Coverage for local/lib/python2.7/site-packages/sage/combinat/designs/database.py: 79%

[(0 ,None),(0 ,None),(0 ,None),(0 ,None),(0 ,None),(0 ,None),(0 ,None),(0 ,None),(0 ,None),(0 ,None),(0 ,None),(0 ,None),(1 ,None),(4 ,None),(9 ,None)],

[(0 ,None),(1 ,None),(2 ,None),(3 , 0),(4 , 2),(5 , 12),(6 , 5),(7 , 6),(8 , 13),(9 , 9),(10, 11),(1 , 3),(0 , 6),(1 , 14),(4 , 12)],

[(0 ,None),(2 ,None),(4 , 4),(6 , 4),(8 , 7),(10, 2),(1 , 2),(3 , 13),(5 , 0),(7 , 3),(9 , 7),(4 , 6),(1 , 12),(0 , 1),(1 , 10)], # 5,1 became 5,0

[(0 ,None),(3 ,None),(6 , 3),(9 , 4),(1 , 6),(4 , 13),(7 , 1),(10, 1),(2 , 7),(5 , 1),(8 , 0),(9 , 6),(4 , 4),(1 , 5),(0 , 1)],

[(0 ,None),(4 ,None),(8 , 13),(1 , 8),(5 , 0),(9 , 5),(2 , 14),(6 ,None),(10, 5),(3 , 7),(7 , 10),(5 , 3),(9 , 10),(4 , 11),(1 , 14)],

[(0 ,None),(5 ,None),(10, 10),(4 , 2),(9 , 7),(3 , 2),(8 , 3),(2 , 13),(7 , 7),(1 , 9),(6 ,None),(3 , 7),(5 , 1),(9 , 10),(4 , 11)],

[(0 ,None),(6 ,None),(1 , 8),(7 , 14),(2 , 2),(8 , 3),(3 , 11),(9 , 12),(4 , 8),(10, 13),(5 , 1),(3 , 6),(3 , 5),(5 , 10),(9 , 9)],

[(0 ,None),(7 ,None),(3 , 3),(10,None),(6 , 14),(2 , 4),(9 , 1),(5 , 7),(1 , 5),(8 , 7),(4 , 13),(5 , 6),(3 , 6),(3 , 11),(5 , 3)],

[(0 ,None),(8 ,None),(5 , 14),(2 , 11),(10, 14),(7 , 8),(4 , 14),(1 , 14),(9 , 9),(6 , 14),(3 , 9),(9 , 2),(5 , 6),(3 , 3),(3 , 10)],

[(0 ,None),(9 ,None),(7 , 5),(5 , 5),(3 , 8),(1 , 8),(10,None),(8 , 12),(6 , 9),(4 , 12),(2 , 9),(4 , 7),(9 , 2),(5 , 0),(3 , 7)],

[(0 ,None),(10,None),(9 , 11),(8 , 7),(7 , 6),(6 , 12),(5 ,None),(4 , 1),(3 , 13),(2 , 8),(1 , 9),(1 ,None),(4 , 3),(9 , 7),(5 , 13)],

[(0 ,None),(6 , 3),(2 , 0),(10, 8),(8 , 12),(7 , 9),(7 , 2),(8 , 0),(10, 7),(2 , 10),(6 , 4),(0 , 7),(10, 10),(7 , 3),(2 , 11)],

[(0 ,None),(7 , 3),(4 ,None),(2 , 12),(1 , 10),(1 , 3),(2 , 8),(4 , 9),(7 , 0),(0 , 1),(5 , 6),(10, 3),(0 , 9),(10, 13),(7 , 11)],

[(0 ,None),(8 , 3),(6 , 8),(5 , 2),(5 , 13),(6 , 1),(8 , 9),(0 , 2),(4 , 10),(9 , 8),(4 , 12),(7 , 7),(10, 2),(0 , 12),(10, 4)],

