Orthogonal arrays

This module gathers some construction related to orthogonal arrays (or transversal designs). One can build an \(OA(k,n)\) (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 \(k,n,t\).
transversal_design() Return a transversal design of parameters \(k,n\).
incomplete_orthogonal_array() Return an \(OA(k,n)-\sum_{1\leq i\leq x} OA(k,s_i)\).
is_transversal_design() Check that a given set of blocks B is a transversal design.
is_orthogonal_array() Check that the integer matrix \(OA\) is an \(OA(k,n,t)\).
wilson_construction() Return a \(OA(k,rm+u)\) from a truncated \(OA(k+s,r)\) by Wilson’s construction.
TD_product() Return the product of two transversal designs.
OA_find_disjoint_blocks() Return \(x\) disjoint blocks contained in a given \(OA(k,n)\).
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 \(V(m,t)\)
OA_from_PBD() Return an \(OA(k,n)\) from a PBD
OA_n_times_2_pow_c_from_matrix() Return an \(OA(k, \vert G\vert \cdot 2^c)\) from a constrained \((G,k-1,2)\)-difference matrix.
OA_from_wider_OA() Return the first \(k\) columns of \(OA\).
QDM_from_Vmt() Return a QDM a \(V(m,t)\)

REFERENCES:

[CD96]Making the MOLS table Charles Colbourn and Jeffrey Dinitz Computational and constructive design theory vol 368,pages 67-134 1996

Functions

class sage.combinat.designs.orthogonal_arrays.OAMainFunctions(*args, **kwds)

Functions related to orthogonal arrays.

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.

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 \(k\) for which Sage knows how to compute a \(OA(k,14,2)\)?:

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)!
static build(k, n, t=2, resolvable=False)

Return an \(OA(k,n)\) of strength \(t\)

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 article Orthogonal_array.

INPUT:

  • k,n,t (integers) – parameters of the orthogonal array.
  • resolvable (boolean) – set to True if you want the design to be resolvable. The \(n\) classes of the resolvable design are obtained as the first \(n\) blocks, then the next \(n\) blocks, etc ... Set to False by default.

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
static exists(k, n, t=2)

Return the existence status of an \(OA(k,n)\)

INPUT:

  • k,n,t (integers) – parameters of the orthogonal array.

Warning

The function does not only return booleans, but True, False, or Unknown.

See also

is_available()

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
static explain_construction(k, n, t=2)

Return a string describing how to builds an \(OA(k,n)\)

INPUT:

  • k,n,t (integers) – parameters of the orthogonal array.

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'
static is_available(k, n, t=2)

Return whether Sage can build an \(OA(k,n)\).

INPUT:

  • k,n,t (integers) – parameters of the orthogonal array.

See also

exists()

EXAMPLE:

sage: designs.orthogonal_arrays.is_available(3,6) # indirect doctest
True
sage: designs.orthogonal_arrays.is_available(4,6) # indirect doctest
False
static largest_available_k(n, t=2)

Return the largest \(k\) such that Sage can build an \(OA(k,n)\).

INPUT:

  • n (integer)
  • t – (integer; default: 2) – strength of the array

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
sage.combinat.designs.orthogonal_arrays.OA_find_disjoint_blocks(OA, k, n, x)

Return \(x\) disjoint blocks contained in a given \(OA(k,n)\).

\(x\) blocks of an \(OA\) 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 \(TD\) assciated with the \(OA\).

INPUT:

  • OA – an orthogonal array
  • k,n,x (integers)

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)
sage.combinat.designs.orthogonal_arrays.OA_from_PBD(k, n, PBD, check=True)

Return an \(OA(k,n)\) from a PBD

Construction

Let \(\mathcal B\) be a \((n,K,1)\)-PBD. If there exists for every \(i\in K\) a \(TD(k,i)-i\times TD(k,1)\) (i.e. if there exist \(k\) idempotent MOLS), then one can obtain a \(OA(k,n)\) by concatenating:

  • A \(TD(k,i)-i\times TD(k,1)\) defined over the elements of \(B\) for every \(B \in \mathcal B\).
  • The rows \((i,...,i)\) of length \(k\) for every \(i\in [n]\).

Note

This function raises an exception when Sage is unable to build the necessary designs.

INPUT:

  • k,n (integers)
  • PBD – a PBD on \(0,...,n-1\).

