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trac #16763: New OA for n=189, plus some others through Vmt vectors
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@nathanncohen
nathanncohen committed Aug 20, 2014
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181 changes: 174 additions & 7 deletions src/sage/combinat/designs/database.py
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Expand Up @@ -55,6 +55,7 @@
:func:`OA(9,135) <OA_9_135>`,
:func:`OA(11,160) <OA_11_160>`,
:func:`OA(16,176) <OA_16_176>`,
:func:`OA(11,185) <OA_11_185>`,
:func:`OA(16,208) <OA_16_208>`,
:func:`OA(15,224) <OA_15_224>`,
:func:`OA(18,273) <OA_18_273>`,
Expand Down Expand Up @@ -2455,6 +2456,74 @@ def OA_16_176():
Y = [None, 0, 1, 2, 8, 6, 9, 4, 10, 3, 5, 11, 13, 14, 12]
return OA_n_times_2_pow_c_from_matrix(16,4,FiniteField(11),zip(*A),Y,check=False)

def OA_11_185():
r"""
Returns an OA(11,185)
The construction is given in [Greig99]_. In Julian R. Abel's words:
Start with a `PG(2,16)` with a `7` points Fano subplane; outside this
plane there are `7(17-3) = 98` points on a line of the subplane and
`273-98-7 = 168` other points. Greig notes that the subdesign
consisting of these `168` points is a `(168, \{10,12\})-PBD`. Now add
the `17` points of a line disjoint from this subdesign (e.g. a line of
the Fano subplane). This line will intersect every line of the `168`
point subdesign in `1` point. Thus the new line sizes are `11` and
`13`, plus a unique line of size `17`, giving a `(185,\{11,13,17\}`-PBD
and an `OA(11,185)`.
EXAMPLES::
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_11_185
sage: OA = OA_11_185()
sage: print is_orthogonal_array(OA,11,185,2)
True
The design is available from the general constructor::
sage: designs.orthogonal_array(11,185,existence=True)
True
REFERENCE:
.. [Greig99] Designs from projective planes and PBD bases,
Malcolm Greig,
Journal of Combinatorial Designs,
Volume 7, Issue 5, pages 341–374, 1999
"""
from sage.combinat.designs.difference_family import difference_family
G,(B,) = difference_family(273,17) # a cyclic PG(2,16)
BIBD = [[int(x+i) for x in B] for i in G]

# All points congruent to 0 mod[39] are a Fano subplane
assert set([sum([x%39==0 for x in B]) for B in BIBD]) == set([0,1,3])

# Lines of the Fano subplane
fano_lines = [B for B in BIBD if sum([x%39==0 for x in B])>1]

# Points on a line of the Fano sublane
on_a_fano_line = set(sum(fano_lines,[]))

# Not on a line of the Fano plane
not_on_a_fano_line = set(range(273)).difference(on_a_fano_line)

# The PBD
ground_set = not_on_a_fano_line.union(fano_lines[0])
PBD = [ground_set.intersection(B) for B in BIBD]
relabel = {v:i for i,v in enumerate(ground_set)}
PBD = [[relabel[x] for x in B] for B in PBD if len(B)>1]
special_set = [relabel[x] for x in fano_lines[0]]

# Check that everything is fine
assert all([len(B) in (11,13) or set(B) == set(special_set) for B in PBD])

OA = OA_from_PBD(11,185,[B for B in PBD if len(B)<17],check=False)[:-185]
OA.extend([[i]*11 for i in range(185) if i not in special_set])
OA.extend([[special_set[x] for x in B] for B in orthogonal_array(11,17)])
return OA

def OA_16_208():
r"""
Returns an OA(16,208)
Expand Down Expand Up @@ -3115,6 +3184,7 @@ def OA_33_993():
135 : (9 , OA_9_135),
160 : (11 , OA_11_160),
176 : (16 , OA_16_176),
185 : (11 , OA_11_185),
208 : (16 , OA_16_208),
224 : (15 , OA_15_224),
273 : (18 , OA_18_273),
Expand All @@ -3130,10 +3200,10 @@ def OA_33_993():
}

Vmt_vectors = {
(4,9) : ((0, 1, 3, 2, 8),
(4,9) : ((0,1,3,2,8),
"As explained in the Handbook III.3.66 [DesignHandbook]_."),

(6,7) : ((0, 1, 3, 16, 35, 26, 36),
(6,7) : ((0,1,3,16,35,26,36),
"As explained in the Handbook III.3.68 [DesignHandbook]_."),

(8,9) : ((0,1,20,70,23,59,3,8,19),
Expand All @@ -3142,13 +3212,13 @@ def OA_33_993():
(8,11) : ((0,1,6,56,22,35,47,23,60),
"Given by Julian R. Abel, using a `V(m,t)` from the Handbook [DesignHandbook]_."),

(10,13) : ((0, 1, 5, 10, 22, 6, 14, 9, 53, 129, 84),
(10,13) : ((0,1,5,10,22,6,14,9,53,129,84),
"Given by Julian R. Abel, using a `V(m,t)` from the Handbook [DesignHandbook]_."),

(8,17) : ((0,1,3,2,133,126,47,109,74),
"Given by Julian R. Abel, using a `V(m,t)` from the Handbook [DesignHandbook]_."),

(10,19) : ((0, 1, 3, 96, 143, 156, 182, 142, 4, 189, 25),
(10,19) : ((0,1,3,96,143,156,182,142,4,189,25),
"Given by Julian R. Abel, using a `V(m,t)` from the Handbook [DesignHandbook]_."),

(8,29) : ((0,1,4,11,94,60,85,16,198),
Expand All @@ -3166,10 +3236,77 @@ def OA_33_993():
(10,43) : ((0,1,6,29,170,207,385,290,375,32,336),
"Given by Julian R. Abel, using a `V(m,t)` from the Handbook [DesignHandbook]_."),