[(0 ,None),(9 , 3),(8 , 3),(8 , 9),(9 , 1),(0 , 4),(3 , 3),(7 , 11),(1 , 9),(7 , 10),(3 , 8),(2 , 10),(7 , 6),(10, 14),(0 , 3)],

[(0 ,None),(10, 3),(10, 5),(0 , 1),(2 , 1),(5 , 8),(9 , 2),(3 , 5),(9 , 5),(5 , 3),(2 , 4),(6 , 12),(2 , 6),(7 , 11),(10, 7)],

[(0 ,None),(0 , 3),(1 ,None),(3 , 2),(6 , 8),(10, 11),(4 , 6),(10,None),(6 ,None),(3 , 1),(1 , 1),(8 , 0),(6 , 14),(2 , 0),(7 , 14)],

[(0 ,None),(1 , 3),(3 , 8),(6 , 9),(10, 8),(4 , 10),(10, 1),(6 , 10),(3 , 0),(1 , 8),(0 , 11),(8 , 10),(8 , 14),(6 , 10),(2 , 14)],

[(0 ,None),(2 , 3),(5 , 1),(9 , 8),(3 , 4),(9 , 14),(5 , 5),(2 , 4),(0 , 2),(10, 2),(10,None),(6 , 2),(8 , 5),(8 , 1),(6 , 9)],

[(0 ,None),(3 , 3),(7 , 0),(1 ,None),(7 , 1),(3 , 10),(0 , 8),(9 , 13),(8 ,None),(8 , 10),(9 , 14),(2 , 0),(6 , 5),(8 , 5),(8 , 7)], # 2,None became 2,0

[(0 ,None),(4 , 3),(9 , 10),(4 , 14),(0 , 14),(8 , 14),(6 , 14),(5 , 6),(5 , 13),(6 , 5),(8 , 12),(7 , 1),(2 , 4),(6 , 3),(8 , 6)],

[(0 ,None),(5 , 3),(0 , 8),(7 , 3),(4 , 10),(2 , 1),(1 , 3),(1 , 10),(2 ,None),(4 , 8),(7 , 12),(10, 6),(7 , 10),(2 , 6),(6 , 1)], # 7,12 became 4,8

]

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),list(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: is_orthogonal_array(OA,11,185,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(11,185)

True

"""

from sage.combinat.designs.difference_family import difference_family

G,(B,) = difference_family(273,17)

BIBD = [[int(x+i) for x in B] for i in G] # a cyclic PG(2,16)

# All points congruent to 0 mod[39] form a Fano subplane with the property

# that each block of the PG(2,16) intersect the Fano subplane in either 0,1

# or 3 points

assert all(sum(x%39==0 for x in B) in [0,1,3] for B in BIBD)

# Lines of the Fano subplane that are contained in blocks

fano_lines = [B for B in BIBD if sum(x%39==0 for x in B) == 3]

# Points on a line of the Fano subplane

on_a_fano_line = set().union(*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_10_205():

r"""

Return an OA(10,205)

Julian R. Abel shared the following construction, which originally appeared

in Theorem 8.7 of [Greig99]_, and can in Lemmas 5.14-5.16 of [ColDin01]_:

Consider a `PG(2,4^2)` containing a Baer subplane (i.e. a `PG(2,4)`) `B`

and a point `p\in B`. Among the `4^2+1=17` lines of `PG(2,4^2)`

containing `p`:

* `4+1=5` lines intersect `B` on `5` points

* `4^2-4=12` lines intersect `B` on `1` point

As those lines are disjoint outside of `B` we can use them as groups to

build a GDD on `16^2+16+1-(4^4+4+1)=252` points. By keeping only 9 lines

of the second kind, however, we obtain a `(204,\{9,13,17\})`-GDD of type

12^5.16^9.

We complete it into a PBD by adding a block `g\cup \{204\}` for each

group `g`. We then build an OA from this PBD using the fact that all

blocks of size 9 are disjoint.

