sage: total = 0
sage: for k in range(3,8):
....: for n in range(1,30):
....: if find_wilson_decomposition_with_one_truncated_group(k,n):
....: total += 1
....: f, args = find_wilson_decomposition_with_one_truncated_group(k,n)
....: _ = f(*args)
sage: print total
41
sage: print designs.orthogonal_arrays.explain_construction(7,58)
Wilson's construction n=8.7+1+1 with master design OA(7+2,8)
sage: print designs.orthogonal_arrays.explain_construction(9,115)
Wilson's construction n=13.8+11 with master design OA(9+1,13)
sage: print wilson_construction(None,5,11,21,[[(5,5)]],explain_construction=True)
Brouwer-van Rees construction n=11.21+(5.5) with master design OA(5+1,11)
sage: print wilson_construction(None,71,17,21,[[(4,9),(1,1)],[(9,9),(1,1)]],explain_construction=True)
Brouwer-van Rees construction n=17.21+(9.4+1.1)+(9.9+1.1) with master design OA(71+2,17)
An example using the Brouwer-van Rees generalization:
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.orthogonal_arrays import wilson_construction
sage: OA = designs.orthogonal_arrays.build(6,11)
sage: OA = [[x if (i<5 or x<5) else None for i,x in enumerate(R)] for R in OA]
sage: OAb = wilson_construction(OA,5,11,21,[[(5,5)]])
sage: is_orthogonal_array(OAb,5,256)
True