(12,73) : ((0, 1, 607, 719, 837, 496, 240, 645, 184, 829, 451, 830, 770),
(12,73) : ((0,1,607,719,837,496,240,645,184,829,451,830,770),
"Given by Julian R. Abel."),
}

(12, 83) : ((0,1,627,898,836,939,742,42,847,531,173,607,361),
"Given by Julian R. Abel."),

(12, 89) : ((0,1,602,894,827,661,350,647,304,47,430,533,550),
"Given by Julian R. Abel."),

(12,101) : ((0,1,787,1049,818,1064,288,346,464,958,1188,340,1192),
"Given by Julian R. Abel."),

(12,103) : ((0,1,770,1027,806,1082,515,436,1096,1060,57,1135,1144),
"Given by Julian R. Abel."),

(12,121) : ((0,1,713,1265,848,421,998,69,874,1126,693,467,1164),
"Given by Julian R. Abel."),

(12,169) : ((0,1,425,326,951,1211,1881,1063,1631,1363,1554,665,1600),
"Given by Julian R. Abel."),

(12,191) : ((0,1,491,527,939,377,1685,1735,1967,1176,391,2192,681),
"Given by Julian R. Abel."),

(12,199) : ((0,1,377,524,946,560,316,1591,2036,273,1841,2091,713),
"Given by Julian R. Abel."),

(12,229) : ((0,1,338,312,933,380,401,2398,612,1279,1514,268,528),
"Given by Julian R. Abel."),

(10, 81) : ((0,1,3,2,27,438,615,708,168,410,656),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10, 97) : ((0,1,3,6,11,274,772,340,707,157,556),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10,103) : ((0,1,3,2,7,744,342,797,468,46,561),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10,181) : ((0,1,3,8,5,68,514,16,1168,225,929),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10,187) : ((0,1,3,7,2,325,1138,730,1013,534,366),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10,259) : ((0,1,3,7,2,15,324,1956,1353,2041,1616),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10,273) : ((0,1,3,6,11,28,2573,38,1215,1299,2468),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10,319) : ((0,1,3,7,2,43,239,1335,1586,2724,63),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10,391) : ((0,1,3,2,5,32,555,3450,1242,1823,3833),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""),

(10,409) : ((0,1,3,2,5,11,505,3202,1502,2521,3023),
"""Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104""")

}
# Translate all V(m,t) into OA constructors
for (m,t),(vec,source) in Vmt_vectors.iteritems():
OA_constructions[(m+1)*t+1] = (m+2, lambda m=m,t=t,vec=vec:OA_from_Vmt(m,t,vec))
Expand All @@ -3184,7 +3321,8 @@ def OA_33_993():
sage: from sage.combinat.designs.database import Vmt_vectors
sage: for (m,t),(vec,source) in sorted(Vmt_vectors.items()):
....: k,n = m+2,(m+1)*t+1
....: assert is_orthogonal_array(OA_from_Vmt(m,t,vec),k,n)
....: if n < 1000:
....: assert is_orthogonal_array(OA_from_Vmt(m,t,vec),k,n)
....: print "{:11}{}".format("V({},{}):".format(m,t),source)
V(4,9): As explained in the Handbook III.3.66 [DesignHandbook]_.
V(6,7): As explained in the Handbook III.3.68 [DesignHandbook]_.
Expand All @@ -3198,7 +3336,36 @@ def OA_33_993():
V(10,27): Given by Julian R. Abel, using a `V(m,t)` from the Handbook [DesignHandbook]_.
V(10,31): Given by Julian R. Abel, using a `V(m,t)` from the Handbook [DesignHandbook]_.
V(10,43): Given by Julian R. Abel, using a `V(m,t)` from the Handbook [DesignHandbook]_.
V(10,81): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,97): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,103): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,181): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,187): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,259): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,273): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,319): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,391): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(10,409): Charles J. Colbourn, Some direct constructions for incomplete transversal designs,
Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104
V(12,73): Given by Julian R. Abel.
V(12,83): Given by Julian R. Abel.
V(12,89): Given by Julian R. Abel.
V(12,101): Given by Julian R. Abel.
V(12,103): Given by Julian R. Abel.
V(12,121): Given by Julian R. Abel.
V(12,169): Given by Julian R. Abel.
V(12,191): Given by Julian R. Abel.
V(12,199): Given by Julian R. Abel.
V(12,229): Given by Julian R. Abel.
"""

def CDF_21_5_1():
Expand Down
4 changes: 2 additions & 2 deletions src/sage/combinat/designs/latin_squares.py
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Expand Up @@ -44,7 +44,7 @@
120| 7 120 6 6 6 124 6 126 127 7 6 130 6 7 6 7 7 136 6 138
140| 6 7 6 10 10 7 6 7 6 148 6 150 7 8 8 7 6 156 7 6
160| 9 7 6 162 6 7 6 166 7 168 6 8 6 172 6 6 14 9 6 178
180| 6 180 6 6 7 8 6 10 6 7 6 190 7 192 6 7 6 196 6 198
180| 6 180 6 6 7 9 6 10 6 7 6 190 7 192 6 7 6 196 6 198
200| 7 7 6 7 6 7 6 8 14 11 10 210 6 7 6 7 7 8 6 10
220| 6 12 6 222 13 8 6 226 6 228 6 7 7 232 6 7 6 7 6 238
240| 7 240 6 242 6 7 6 12 7 7 6 250 6 10 7 7 255 256 6 12
Expand Down Expand Up @@ -82,7 +82,7 @@
120|
140|
160|
180| - -
180| -
200| - -
220|
240| - -
Expand Down

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