### Navigation

- index
- modules |
- next |
- previous |
- Combinatorics »
- Designs and Incidence Structures »

# Orthogonal arrays (find recursive constructions)

Orthogonal arrays (find recursive constructions)

This module implements several functions to find recursive constructions of `Orthogonal Arrays`.

The main function of this module, i.e. `find_recursive_construction()`, queries all implemented recursive constructions of designs implemented in `orthogonal_arrays_build_recursive` in order to obtain an \(OA(k,n)\).

`find_recursive_construction()` is called by the `orthogonal_array()` function.

`find_recursive_construction()` | Find a recursive construction of an \(OA(k,n)\) (calls all others `find_*` functions)
`find_product_decomposition()` | Find \(n_1n_2=n\) to obtain an \(OA(k,n)\) by the product construction
`find_wilson_decomposition_with_one_truncated_group()` | Find \(rm+u=n\) to obtain an \(OA(k,n)\) by Wilson’s construction with one truncated column.
`find_wilson_decomposition_with_two_truncated_groups()` | Find \(rm+r_1+r_2=n\) to obtain an \(OA(k,n)\) by Wilson’s construction with two truncated columns.
`find_construction_3_3()` | Find a decomposition for construction 3.3 from [AC07].
`find_construction_3_4()` | Find a decomposition for construction 3.4 from [AC07].
`find_construction_3_5()` | Find a decomposition for construction 3.5 from [AC07].
`find_construction_3_6()` | Find a decomposition for construction 3.6 from [AC07].
`find_q_x()` | Find integers \(q,x\) such that the \(q-x\) construction yields an \(OA(k,n)\).
`find_thwart_lemma_3_5()` | Find the values on which Lemma 3.5 from [Thwarts] applies.
`find_thwart_lemma_4_1()` | Find a decomposition for Lemma 4.1 from [Thwarts].
`find_three_factor_product()` | Find \(n_1n_2n_3=n\) to obtain an \(OA(k,n)\) by the three-factor product from [DukesLing14]
`find_brouwer_separable_design()` | Find \(t(q^2+q+1)+x=n\) to obtain an \(OA(k,n)\) by Brouwer’s separable design construction.
`find_brouwer_van_rees_with_one_truncated_column()` | Find \(rm+x_1+...+x_c=n\) such that the Brouwer-van Rees constructions yields a \(OA(k,n)\).

REFERENCES:

[AC07] | (1, 2, 3, 4, 5, 6, 7, 8) Concerning eight mutually orthogonal latin squares Julian R. Abel, Nicholas Cavenagh Journal of Combinatorial Designs Vol. 15, n.3, pp. 255-261 2007

## Functions

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_brouwer_separable_design`(k, n)

  Find \(t(q^2+q+1)+x=n\) to obtain an \(OA(k,n)\) by Brouwer’s separable design construction.

  INPUT:

  - `k,n` (integers)

  The assumptions made on the parameters \(t,q,x\) are explained in the documentation of `brouwer_separable_design()`.

  EXAMPLE:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_brouwer_separable_design
      sage: find_brouwer_separable_design(5,13)[1]
      (5, 1, 3, 0)
      sage: find_brouwer_separable_design(5,14)
      False

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_brouwer_van_rees_with_one_truncated_column`(k, n)

  Find \(rm+x_1+...+x_c=n\) such that the Brouwer-van Rees constructions yields a \(OA(k,n)\).

  Let \(n=rm+\sum_{1\leq i\leq c}\) such that \(c\leq r\). The generalization of Wilson’s construction found by Brouwer and van Rees (with one truncated column) ensures that an \(OA(k,n)\) exists if the following designs exist: \(OA(k+1,r)\), \(OA(k,m)\), \(OA(k,\sum_{1\leq i\leq c} u_i)\), \(OA(k,m+x_1)-OA(k,x_1)\), ..., \(OA(k,m+x_c)-OA(k,x_c)\).

