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trac #16361: OA(7,66), OA(7,68), OA(8,69), OA(7,74) and OA(8,76)
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nathanncohen committed May 16, 2014
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279 changes: 276 additions & 3 deletions src/sage/combinat/designs/database.py
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Expand Up @@ -8,6 +8,9 @@
<sage.combinat.designs.latin_squares>` from the Handbook of Combinatorial
Designs.
Most of this would only be a dream without the mathematical knowledge and help
of Julian R. Abel.
All the designs returned by these functions can be obtained through the
``designs.<tab>`` functions.
Expand Down Expand Up @@ -42,7 +45,12 @@
:func:`OA(9,57) <OA_9_57>`,
:func:`OA(7,60) <OA_7_60>`,
:func:`OA(7,62) <OA_7_62>`,
:func:`OA(7,66) <OA_7_66>`,
:func:`OA(7,68) <OA_7_68>`,
:func:`OA(8,69) <OA_8_69>`,
:func:`OA(7,74) <OA_7_74>`,
:func:`OA(9,75) <OA_9_75>`,
:func:`OA(8,76) <OA_8_76>`,
:func:`OA(11,80) <OA_11_80>`,
:func:`OA(10,82) <OA_10_82>`,
:func:`OA(10,100) <OA_10_100>`,
Expand Down Expand Up @@ -84,7 +92,9 @@

from sage.combinat.designs.orthogonal_arrays import (OA_from_quasi_difference_matrix,
OA_from_Vmt,
OA_from_wider_OA)
OA_from_wider_OA,
OA_from_PBD,
orthogonal_array)

# Cyclic shift of a list
cyclic_shift = lambda l,i : l[-i:]+l[:-i]
Expand Down Expand Up @@ -1710,7 +1720,6 @@ def OA_9_57():
sage: designs.orthogonal_array(9,57,existence=True)
True
"""
from orthogonal_arrays import orthogonal_array
M = orthogonal_array(8,8)
M = [R for R in M if any(x!=R[0] for x in R)] # removing the 0..0, 1..1, 7..7 rows.
B = (1,6,7,9,19,38,42,49) # base block of a (57,8,1) BIBD
Expand Down Expand Up @@ -1849,7 +1858,6 @@ def OA_9_65():
True
"""
from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as G
from orthogonal_arrays import orthogonal_array

B = [None,1, 6, 7, 9, 19, 38, 42, 49] # Base block of a (57,8,1)-BIBD
OA = orthogonal_array(9,9,2)
Expand All @@ -1861,6 +1869,199 @@ def OA_9_65():
M = OA_from_quasi_difference_matrix(zip(*M), G(57),add_col=False)
return M

def OA_7_66():
r"""
Returns an OA(7,66)
Construction shared by Julian R. Abel.
.. SEEALSO::
:func:`sage.combinat.designs.orthogonal_arrays.OA_from_PBD`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_7_66
sage: OA = OA_7_66()
sage: print is_orthogonal_array(OA,7,66,2)
True
The design is available from the general constructor::
sage: designs.orthogonal_array(7,66,existence=True)
True
"""

# base block of a (73,9,1) BIBD
B = [0, 19, 26, 14, 63, 15, 32, 35, 65]
# The corresponding BIBD
BIBD= [[(x+i)%73 for x in B] for i in range(73)]
# the first 7 elements of an oval
#
# (this is the only difference with the OA(7,68) construction)
oval = [(-x)%73 for x in B][:7]
# PBD minus the oval
PBD = [[x for x in B if x not in oval] for B in BIBD]
# We relabel the points to 0,1,2,...
V = [x for x in range(73) if x not in oval]
rel = dict(zip(V,range(len(V))))
PBD = [[rel[x] for x in B] for B in PBD]
return OA_from_PBD(7,66,PBD,check=False)

def OA_7_68():
r"""
Returns an OA(7,68)
Construction shared by Julian R. Abel.
.. SEEALSO::
:func:`sage.combinat.designs.orthogonal_arrays.OA_from_PBD`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_7_68
sage: OA = OA_7_68()
sage: print is_orthogonal_array(OA,7,68,2)
True
The design is available from the general constructor::
sage: designs.orthogonal_array(7,68,existence=True)
True
"""