EXAMPLES:

We start from the example VI.1.2 from the [DesignHandbook] to build an \(OA(3,10)\):

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 \(OA(4,10)\) for this PBD (although there exists an \(OA(4,10)\):

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 \(OA(3,6)\) (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]
sage.combinat.designs.orthogonal_arrays.OA_from_Vmt(m, t, V)

Return an Orthogonal Array from a \(V(m,t)\)

INPUT:

  • m,t (integers)
  • V – the vector \(V(m,t)\).

EXAMPLES:

sage: _ = designs.orthogonal_arrays.build(6,46) # indirect doctest
sage.combinat.designs.orthogonal_arrays.OA_from_quasi_difference_matrix(M, G, add_col=True, fill_hole=True)

Return an Orthogonal Array from a Quasi-Difference matrix

Difference Matrices

Let \(G\) be a group of order \(g\). A difference matrix \(M\) is a \(g\times k\) matrix with entries from \(G\) such that for any \(1\leq i < j < k\) the set \(\{d_{li}-d_{lj}:1\leq l \leq g\}\) is equal to \(G\).

By concatenating the \(g\) matrices \(M+x\) (where \(x\in G\)), one obtains a matrix of size \(g^2\times x\) which is also an \(OA(k,g)\).

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 \(M\) are replaced by unique values on which \(G\) 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 \(M\)). If there exists an \(OA(k,u)\), then adding the rows of this \(OA(k,u)\) to the incomplete orthogonal array should lead to an OA...

Formal definition (from the Handbook of Combinatorial Designs [DesignHandbook])

Let \(G\) be an abelian group of order \(n\). A \((n,k;\lambda,\mu;u)\)-quasi-difference matrix (QDM) is a matrix \(Q=(q_{ij})\) with \(\lambda(n-1+2u)+\mu\) rows and \(k\) columns, with each entry either empty or containing an element of \(G\). Each column contains exactly \(\lambda u\) entries, and each row contains at most one empty entry. Furthermore, for each \(1 \leq i < j \leq k\) the multiset

\[\{ q_{li} - q_{lj}: 1 \leq l \leq \lambda (n-1+2u)+\mu, \text{ with }q_{li}\text{ and }q_{lj}\text{ not empty}\}\]

contains every nonzero element of \(G\) exactly \(\lambda\) times, and contains 0 exactly \(\mu\) times.

Construction

If a \((n,k;\lambda,\mu;u)\)-QDM exists and \(\mu \leq \lambda\), then an \(ITD_\lambda (k,n+u;u)\) exists. Start with a \((n,k;\lambda,\mu;u)\)-QDM \(A\) over the group \(G\). Append \(\lambda-\mu\) rows of zeroes. Then select \(u\) elements \(\infty_1,\dots,\infty_u\) not in \(G\), and replace the empty entries, each by one of these infinite symbols, so that \(\infty_i\) appears exactly once in each column. Develop the resulting matrix over the group \(G\) (leaving infinite symbols fixed), to obtain a \(\lambda (n^2+2nu)\times k\) matrix \(T\). Then \(T\) is an orthogonal array with \(k\) columns and index \(\lambda\), having \(n+u\) symbols and one hole of size \(u\).

Adding to \(T\) an \(OA(k,u)\) with elements \(\infty_1,\dots,\infty_u\) yields the \(ITD_\lambda(k,n+u;u)\).

For more information, see the Handbook of Combinatorial Designs [DesignHandbook] or http://web.cs.du.edu/~petr/milehigh/2013/Colbourn.pdf.

INPUT:

  • M – the difference matrix whose entries belong to G
  • G – a group
  • add_col (boolean) – whether to add a column to the final OA equal to \((x_1,\dots,x_g,x_1,\dots,x_g,\dots)\) where \(G=\{x_1,\dots,x_g\}\).
  • fill_hole (boolean) – whether to return the incomplete orthogonal array, or complete it with the \(OA(k,u)\) (default). When fill_hole is None, no block of the incomplete OA contains more than one value \(\geq |G|\).

EXAMPLES:

sage: _ = designs.orthogonal_arrays.build(6,20) # indirect doctest
sage.combinat.designs.orthogonal_arrays.OA_from_wider_OA(OA, k)

Return the first \(k\) columns of \(OA\).

If \(OA\) has \(k\) columns, this function returns \(OA\) immediately.

INPUT:

  • OA – an orthogonal array.
  • k (integer)

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
sage.combinat.designs.orthogonal_arrays.OA_n_times_2_pow_c_from_matrix(k, c, G, A, Y, check=True)

Return an \(OA(k, |G| \cdot 2^c)\) from a constrained \((G,k-1,2)\)-difference matrix.