  For more information, see the documentation of `wilson_construction()`.

  INPUT:

  - `k,n` (integers)

  EXAMPLE:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_brouwer_van_rees_with_one_truncated_column
      sage: find_brouwer_van_rees_with_one_truncated_column(5,53)[1]
      (None, 5, 7, 7, [[(2, 1), (2, 1)]])
      sage: find_brouwer_van_rees_with_one_truncated_column(6,96)[1]
      (None, 6, 7, 13, [[(3, 1), (1, 1), (1, 1)]])

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_construction_3_3`(k, n)

  Find a decomposition for construction 3.3 from [AC07]

  INPUT:

  - `k,n` (integers)

  See also

  `construction_3_3()`

  OUTPUT:

  A pair `f,args` such that `f(*args)` returns the requested OA.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_construction_3_3
      sage: find_construction_3_3(11,177)[1]
      (11, 11, 16, 1)
      sage: find_construction_3_3(12,11)

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_construction_3_4`(k, n)

  Find a decomposition for construction 3.4 from [AC07]

  INPUT:

  - `k,n` (integers)

  See also

  `construction_3_4()`

  OUTPUT:

  A pair `f,args` such that `f(*args)` returns the requested OA.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_construction_3_4
      sage: find_construction_3_4(8,196)[1]
      (8, 25, 7, 12, 9)
      sage: find_construction_3_4(9,24)

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_construction_3_5`(k, n)

  Find a decomposition for construction 3.5 from [AC07]

  INPUT:

  - `k,n` (integers)

  See also

  `construction_3_5()`

  OUTPUT:

  A pair `f,args` such that `f(*args)` returns the requested OA.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_construction_3_5
      sage: find_construction_3_5(8,111)[1]
      (8, 13, 6, 9, 11, 13)
      sage: find_construction_3_5(9,24)

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_construction_3_6`(k, n)

  Find a decomposition for construction 3.6 from [AC07]

  INPUT:

  - `k,n` (integers)

  See also

  `construction_3_6()`

  OUTPUT:

  A pair `f,args` such that `f(*args)` returns the requested OA.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_construction_3_6
      sage: find_construction_3_6(8,95)[1]
      (8, 13, 7, 4)
      sage: find_construction_3_6(8,98)

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_product_decomposition`(k, n)

  Find \(n_1n_2=n\) to obtain an \(OA(k,n)\) by the product construction.

  If Sage can build a \(OA(k,n_1)\) and a \(OA(k,n_2)\) such that \(n=n_1\times n_2\) then a \(OA(k,n)\) can be built by a product construction (which correspond to Wilson’s construction with no truncated column). This function look for a pair of integers \((n_1,n_2)\) with \(n1 \leq n_2\), \(n_1 \times n_2 = n\) and such that both an \(OA(k,n_1)\) and an \(OA(k,n_2)\) are available.

  INPUT:

  - `k,n` (integers) – see above.

  OUTPUT:

  A pair `f,args` such that `f(*args)` is an \(OA(k,n)\) or `False` if no product decomposition was found.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_product_decomposition
      sage: f,args = find_product_decomposition(6, 84)
      sage: args
      (None, 6, 7, 12, (), False)
      sage: _ = f(*args)

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_q_x`(k, n)

  Find integers \(q,x\) such that the \(q-x\) construction yields an \(OA(k,n)\).

  See the documentation of `construction_q_x()` to find out what hypotheses the integers \(q,x\) must satisfy.

  Warning

  For efficiency reasons, this function checks that Sage can build an \(OA(k+1,q-x-1)\) and an \(OA(k+1,q-x+1)\), which is stronger than checking that Sage can build a \(OA(k,q-x-1)-(q-x-1).OA(k,1)\) and a \(OA(k,q-x+1)-(q-x+1).OA(k,1)\). The latter would trigger a lot of independent set computations in `sage.combinat.designs.orthogonal_arrays.incomplete_orthogonal_array()`.