# base block of a (73,9,1) BIBD
B = [0, 19, 26, 14, 63, 15, 32, 35, 65]
# The corresponding BIBD
BIBD= [[(x+i)%73 for x in B] for i in range(73)]
# the first 5 elements of an oval
#
# (this is the only difference with the OA(7,66) construction)
oval = [(-x)%73 for x in B][:5]
# PBD minus the oval
PBD = [[x for x in B if x not in oval] for B in BIBD]
# We relabel the points to 0,1,2,...
V = [x for x in range(73) if x not in oval]
rel = dict(zip(V,range(len(V))))
PBD = [[rel[x] for x in B] for B in PBD]
return OA_from_PBD(7,68,PBD,check=False)

def OA_8_69():
r"""
Returns an OA(8,69)
Construction shared by Julian R. Abel.
.. SEEALSO::
:func:`sage.combinat.designs.orthogonal_arrays.OA_from_PBD`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_8_69
sage: OA = OA_8_69()
sage: print is_orthogonal_array(OA,8,69,2)
True
The design is available from the general constructor::
sage: designs.orthogonal_array(8,69,existence=True)
True
"""
# base block of a (73,9,1) BIBD
B = [1,2,4,8,16,32,37,55,64]
# The corresponding BIBD
BIBD= [[(x+i)%73 for x in B] for i in range(73)]
oval = [72,71,69,65]
# PBD minus the oval
PBD = [[x for x in B if x not in oval] for B in BIBD]

sets_of_size_seven = [R for R in PBD if len(R) == 7]
others = [R for R in PBD if len(R) != 7]

# 68, 27, and 52 are the only elements appearing twice in the rows of
# sets_of_size_seven, and each row contains exactly one of them.

# We split them into "balanced" halves.
O1 = sets_of_size_seven[:3]
O2 = sets_of_size_seven[-3:]
assert all(x in sum(O1,[]) for x in (68,27,52))
assert all(x in sum(O2,[]) for x in (68,27,52))

# Blocks of "others", without the 0..0,1..1,2..2 ... rows
OA = OA_from_PBD(8,69,others,check=False)[:-69]

# Blocks of O1
OA_8_7 = orthogonal_array(8,7,check=False)
for B in O1:
for BB in OA_8_7:
OA.append([B[i] for i in BB])

# Blocks of O2
OA_8_7_minus_TD_8_1 = OA_8_7
OA_8_7_minus_TD_8_1.remove([0]*8)
for B in O2:
# Making sure the double element is the first one
B.sort(key=lambda x: int(bool(x not in (68,27,52))))
for BB in OA_8_7:
OA.append([B[i] for i in BB])


# Adding the missing 0..0,1..1,... rows
done = sum(O1,[])+sum(O2,[])
missing = [x for x in range(73) if x not in done and x not in oval]
for x in missing:
OA.append([x]*8)

# Relabelling everything to 0..68
relabel = dict(zip([x for x in range(73) if x not in oval],range(69)))
OA = [[relabel[x] for x in B] for B in OA]
return OA

def OA_7_74():
r"""
Returns an OA(7,74)
Construction shared by Julian R. Abel.
.. SEEALSO::
:func:`sage.combinat.designs.orthogonal_arrays.OA_from_PBD`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_7_74
sage: OA = OA_7_74()
sage: print is_orthogonal_array(OA,7,74,2)
True
The design is available from the general constructor::
sage: designs.orthogonal_array(7,74,existence=True)
True
"""

