Orthogonal arrays (OA)#
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:
Return an orthogonal array of parameters \(k,n,t\). 

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

Return an \(OA(k,n)\sum_{1\leq i\leq x} OA(k,s_i)\). 
Check that a given set of blocks 

Check that the integer matrix \(OA\) is an \(OA(k,n,t)\). 

Return a \(OA(k,rm+u)\) from a truncated \(OA(k+s,r)\) by Wilson’s construction. 

Return the product of two transversal designs. 

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

Return a relabelled version of the OA. 

Return an Orthogonal Array from a QuasiDifference matrix 

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

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

Return an \(OA(k, \vert G\vert \cdot 2^c)\) from a constrained \((G,k1,2)\)difference matrix. 

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

Return a QDM a \(V(m,t)\) 
REFERENCES:
– [CD1996]
Functions#
 class sage.combinat.designs.orthogonal_arrays.OAMainFunctions(*args, **kwds)#
Bases:
object
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:
build(k,n,t=2)
: return an orthogonal array with the given parameters.is_available(k,n,t=2)
: answer whether there is a construction available in Sage for a given set of parameters.exists(k,n,t=2)
: answer whether an orthogonal array with these parameters exist.largest_available_k(n,t=2)
: return the largest integer \(k\) such that Sage knows how to build an \(OA(k,n)\).explain_construction(k,n,t=2)
: return a string that explains the construction that Sage uses to build an \(OA(k,n)\).
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 aNotImplementedError
:sage: designs.orthogonal_arrays.build(4,2) Traceback (most recent call last): ... EmptySetError: There exists no OA(4,2) as k(=4)>n+t1=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 toTrue
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 toFalse
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
, orUnknown
.See also
EXAMPLES:
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.
EXAMPLES:
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
EXAMPLES:
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
EXAMPLES:
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, solver, integrality_tolerance)#
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 \(TD\) associated with the \(OA\).
INPUT:
OA
– an orthogonal arrayk
,n
,x
(integers)solver
– (default:None
) Specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.integrality_tolerance
– parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
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)
 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 \(0,...,n1\).
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 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 QuasiDifference 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 \(OA(k,g)\).
Quasidifference Matrices
A quasidifference 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 \((n,k;\lambda,\mu;u)\)quasidifference matrix (QDM) is a matrix \(Q=(q_{ij})\) with \(\lambda(n1+2u)+\mu\) 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
\[\{ q_{li}  q_{lj}: 1 \leq l \leq \lambda (n1+2u)+\mu, \text{ with }q_{li}\text{ and }q_{lj}\text{ not empty}\}\]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 toG
G
– a groupadd_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). Whenfill_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 \((G,k1,2)\)difference matrix.
This construction appears in [AC1994] and [Ab1995].
Let \(G\) be an additive Abelian group. We denote by \(H\) a \(GF(2)\)hyperplane in \(GF(2^c)\).
Let \(A\) be a \((k1) \times 2G\) array with entries in \(G \times GF(2^c)\) and \(Y\) be a vector with \(k1\) 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 \((G,k1,2)\)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 implicitly assumed that \(Y_i \not= Y_j\)).
Under these conditions, it is easy to check that the array whose \(k1\) rows of length \(G\cdot 2^c\) indexed by \(1 \leq i \leq k1\) given by \(A_{i,s} + (0, Y_i \cdot v)\) where \(1\leq s \leq 2G,v\in H\) is a \((G \times GF(2^c),k1,1)\)difference matrix.
INPUT:
k,c
(integers) – integersG
– an additive Abelian groupA
– 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 toTrue
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 \(w^0, w^1, \ldots, w^{c1}\). 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 \(0\).See also
Several examples use this construction:
EXAMPLES:
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,k1,2)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
 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 \(n1\).k,n
(integers)blocks
(list of blocks) – relabels the integers of the OA from \([0..n1]\) into \([0..n1]\) in such a way that the \(i\) blocks fromblock
are respectively relabeled as[ni,...,ni]
, …,[n1,...,n1]
. 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 withmatrix[i][x]
. This is not necessarily an integer between \(0\) and \(n1\), 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 aNone
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]]
 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 \(\GF{q}\). 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 \(\GF{mt+1}\) (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 quasidifference 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 \(V(m,t)\).