This construction appears in [AbelCheng1994] and [AbelThesis].

Let \(G\) be an additive Abelian group. We denote by \(H\) a \(GF(2)\)-hyperplane in \(GF(2^c)\).

Let \(A\) be a \((k-1) \times 2|G|\) array with entries in \(G \times GF(2^c)\) and \(Y\) be a vector with \(k-1\) entries in \(GF(2^c)\). Let \(B\) and \(C\) be respectively the part of the array that belong to \(G\) and \(GF(2^c)\).

The input \(A\) and \(Y\) must satisfy the following conditions. For any \(i \neq j\) and \(g \in G\):

  • there are exactly two values of \(s\) such that \(B_{i,s} - B_{j,s} = g\) (i.e. \(B\) is a \((G,k-1,2)\)-difference matrix),
  • let \(s_1\) and \(s_2\) denote the two values of \(s\) given above, then exactly one of \(C_{i,s_1} - C_{j,s_1}\) and \(C_{i,s_2} - C_{j,s_2}\) belongs to the \(GF(2)\)-hyperplane \((Y_i - Y_j) \cdot H\) (we implicitely assumed that \(Y_i \not= Y_j\)).

Under these conditions, it is easy to check that the array whose \(k-1\) rows of length \(|G|\cdot 2^c\) indexed by \(1 \leq i \leq k-1\) given by \(A_{i,s} + (0, Y_i \cdot v)\) where \(1\leq s \leq 2|G|,v\in H\) is a \((G \times GF(2^c),k-1,1)\)-difference matrix.

INPUT:

  • k,c (integers) – integers
  • G – an additive Abelian group
  • A – a matrix with entries in \(G \times GF(2^c)\)
  • Y – a vector with entries in \(GF(2^c)\)
  • 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.

Note

By convention, a multiplicative generator \(w\) of \(GF(2^c)^*\) is fixed (inside the function). The hyperplane \(H\) is the one spanned by \(w^0, w^1, \ldots, w^{c-1}\). The \(GF(2^c)\) part of the input matrix \(A\) and vector \(Y\) are given in the following form: the integer \(i\) corresponds to the element \(w^i\) and None corresponds to \(0\).

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)
sage.combinat.designs.orthogonal_arrays.OA_relabel(OA, k, n, blocks=(), matrix=None)

Return a relabelled version of the OA.

INPUT:

  • OA – an OA, or rather a list of blocks of length \(k\), each of which contains integers from \(0\) to \(n-1\).

  • k,n (integers)

  • blocks (list of blocks) – relabels the integers of the OA from \([0..n-1]\) into \([0..n-1]\) in such a way that the \(i\) 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 \(k,n\) such that if the i th coordinate of a block is \(x\), this \(x\) will be relabelled with matrix[i][x]. This is not necessarily an integer between \(0\) and \(n-1\), 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 \(OA(4,3)\). 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
sage.combinat.designs.orthogonal_arrays.QDM_from_Vmt(m, t, V)

Return a QDM from a \(V(m,t)\)

Definition

Let \(q\) be a prime power and let \(q=mt+1\) for \(m,t\) integers. Let \(\omega\) be a primitive element of \(\mathbb{F}_q\). A \(V(m,t)\) vector is a vector \((a_1,\dots,a_{m+1}\) for which, for each \(1\leq k < m\), the differences

\[\{a_{i+k}-a_i:1\leq i \leq m+1,i+k\neq m+2\}\]

represent the \(m\) cyclotomic classes of \(\mathbb{F}_{mt+1}\) (compute subscripts modulo \(m+2\)). In other words, for fixed \(k\), is \(a_{i+k}-a_i=\omega^{mx+\alpha}\) and \(a_{j+k}-a_j=\omega^{my+\beta}\) then \(\alpha\not\equiv\beta \mod{m}\)

Construction of a quasi-difference matrix from a `V(m,t)` vector

Starting with a \(V(m,t)\) vector \((a_1,\dots,a_{m+1})\), form a single row of length \(m+2\) whose first entry is empty, and whose remaining entries are \((a_1,\dots,a_{m+1})\). Form \(t\) rows by multiplying this row by the \(t\) th roots, i.e. the powers of \(\omega^m\). From each of these \(t\) rows, form \(m+2\) rows by taking the \(m+2\) cyclic shifts of the row. The result is a \((a,m+2;1,0;t)-QDM\).

For more information, refer to the Handbook of Combinatorial Designs [DesignHandbook].