  INPUT:

  - `k,n` (integers)

  See also

  `construction_q_x()`

  EXAMPLE:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_q_x
      sage: find_q_x(10,9)
      False
      sage: find_q_x(9,158)[1]
      (9, 16, 6)

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_recursive_construction`(k, n)

  Find a recursive construction of an \(OA(k,n)\) (calls all others `find_*` functions)

  This determines whether an \(OA(k,n)\) can be built through the following constructions:

  - `wilson_construction()`
  - `construction_3_3()`
  - `construction_3_4()`
  - `construction_3_5()`
  - `construction_3_6()`
  - `construction_q_x()`
  - `thwart_lemma_3_5()`
  - `thwart_lemma_4_1()`
  - `three_factor_product()`
  - `brouwer_separable_design()`
  - `find_brouwer_van_rees_with_one_truncated_column()`

  INPUT:

  - `k,n` (integers)

  OUTPUT:

  Return a pair `f,args` such that `f(*args)` returns the requested \(OA\) if possible, and `False` otherwise.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_recursive_construction
      sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
      sage: count = 0
      sage: for n in range(10,150):
      ....:     k = designs.orthogonal_arrays.largest_available_k(n)
      ....:     if find_recursive_construction(k,n):
      ....:         count = count + 1
      ....:         f,args = find_recursive_construction(k,n)
      ....:         OA = f(*args)
      ....:         assert is_orthogonal_array(OA,k,n,2,verbose=True)
      sage: print count
      56

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_three_factor_product`(k, n)

  Find \(n_1n_2n_3=n\) to obtain an \(OA(k,n)\) by the three-factor product from [DukesLing14]

  INPUT:

  - `k,n` (integers)

  See also

  `three_factor_product()`

  OUTPUT:

  A pair `f,args` such that `f(*args)` returns the requested OA.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_three_factor_product
      sage: find_three_factor_product(10,648)[1]
      (9, 8, 9, 9)
      sage: find_three_factor_product(10,50)
      False

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_thwart_lemma_3_5`(k, N)

  Find the values on which Lemma 3.5 from [Thwarts] applies.

  OUTPUT:

  A pair `(f,args)` such that `f(*args)` returns an \(OA(k,n)\) or `False` if the construction is not available.

  See also

  `thwart_lemma_3_5()`

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_thwart_lemma_3_5
      sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array

      sage: f,args = find_thwart_lemma_3_5(7,66)
      sage: args
      (7, 9, 7, 1, 1, 1, 0, False)
      sage: OA = f(*args)
      sage: is_orthogonal_array(OA,7,66,2)
      True

      sage: f,args = find_thwart_lemma_3_5(6,100)
      sage: args
      (6, 8, 10, 8, 7, 5, 0, True)
      sage: OA = f(*args)
      sage: is_orthogonal_array(OA,6,100,2)
      True

  Some values from [Thwarts]:

      sage: kn = ((10,1046), (10,1048), (10,1059), (11,1524),
      ....:       (11,2164), (12,3362), (12,3992),  (12,3994))
      sage: for k,n in kn:
      ....:     print k,n,find_thwart_lemma_3_5(k,n)[1]
      10 1046 (10, 13, 79, 9, 1, 0, 9, False)
      10 1048 (10, 13, 79, 9, 1, 0, 11, False)
      10 1059 (10, 13, 80, 9, 1, 0, 9, False)
      11 1524 (11, 19, 78, 16, 13, 13, 0, True)
      11 2164 (11, 27, 78, 23, 19, 16, 0, True)
      12 3362 (12, 16, 207, 13, 13, 11, 13, True)
      12 3992 (12, 19, 207, 16, 13, 11, 19, True)
      12 3994 (12, 19, 207, 16, 13, 13, 19, True)

      sage: for k,n in kn:                                                     # not tested -- too long
      ....:     assert designs.orthogonal_array(k,n,existence=True) is True    # not tested -- too long

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_thwart_lemma_4_1`(k, n)

  Find a decomposition for Lemma 4.1 from [Thwarts].

