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 a recursive construction of an \(OA(k,n)\) (calls all others  | |
| Find \(n_1n_2=n\) to obtain an \(OA(k,n)\) by the product construction | |
| Find \(rm+u=n\) to obtain an \(OA(k,n)\) by Wilson’s construction with one truncated column. | |
| Find \(rm+r_1+r_2=n\) to obtain an \(OA(k,n)\) by Wilson’s construction with two truncated columns. | |
| Find a decomposition for construction 3.3 from [AC07]. | |
| Find a decomposition for construction 3.4 from [AC07]. | |
| Find a decomposition for construction 3.5 from [AC07]. | |
| Find a decomposition for construction 3.6 from [AC07]. | |
| Find integers \(q,x\) such that the \(q-x\) construction yields an \(OA(k,n)\). | |
| Find the values on which Lemma 3.5 from [Thwarts] applies. | |
| Find a decomposition for Lemma 4.1 from [Thwarts]. | |
| Find \(n_1n_2n_3=n\) to obtain an \(OA(k,n)\) by the three-factor product from [DukesLing14] | |
| Find \(t(q^2+q+1)+x=n\) to obtain an \(OA(k,n)\) by Brouwer’s separable design construction. | |
| Find \(rm+x_1+...+x_c=n\) such that the Brouwer-van Rees constructions yields a \(OA(k,n)\). | 
REFERENCES:
Functions¶
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_brouwer_separable_design(k, n)[source]¶
- 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().- EXAMPLES: - 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 - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_brouwer_separable_design >>> find_brouwer_separable_design(Integer(5),Integer(13))[Integer(1)] (5, 1, 3, 0) >>> find_brouwer_separable_design(Integer(5),Integer(14)) False 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_brouwer_van_rees_with_one_truncated_column(k, n)[source]¶
- 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
 - EXAMPLES: - 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)]]) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_brouwer_van_rees_with_one_truncated_column >>> find_brouwer_van_rees_with_one_truncated_column(Integer(5),Integer(53))[Integer(1)] (None, 5, 7, 7, [[(2, 1), (2, 1)]]) >>> find_brouwer_van_rees_with_one_truncated_column(Integer(6),Integer(96))[Integer(1)] (None, 6, 7, 13, [[(3, 1), (1, 1), (1, 1)]]) 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_construction_3_3(k, n)[source]¶
- Find a decomposition for construction 3.3 from [AC07]. - INPUT: - k,- n– integers
 - See also - OUTPUT: - A pair - f,argssuch 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) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_construction_3_3 >>> find_construction_3_3(Integer(11),Integer(177))[Integer(1)] (11, 11, 16, 1) >>> find_construction_3_3(Integer(12),Integer(11)) 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_construction_3_4(k, n)[source]¶
- Find a decomposition for construction 3.4 from [AC07]. - INPUT: - k,- n– integers
 - See also - OUTPUT: - A pair - f,argssuch 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) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_construction_3_4 >>> find_construction_3_4(Integer(8),Integer(196))[Integer(1)] (8, 25, 7, 12, 9) >>> find_construction_3_4(Integer(9),Integer(24)) 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_construction_3_5(k, n)[source]¶
- Find a decomposition for construction 3.5 from [AC07]. - INPUT: - k,- n– integers
 - See also - OUTPUT: - A pair - f,argssuch 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) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_construction_3_5 >>> find_construction_3_5(Integer(8),Integer(111))[Integer(1)] (8, 13, 6, 9, 11, 13) >>> find_construction_3_5(Integer(9),Integer(24)) 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_construction_3_6(k, n)[source]¶
- Find a decomposition for construction 3.6 from [AC07]. - INPUT: - k,- n– integers
 - See also - OUTPUT: - A pair - f,argssuch 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) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_construction_3_6 >>> find_construction_3_6(Integer(8),Integer(95))[Integer(1)] (8, 13, 7, 4) >>> find_construction_3_6(Integer(8),Integer(98)) 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_product_decomposition(k, n)[source]¶
- 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
 - OUTPUT: - A pair - f,argssuch that- f(*args)is an \(OA(k,n)\) or- Falseif 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) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_product_decomposition >>> f,args = find_product_decomposition(Integer(6), Integer(84)) >>> args (None, 6, 7, 12, (), False) >>> _ = f(*args) 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_q_x(k, n)[source]¶
- 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 - EXAMPLES: - 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) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_q_x >>> find_q_x(Integer(10),Integer(9)) False >>> find_q_x(Integer(9),Integer(158))[Integer(1)] (9, 16, 6) 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_recursive_construction(n)[source]¶