INPUT:

  • m,t (integers)
  • V – the vector \(V(m,t)\).

EXAMPLES:

sage: _ = designs.orthogonal_arrays.build(6,46) # indirect doctest
sage.combinat.designs.orthogonal_arrays.TD_product(k, TD1, n1, TD2, n2, check=True)

Return the product of two transversal designs.

From a transversal design \(TD_1\) of parameters \(k,n_1\) and a transversal design \(TD_2\) of parameters \(k,n_2\), this function returns a transversal design of parameters \(k,n\) where \(n=n_1\times n_2\).

Formally, if the groups of \(TD_1\) are \(V^1_1,\dots,V^1_k\) and the groups of \(TD_2\) are \(V^2_1,\dots,V^2_k\), the groups of the product design are \(V^1_1\times V^2_1,\dots,V^1_k\times V^2_k\) and its blocks are the \(\{(x^1_1,x^2_1),\dots,(x^1_k,x^2_k)\}\) where \(\{x^1_1,\dots,x^1_k\}\) is a block of \(TD_1\) and \(\{x^2_1,\dots,x^2_k\}\) is a block of \(TD_2\).

INPUT:

  • TD1, TD2 – transversal designs.
  • k,n1,n2 (integers) – see above.
  • 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.

Note

This function uses transversal designs with \(V_1=\{0,\dots,n-1\},\dots,V_k=\{(k-1)n,\dots,kn-1\}\) 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)
class sage.combinat.designs.orthogonal_arrays.TransversalDesign(blocks, k=None, n=None, check=True, **kwds)

Bases: sage.combinat.designs.incidence_structures.GroupDivisibleDesign

Class for Transversal Designs

INPUT:

  • blocks – collection of blocks
  • k,n (integers) – parameters of the transversal design. They can be set to None (default) in which case their value is determined by the blocks.
  • check (boolean) – whether to check that the design is indeed a transversal design with the right parameters. Set to True by default.

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)
sage.combinat.designs.orthogonal_arrays.incomplete_orthogonal_array(k, n, holes_sizes, resolvable=False, existence=False)

Return an \(OA(k,n)-\sum_{1\leq i\leq x} OA(k,s_i)\).

An \(OA(k,n)-\sum_{1\leq i\leq x} OA(k,s_i)\) is an orthogonal array from which have been removed disjoint \(OA(k,s_1),...,OA(k,s_x)\). If there exist \(OA(k,s_1),...,OA(k,s_x)\) they can be used to fill the holes and give rise to an \(OA(k,n)\).

A very useful particular case (see e.g. the Wilson construction in wilson_construction()) is when all \(s_i=1\). In that case the incomplete design is a \(OA(k,n)-x.OA(k,1)\). Such design is equivalent to transversal design \(TD(k,n)\) from which has been removed \(x\) 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. \(s_i=1\).

  • resolvable (boolean) – set to True if you want the design to be resolvable. The classes of the resolvable design are obtained as the first \(n\) blocks, then the next \(n\) 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 \(V = \{0, ..., n-1\}\) and the holes are \(\{n-1, ..., n-s_1\}^k\), \(\{n-s_1-1,...,n-s_1-s_2\}^k\), etc.

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 \(OA(k,n)-n.OA(k,1)\). We check that extending each class and adding the \([i,i,...]\) blocks turns it into an \(OA(k+1,n)\).:

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
sage.combinat.designs.orthogonal_arrays.is_transversal_design(B, k, n, verbose=False)

Check that a given set of blocks B is a transversal design.

See transversal_design() for a definition.

INPUT:

  • B – the list of blocks
  • k, n – integers
  • verbose (boolean) – whether to display information about what is going wrong.

Note

The tranversal design must have \(\{0, \ldots, kn-1\}\) as a ground set, partitioned as \(k\) sets of size \(n\): \(\{0, \ldots, k-1\} \sqcup \{k, \ldots, 2k-1\} \sqcup \cdots \sqcup \{k(n-1), \ldots, kn-1\}\).

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
sage.combinat.designs.orthogonal_arrays.largest_available_k(n, t=2)

Return the largest \(k\) such that Sage can build an \(OA(k,n)\).

INPUT:

  • n (integer)
  • t – (integer; default: 2) – strength of the array

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
sage.combinat.designs.orthogonal_arrays.orthogonal_array(k, n, t=2, resolvable=False, check=True, existence=False, explain_construction=False)

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 \(\lambda=1\).

