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trac #16662: find_thwart_lemma_3_5
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videlec committed Aug 15, 2014
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120 changes: 0 additions & 120 deletions src/sage/combinat/designs/database.py
View file
Expand Up @@ -77,11 +77,6 @@
:func:`OA(10,796) <OA_10_796>`,
:func:`OA(15,896) <OA_15_896>`,
:func:`OA(33,993) <OA_33_993>`,
:func:`OA(10,1046) <OA_10_1046>`,
:func:`OA(10,1059) <OA_10_1059>`,
:func:`OA(11,2164) <OA_11_2164>`,
:func:`OA(12,3992) <OA_12_3992>`,
:func:`OA(12,3994) <OA_12_3994>`
- :func:`two MOLS of order 10 <MOLS_10_2>`,
:func:`five MOLS of order 12 <MOLS_12_5>`,
Expand Down Expand Up @@ -3268,116 +3263,6 @@ def OA_33_993():
M = OA_from_quasi_difference_matrix(Mb,G,add_col=True)
return M

def OA_10_1046():
r"""
Returns an `OA(10,1046)`.
Proved by Lemma 3.5 from [Thwarts]_.
EXAMPLES::
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_10_1046
sage: OA = OA_10_1046() # not tested -- around 5s
sage: print is_orthogonal_array(OA,10,1046,2) # not tested
True
The design is available from the general constructor::
sage: designs.orthogonal_array(10,1046,existence=True)
True
"""
from sage.combinat.designs.orthogonal_arrays_recursive import thwart_lemma_3_5
return thwart_lemma_3_5(10, 13, 78, 9, 9, 13, 1, complement=True)

def OA_10_1059():
r"""
Returns an `OA(10,1059)`.
Proved by Lemma 3.5 from [Thwarts]_.
EXAMPLES::
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_10_1059
sage: OA = OA_10_1059() # not tested -- around 6s
sage: print is_orthogonal_array(OA,10,1059,2) # not tested
True
The design is available from the general constructor::
sage: designs.orthogonal_array(10,1059,existence=True)
True
"""
from sage.combinat.designs.orthogonal_arrays_recursive import thwart_lemma_3_5
return thwart_lemma_3_5(10, 13, 79, 9, 9, 13, 1, complement=True)

def OA_11_2164():
r"""
Returns an `OA(11,2164)`.
Proved by Lemma 3.5 from [Thwarts]_.
EXAMPLES::
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_11_2164
sage: OA = OA_11_2164() # not tested -- around 15s
sage: print is_orthogonal_array(OA,11,2164,2) # not tested
True
The design is available from the general constructor::
sage: designs.orthogonal_array(10,2164,existence=True)
True
"""
from sage.combinat.designs.orthogonal_arrays_recursive import thwart_lemma_3_5
return thwart_lemma_3_5(11, 27, 78, 16, 17, 25, 0, complement=True)

def OA_12_3992():
r"""
Returns an `OA(12,3992)`.
Proved by Lemma 3.5 from [Thwarts]_.
EXAMPLES::
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_12_3992
sage: OA = OA_12_3992() # not tested -- around 60s
sage: print is_orthogonal_array(OA,12,3992,2) # not tested
True
The design is available from the general constructor::
sage: designs.orthogonal_array(12,3992,existence=True)
True
"""
from sage.combinat.designs.orthogonal_arrays_recursive import thwart_lemma_3_5
return thwart_lemma_3_5(12, 19, 208, 11, 13, 16, 0, complement=True)

def OA_12_3994():
r"""
Returns an `OA(12,3994)`.
Proved by Lemma 3.5 from [Thwarts]_.
EXAMPLES::
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_12_3994
sage: OA = OA_12_3994() # not tested -- around 60s
sage: print is_orthogonal_array(OA,12,3994,2) # not tested
True
The design is available from the general constructor::
sage: designs.orthogonal_array(12,3994,existence=True)
True
"""
from sage.combinat.designs.orthogonal_arrays_recursive import thwart_lemma_3_5
return thwart_lemma_3_5(12, 19, 208, 13, 13, 16, 0, complement=True)

def _helper_function_when_n_is_prime_times_power_of_2(k,n,A,Y):
r"""
This is an helper function to build `OA(k,p2^c)`
Expand Down Expand Up @@ -3503,11 +3388,6 @@ def _helper_function_when_n_is_prime_times_power_of_2(k,n,A,Y):
796 : (10 , OA_10_796),
896 : (15 , OA_15_896),
993 : (33 , OA_33_993),
1046: (10, OA_10_1046),
1059: (10, OA_10_1059),
2164: (11, OA_11_2164),
3992: (12, OA_12_3992),
3994: (12, OA_12_3994)
}

def CDF_21_5_1():
Expand Down
144 changes: 129 additions & 15 deletions src/sage/combinat/designs/orthogonal_arrays_recursive.py
View file
Expand Up @@ -38,6 +38,7 @@ def find_recursive_construction(k,n):
- :func:`construction_3_5`
- :func:`construction_3_6`
- :func:`construction_q_x`
- :func:`thwart_lemma_3_5`
INPUT:
Expand Down Expand Up @@ -72,7 +73,8 @@ def find_recursive_construction(k,n):
find_construction_3_4,
find_construction_3_5,
find_construction_3_6,
find_q_x]:
find_q_x,
find_thwart_lemma_3_5]:
res = find_c(k,n)
if res:
return res
Expand Down Expand Up @@ -923,6 +925,130 @@ def find_q_x(k,n):
return construction_q_x, (k,q,x)
return False