- 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: - INPUT: - k,- n– integers
 - OUTPUT: - Return a pair - f,argssuch that- f(*args)returns the requested \(OA\) if possible, and- Falseotherwise.- 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: count 56 - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_recursive_construction >>> from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array >>> count = Integer(0) >>> for n in range(Integer(10),Integer(150)): ... k = designs.orthogonal_arrays.largest_available_k(n) ... if find_recursive_construction(k,n): ... count = count + Integer(1) ... f,args = find_recursive_construction(k,n) ... OA = f(*args) ... assert is_orthogonal_array(OA,k,n,Integer(2),verbose=True) >>> count 56 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_three_factor_product(k, n)[source]¶
- 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 - OUTPUT: - A pair - f,argssuch 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 - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_three_factor_product >>> find_three_factor_product(Integer(10),Integer(648))[Integer(1)] (9, 8, 9, 9) >>> find_three_factor_product(Integer(10),Integer(50)) False 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_thwart_lemma_3_5(k, N)[source]¶
- 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- Falseif the construction is not available.- See also - 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 - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_thwart_lemma_3_5 >>> from sage.combinat.designs.designs_pyx import is_orthogonal_array >>> f,args = find_thwart_lemma_3_5(Integer(7),Integer(66)) >>> args (7, 9, 7, 1, 1, 1, 0, False) >>> OA = f(*args) >>> is_orthogonal_array(OA,Integer(7),Integer(66),Integer(2)) True >>> f,args = find_thwart_lemma_3_5(Integer(6),Integer(100)) >>> args (6, 8, 10, 8, 7, 5, 0, True) >>> OA = f(*args) >>> is_orthogonal_array(OA,Integer(6),Integer(100),Integer(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("{} {} {}".format(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 - >>> from sage.all import * >>> kn = ((Integer(10),Integer(1046)), (Integer(10),Integer(1048)), (Integer(10),Integer(1059)), (Integer(11),Integer(1524)), ... (Integer(11),Integer(2164)), (Integer(12),Integer(3362)), (Integer(12),Integer(3992)), (Integer(12),Integer(3994))) >>> for k,n in kn: ... print("{} {} {}".format(k,n,find_thwart_lemma_3_5(k,n)[Integer(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) >>> for k,n in kn: # not tested -- too long ... assert designs.orthogonal_array(k,n,existence=True) is True 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_thwart_lemma_4_1(k, n)[source]¶
- Find a decomposition for Lemma 4.1 from [Thwarts]. - INPUT: - k,- n– integers
 - See also - OUTPUT: - A pair - f,argssuch 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 - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_thwart_lemma_4_1 >>> find_thwart_lemma_4_1(Integer(10),Integer(408))[Integer(1)] (10, 13, 28) >>> find_thwart_lemma_4_1(Integer(10),Integer(50)) False 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_wilson_decomposition_with_one_truncated_group(k, n)[source]¶
- 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
 - OUTPUT: - A pair - f,argssuch that- f(*args)is an \(OA(k,n)\) or- Falseif 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 - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_wilson_decomposition_with_one_truncated_group >>> f,args = find_wilson_decomposition_with_one_truncated_group(Integer(4),Integer(38)) >>> args (None, 4, 5, 7, (3,), False) >>> _ = f(*args) >>> find_wilson_decomposition_with_one_truncated_group(Integer(4),Integer(20)) False 
- sage.combinat.designs.orthogonal_arrays_find_recursive.find_wilson_decomposition_with_two_truncated_groups(k, n)[source]¶
- 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
 - OUTPUT: - A pair - f,argssuch that- f(*args)is an \(OA(k,n)\) or- Falseif 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) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import find_wilson_decomposition_with_two_truncated_groups >>> f,args = find_wilson_decomposition_with_two_truncated_groups(Integer(5),Integer(58)) >>> args (None, 5, 7, 7, (4, 5), False) >>> _ = f(*args) 
- sage.combinat.designs.orthogonal_arrays_find_recursive.int_as_sum(value, S, k_max)[source]¶
- 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– list of integers
- k_max– integer
 - EXAMPLES: - 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("{} {}".format(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) - >>> from sage.all import * >>> from sage.combinat.designs.orthogonal_arrays_find_recursive import int_as_sum >>> D = int_as_sum(Integer(21),[Integer(5),Integer(12)],Integer(100)) >>> for k in range(Integer(20),Integer(40)): ... print("{} {}".format(k, int_as_sum(k,[Integer(5),Integer(12)],Integer(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)