See also
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 toTrue
by default.
Note
This function uses transversal designs with \(V_1=\{0,\dots,n1\},\dots,V_k=\{(k1)n,\dots,kn1\}\) both as input and output.
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.group_divisible_designs.GroupDivisibleDesign
Class for Transversal Designs
INPUT:
blocks
– collection of blocksk,n
(integers) – parameters of the transversal design. They can be set toNone
(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 toTrue
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, 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
(list of integers) – respective sizes of the holes to be found.resolvable
(boolean) – set toTrue
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 toFalse
by default.existence
(boolean) – instead of building the design, return:True
– meaning that Sage knows how to build the designUnknown
– meaning that Sage does not know how to build the design, but that the design may exist (seesage.misc.unknown
).False
– meaning that the design does not exist.
Note
By convention, the ground set is always \(V = \{0, ..., n1\}\).
If all holes have size 1, in the incomplete orthogonal array returned by this function the holes are \(\{n1, ..., ns_1\}^k\), \(\{ns_11,...,ns_1s_2\}^k\), etc.
More generally, if
holes
is equal to \(u1,...,uk\), the \(i\)th hole is the set of points \(\{n\sum_{j\geq i}u_j,...,n\sum_{j\geq i+1}u_j\}^k\).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
 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 blocksk, n
– integersverbose
(boolean) – whether to display information about what is going wrong.
Note
The transversal design must have \(\{0, \ldots, kn1\}\) as a ground set, partitioned as \(k\) sets of size \(n\): \(\{0, \ldots, k1\} \sqcup \{k, \ldots, 2k1\} \sqcup \cdots \sqcup \{k(n1), \ldots, kn1\}\).
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
EXAMPLES:
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. Ifk=None
it is set to the largest value available.n
– (integer) number of symbolst
– (integer; default: 2) – strength of the arrayresolvable
(boolean) – set toTrue
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 toFalse
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 toTrue
by default.existence
(boolean) – instead of building the design, return:True
– meaning that Sage knows how to build the designUnknown
– meaning that Sage does not know how to build the design, but that the design may exist (seesage.misc.unknown
).False
– meaning that the design does not exist.
Note
When
k=None
andexistence=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 parametersk
andn
if
existence=True
andk
is an integer, then the function returns a troolean: eitherTrue
,Unknown
orFalse
if
existence=True
andk=None
then the output is the largest value ofk
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 (seemutually_orthogonal_latin_squares()
).
 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 toTrue
if you want the design to be resolvable (seesage.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 toFalse
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 toTrue
by default.existence
(boolean) – instead of building the design, return:True
– meaning that Sage knows how to build the designUnknown
– meaning that Sage does not know how to build the design, but that the design may exist (seesage.misc.unknown
).False
– meaning that the design does not exist.
Note
When
k=None
andexistence=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 \(V_1=\{0,\dots,n1\},\dots,V_k=\{(k1)n,\dots,kn1\}\)if
existence=True
andk
is an integer, then the function returns a troolean: eitherTrue
,Unknown
orFalse
if
existence=True
andk=None
then the output is the largest value ofk
for which Sage knows how to compute a \(TD(k,n)\).
See also
orthogonal_array()
– a transversal 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 aNotImplementedError
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
orFalse
when the parameterexistence
is set toTrue
: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
andexistence
set toTrue
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
 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].
BrouwerVan 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 [BvR1982].
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 byNone
values. IfOA=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 sizeu[i]
. The untruncated points of columnk+i
are assumed to be[0,...,u[i]1]
.BrouwerVan 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 columnk+i
are assumed to be \([0,...,u_i1]\) 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 toTrue
by default.
REFERENCE:
 HananiBIBD
Balanced incomplete block designs and related designs, Haim Hanani, Discrete Mathematics 11.3 (1975) pages 255369.
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: 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)) Brouwervan 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)) Brouwervan Rees construction n=17.21+(9.4+1.1)+(9.9+1.1) with master design OA(71+2,17)
An example using the Brouwervan 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