  INPUT:

  - `k,n` (integers)

  See also

  `thwart_lemma_4_1()`

  OUTPUT:

  A pair `f,args` such that `f(*args)` returns the requested OA.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_thwart_lemma_4_1
      sage: find_thwart_lemma_4_1(10,408)[1]
      (10, 13, 28)
      sage: find_thwart_lemma_4_1(10,50)
      False

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_wilson_decomposition_with_one_truncated_group`(k, n)

  Find \(rm+u=n\) to obtain an \(OA(k,n)\) by Wilson’s construction with one truncated column.

  This function looks for possible integers \(m,t,u\) satisfying that \(mt+u=n\) and such that Sage knows how to build a \(OA(k,m)\), \(OA(k,m+1)\), \(OA(k+1,t)\) and a \(OA(k,u)\).

  INPUT:

  - `k,n` (integers) – see above

  OUTPUT:

  A pair `f,args` such that `f(*args)` is an \(OA(k,n)\) or `False` if no decomposition with one truncated block was found.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_wilson_decomposition_with_one_truncated_group
      sage: f,args = find_wilson_decomposition_with_one_truncated_group(4,38)
      sage: args
      (None, 4, 5, 7, (3,), False)
      sage: _ = f(*args)

      sage: find_wilson_decomposition_with_one_truncated_group(4,20)
      False

`sage.combinat.designs.orthogonal_arrays_find_recursive.``find_wilson_decomposition_with_two_truncated_groups`(k, n)

  Find \(rm+r_1+r_2=n\) to obtain an \(OA(k,n)\) by Wilson’s construction with two truncated columns.

  Look for integers \(r,m,r_1,r_2\) satisfying \(n=rm+r_1+r_2\) and \(1\leq r_1,r_2<r\) and such that the following designs exist : \(OA(k+2,r)\), \(OA(k,r1)\), \(OA(k,r2)\), \(OA(k,m)\), \(OA(k,m+1)\), \(OA(k,m+2)\).

  INPUT:

  - `k,n` (integers) – see above

  OUTPUT:

  A pair `f,args` such that `f(*args)` is an \(OA(k,n)\) or `False` if no decomposition with two truncated blocks was found.

  EXAMPLES:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_wilson_decomposition_with_two_truncated_groups
      sage: f,args = find_wilson_decomposition_with_two_truncated_groups(5,58)
      sage: args
      (None, 5, 7, 7, (4, 5), False)
      sage: _ = f(*args)

`sage.combinat.designs.orthogonal_arrays_find_recursive.``int_as_sum`(value, S, k_max)

  Return a tuple \((s_1, s_2, \ldots, s_k)\) of less then \(k_max\) elements of \(S\) such that \(value = s_1 + s_2 + \ldots + s_k\). If there is no such tuples then the function returns `None`.

  INPUT:

  - `value` (integer)
  - `S` – a list of integers
  - `k_max` (integer)

  EXAMPLE:

      sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import int_as_sum
      sage: D = int_as_sum(21,[5,12],100)
      sage: for k in range(20,40):
      ....:     print k, int_as_sum(k,[5,12],100)
      20 (5, 5, 5, 5)
      21 None
      22 (12, 5, 5)
      23 None
      24 (12, 12)
      25 (5, 5, 5, 5, 5)
      26 None
      27 (12, 5, 5, 5)
      28 None
      29 (12, 12, 5)
      30 (5, 5, 5, 5, 5, 5)
      31 None
      32 (12, 5, 5, 5, 5)
      33 None
      34 (12, 12, 5, 5)
      35 (5, 5, 5, 5, 5, 5, 5)
      36 (12, 12, 12)
      37 (12, 5, 5, 5, 5, 5)
      38 None
      39 (12, 12, 5, 5, 5)

### Navigation

- index
- modules |
- next |
- previous |
- Combinatorics »
- Designs and Incidence Structures »