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 \(n\) classes of the resolvable design are obtained as the first \(n\) blocks, then the next \(n\) 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 \(k\) such that we can build a \(OA(k,n)\).

  • explain_construction (boolean) – return a string describing the construction.

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.

See also

When \(t=2\) 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 \(n=0,1\):

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 \(t>2\) and \(k=None\):

sage: t = 3
sage: designs.orthogonal_arrays.largest_available_k(5,t=t) == t
True
sage: _ = designs.orthogonal_arrays.build(t,5,t)
sage.combinat.designs.orthogonal_arrays.transversal_design(k, n, resolvable=False, check=True, existence=False)

Return a transversal design of parameters \(k,n\).

A transversal design of parameters \(k, n\) is a collection \(\mathcal{S}\) of subsets of \(V = V_1 \cup \cdots \cup V_k\) (where the groups \(V_i\) are disjoint and have cardinality \(n\)) such that:

  • Any \(S \in \mathcal{S}\) has cardinality \(k\) and intersects each group on exactly one element.
  • Any two elements from distincts groups are contained in exactly one element of \(\mathcal{S}\).

More general definitions sometimes involve a \(\lambda\) parameter, and we assume here that \(\lambda=1\).

For more information on transversal designs, see http://mathworld.wolfram.com/TransversalDesign.html.

INPUT:

  • \(n,k\) – 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 \(n\) classes of the resolvable design are obtained as the first \(n\) blocks, then the next \(n\) 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 \(k\) 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 a \(TD(k,n)\) with \(V_1=\{0,\dots,n-1\},\dots,V_k=\{(k-1)n,\dots,kn-1\}\)
  • 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 \(TD(k,n)\).

See also

orthogonal_array() – a tranversal design \(TD(k,n)\) is equivalent to an orthogonal array \(OA(k,n,2)\).

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 \(TD(n+1,n)\):

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 \(n\) you want to know the largest \(k\) for which Sage is able to build a \(TD(k,n)\) just call the function with \(k\) 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 \(n=1\):

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 \(n=1\):

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
sage.combinat.designs.orthogonal_arrays.wilson_construction(OA, k, r, m, u, check=True, explain_construction=False)

Returns a \(OA(k,rm+\sum_i u_i)\) from a truncated \(OA(k+s,r)\) by Wilson’s construction.

Simple form:

Let \(OA\) be a truncated \(OA(k+s,r)\) with \(s\) truncated columns of sizes \(u_1,...,u_s\), whose blocks have sizes in \(\{k+b_1,...,k+b_t\}\). If there exist:

  • An \(OA(k,m+b_i) - b_i.OA(k,1)\) for every \(1\leq i\leq t\)
  • An \(OA(k,u_i)\) for every \(1\leq i\leq s\)

Then there exists an \(OA(k,rm+\sum u_i)\). The construction is a generalization of Lemma 3.16 in [HananiBIBD].

Brouwer-Van Rees form:

Let \(OA\) be a truncated \(OA(k+s,r)\) with \(s\) truncated columns of sizes \(u_1,...,u_s\). Let the set \(H_i\) of the \(u_i\) points of column \(k+i\) be partitionned into \(\sum_j H_{ij}\). Let \(m_{ij}\) be integers such that:

  • For \(0\leq i <l\) there exists an \(OA(k,\sum_j m_{ij}|H_{ij}|)\)
  • For any block \(B\in OA\) intersecting the sets \(H_{ij(i)}\) there exists an \(OA(k,m+\sum_i m_{ij})-\sum_i OA(k,m_{ij(j)})\).

Then there exists an \(OA(k,rm+\sum_{i,j}m_{ij})\). This construction appears in [BvR82].

INPUT:

  • OA – an incomplete orthogonal array with \(k+s\) columns. The elements of a column of size \(c\) must belong to \(\{0,...,c\}\). The missing entries of a block are represented by None values. If OA=None, it is defined as a truncated orthogonal arrays with \(k+s\) columns.

  • k,r,m (integers)

  • u (list) – two cases depending on the form to use:

    • Simple form: a list of length \(s\) 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 \(s\) such that u[i] is the list of pairs \((m_{i0},|H_{i0}|),...,(m_{ip_i},|H_{ip_i}|)\). The untruncated points of column k+i are assumed to be \([0,...,u_i-1]\) where \(u_i=\sum_j |H_{ip_i}|\). Besides, the first \(|H_{i0}|\) points represent \(H_{i0}\), the next \(|H_{i1}|\) points represent \(H_{i1}\), 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

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