def find_thwart_lemma_3_5(k,N):
r"""
A function to find the values for which one can apply the
Lemma 3.5 from [Thwarts]_.
OUTPUT:
A pair ``(f,args)`` such that ``f(*args)`` returns an `OA(k,n)` or ``False``
if the construction is not available.
.. SEEALSO::
:func:`thwart_lemma_3_5`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays_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
"""
from sage.rings.arith import is_prime_power

k = int(k)
N = int(N)

for n in xrange(k+2, N):
if not is_prime_power(n):
continue

# we look for (m,n,a,b,c,d) with N = mn + a + b + c (+d) and
# 0 <= a,b,c,d <= n
# hence we have N/n-4 <= m <= N/n

# 1. look for m,a,b,c,d with complement=False
# (we restrict to a <= b <= c)
for m in xrange(max(k-1,(N+n-1)//n-4), N//n+1):
if not (orthogonal_array(k,m+0,existence=True) and
orthogonal_array(k,m+1,existence=True) and
orthogonal_array(k,m+2,existence=True)):
continue

NN = N - n*m
# as a >= b >= c and d <= n we can restrict the start of the loops
for a in range(max(0, (NN-n+2)//3), min(n, NN)+1):
if not orthogonal_array(k,a,existence=True):
continue
for b in range(max(0, (NN-n-a+1)//2), min(a, n+1-a, NN-a)+1):
if not orthogonal_array(k,b,existence=True):
continue
for c in range(max(0, NN-n-a-b), min(b, n+1-a-b, NN-a-b)+1):
if not orthogonal_array(k,c,existence=True):
continue

d = NN - (a + b + c) # necessarily 0 <= d <= n
if d == 0:
return thwart_lemma_3_5, (k,n,m,a,b,c,0,False)
elif (k+4 <= n+1 and
orthogonal_array(k,d,existence=True) and
orthogonal_array(k,m+3,existence=True)):
return thwart_lemma_3_5, (k,n,m,a,b,c,d,False)

# 2. look for m,a,b,c,d with complement=True
# (we restrict to a <= b <= c)
for m in xrange(max(k-2,N//n-4), (N+n-1)//n):
if not (orthogonal_array(k,m+1,existence=True) and
orthogonal_array(k,m+2,existence=True) and
orthogonal_array(k,m+3,existence=True)):
continue

NN = N - n*m
for a in range(max(0, (NN-n+2)//3), min(n, NN)+1):
if not orthogonal_array(k,a,existence=True):
continue
na = n-a
for b in range(max(0, (NN-n-a+1)//2), min(a, NN-a)+1):
nb = n-b
if na+nb > n+1 or not orthogonal_array(k,b,existence=True):
continue
for c in range(max(0, NN-n-a-b), min(b, NN-a-b)+1):
nc = n-c
if na+nb+nc > n+1 or not orthogonal_array(k,c,existence=True):
continue

d = NN - (a + b + c) # necessarily d <= n
if d == 0:
return thwart_lemma_3_5, (k,n,m,a,b,c,0,True)
elif (k+4 <= n+1 and
orthogonal_array(k,d,existence=True) and
orthogonal_array(k,m+4,existence=True)):
return thwart_lemma_3_5, (k,n,m,a,b,c,d,True)

return False

def thwart_lemma_3_5(k,n,m,a,b,c,d=0,complement=False):
r"""
Returns an `OA(k,nm+a+b+c+d)`
Expand Down Expand Up @@ -988,24 +1114,12 @@ def thwart_lemma_3_5(k,n,m,a,b,c,d=0,complement=False):
- ``complement`` (boolean) -- whether to complement the sets, i.e. follow
the `n-a,n-b,n-c` variant described above.
.. WARNING::
There is no "find" function associated with this recursive construction
as I was not able to write one which would not be unnecessarily ugly and
tricky (given that only 5 OA are built with it in the original
paper). Those designs appear in :mod:`sage.combinat.designs.database`
as: :func:`OA(10,1046) <sage.combinat.designs.database.OA_10_1046>`,
:func:`OA(10,1059) <sage.combinat.designs.database.OA_10_1059>`,
:func:`OA(11,2164) <sage.combinat.designs.database.OA_11_2164>`,
:func:`OA(12,3992) <sage.combinat.designs.database.OA_12_3992>`,
:func:`OA(12,3994) <sage.combinat.designs.database.OA_12_3994>`.
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_array_recursive import thwart_lemma_3_5
sage: from sage.combinat.designs.orthogonal_arrays_recursive import thwart_lemma_3_5
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: OA = thwart_lemma_3_5(6,23,7,5,7,8)
sage: is_orthogonal_array(OA,6,23*7+7+7+8)
sage: is_orthogonal_array(OA,6,23*7+5+7+8,2)
True
With sets of parameters from [Thwarts]_::
Expand Down

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