# base block of a (91,10,1) BIBD
B = [0,1,3,9,27,81,61,49,56,77]
# The corresponding BIBD
BIBD= [[(x+i)%91 for x in B] for i in range(91)]
# an oval
oval = [(-x)%91 for x in B][-7:]
# PBD minus the oval+B
to_delete = oval + B
PBD = [[x for x in B if x not in to_delete] for B in BIBD]
PBD.remove([])
# We relabel the points to 0,1,2,...
V = [x for x in range(91) if x not in to_delete]
rel = dict(zip(V,range(len(V))))
PBD = [[rel[x] for x in B] for B in PBD]
return OA_from_PBD(7,74,PBD,check=False)

def OA_9_75():
r"""
Returns an OA(9,75)
Expand Down Expand Up @@ -1916,6 +2117,73 @@ def OA_9_75():
M = OA_from_quasi_difference_matrix(Mb,G,add_col = True)
return M

def OA_8_76():
r"""
Returns an OA(8,76)
Construction shared by Julian R. Abel.
.. SEEALSO::
:func:`sage.combinat.designs.orthogonal_arrays.OA_from_PBD`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_8_76
sage: OA = OA_8_76()
sage: print is_orthogonal_array(OA,8,76,2)
True
The design is available from the general constructor::
sage: designs.orthogonal_array(8,76,existence=True)
True
"""
# base block of a (91,10,1) BIBD
B = [0,1,3,9,27,81,61,49,56,77]
# The corresponding BIBD
BIBD= [[(x+i)%91 for x in B] for i in range(91)]
oval = [2,4,5,12,24]
to_remove = oval + B
# PBD minus the oval
PBD = [[x for x in B if x not in to_remove] for B in BIBD]
PBD.remove([])

sets_of_size_seven = [R for R in PBD if len(R) == 7]
others = [R for R in PBD if len(R) != 7]

# critical_points are the 10 elements appearing twice in the rows of the 10
# sets_of_size_seven, and each row contains exactly two of them
critical_points = [57,83,52,13,15,64,37,50,63,31]

# We reorder the rows such that every element of critical_points is exactly
# once the first element of a row.
for i,x in zip(critical_points,sets_of_size_seven):
x.sort(key=lambda x:-int(x==i))
assert x[0]==i

# Blocks of "others", without the 0..0,1..1,2..2 ... rows
OA = OA_from_PBD(8,76,others,check=False)[:-76]

OA_8_7 = orthogonal_array(8,7,check=False)
OA_8_7_minus_TD_8_1 = OA_8_7
OA_8_7_minus_TD_8_1.remove([0]*8)
for B in sets_of_size_seven:
for BB in OA_8_7:
OA.append([B[i] for i in BB])

# Adding the missing 0..0,1..1,... rows
done = sum(sets_of_size_seven,[])
missing = [x for x in range(91) if x not in done and x not in to_remove]
for x in missing:
OA.append([x]*8)

# Relabelling everything to 0..68
relabel = dict(zip([x for x in range(91) if x not in to_remove],range(91)))
OA = [[relabel[x] for x in B] for B in OA]
return OA

def OA_11_80():
r"""
Returns an OA(11,80)
Expand Down Expand Up @@ -2238,7 +2506,12 @@ def OA_12_342():
60 : (7 , OA_7_60),
62 : (7 , OA_7_62),
65 : (9 , OA_9_65),
66 : (7 , OA_7_66),
68 : (7 , OA_7_68),
69 : (8 , OA_8_69),
74 : (7 , OA_7_74),
75 : (9 , OA_9_75),
76 : (8 , OA_8_76),
80 : (11 , OA_11_80),
82 : (10 , OA_10_82),
100 : (10 , OA_10_100),
Expand Down
2 changes: 1 addition & 1 deletion src/sage/combinat/designs/latin_squares.py
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Expand Up @@ -34,7 +34,7 @@
0| +oo +oo 1 2 3 4 1 6 7 8 2 10 5 12 4 4 15 16 5 18
20| 4 5 3 22 7 24 4 26 5 28 4 30 31 5 4 5 8 36 4 5
40| 7 40 5 42 5 6 4 46 8 48 6 5 5 52 5 6 7 7 2 58
60| 5 60 5 6 63 7 3 66 4 4 6 70 7 72 3 7 3 6 6 78
60| 5 60 5 6 63 7 5 66 5 6 6 70 7 72 5 7 6 6 6 78
80| 9 80 8 82 6 6 6 3 7 88 4 6 6 4 3 6 7 96 6 8
100| 8 100 6 102 7 7 5 106 5 108 4 6 7 112 3 7 5 8 4 6
120| 6 120 5 6 5 124 6 126 127 7 6 130 6 6 6 6 7 136 4 138
Expand Down

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