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_from_PBD`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_10_205

sage: OA = OA_10_205()

sage: is_orthogonal_array(OA,10,205,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(10,205)

True

"""

# Base block of a cyclic PG(2,4^2)

pplane_size = 16**2+16+1

baer_subplane_size = 4**2+4+1

B = [0, 1, 22, 33, 83, 122, 135, 141, 145, 159, 175, 200, 226, 229, 231, 238, 246]

pplane = [[(xx+i)%pplane_size for xx in B] for i in range(pplane_size)]

baer_subplane = set([i*pplane_size/baer_subplane_size for i in range(baer_subplane_size)])

p = list(baer_subplane)[0]

# We want all lines through p, but keep only 9 of those which intersect the

# subplane on 1 point.

lines_through_p = [B for B in pplane if p in B]

lines_through_p.sort(key=lambda s:len(baer_subplane.intersection(s)))

# Remove the Baer subplane and relabel everything to a (204,{9,13})-GDD of type

# 12^5.16^9 whose groups are the subset of (truncated) lines through p

groups = [[xx for xx in l if xx not in baer_subplane] for l in lines_through_p[4**2-4-9:]]

relabel = {v:i for i,v in enumerate(sum(groups,[]))}

GDD = [[relabel[xx] for xx in B if xx in relabel] for B in pplane if p not in B]

# We turn the GDD into a PBD by extending the groups with a new point 204.

GDD.extend([[relabel[xx] for xx in G]+[204] for G in groups])

# We build the OA, knowing that the blocks of size 9 are disjoint

blocks_of_size_9 = [B for B in GDD if len(B) == 9]

blocks_of_size_9_union = sum(blocks_of_size_9,[])

OA = OA_from_PBD(10,205,[B for B in GDD if len(B)!=9],check=False)[:-205]

OA.extend([[B[xx] for xx in R]

for R in orthogonal_array(10,9)

for B in blocks_of_size_9])

# The missing [i,i,...] blocks

OA.extend([[i]*10

for i in set(range(205)).difference(blocks_of_size_9_union)])

return OA

def OA_16_208():

r"""

Returns an OA(16,208)

Published by Julian R. Abel in [AbelThesis]_. Uses the fact that `208 = 2^4

\times 13` is a product of `2` and a prime number.

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_n_times_2_pow_c_from_matrix`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_16_208

sage: OA = OA_16_208() # not tested -- too long

sage: is_orthogonal_array(OA,16,208,2) # not tested -- too long

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(16,208)

True

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

A = [

[(0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (0 ,None), (1 ,None)],

[(0 ,None), (1 ,None), (2 , 0), (3 , 7), (4 , 1), (5 , 11), (6 , 2), (7 , 10), (8 ,None), (9 , 10), (10,None), (11, 3), (12, 3), (1 , 4), (0 , 8)],

[(0 ,None), (2 ,None), (4 , 4), (6 , 3), (8 , 0), (10, 5), (12, 14), (1 ,None), (3 , 10), (5 , 7), (7 , 3), (9 , 12), (11, 6), (4 , 9), (1 , 14)],

[(0 ,None), (3 ,None), (6 , 4), (9 , 6), (12, 10), (2 , 11), (5 , 14), (8 , 3), (11, 13), (1 , 1), (4 , 12), (7 , 14), (10, 1), (9 , 7), (4 , 8)],

[(0 ,None), (4 ,None), (8 , 9), (12, 5), (3 , 10), (7 , 14), (11, 0), (2 , 6), (6 , 11), (10, 11), (1 , 9), (5 , 3), (9 , 9), (3 , 6), (9 , 8)],

[(0 ,None), (5 ,None), (10, 5), (2 , 5), (7 , 3), (12, 3), (4 , 12), (9 , 3), (1 , 2), (6 , 2), (11,None), (3 , 13), (8 , 7), (12, 10), (3 , 1)],

[(0 ,None), (6 ,None), (12, 13), (5 , 5), (11, 13), (4 , 6), (10, 6), (3 , 2), (9 , 4), (2 , 12), (8 , 13), (1 , 13), (7 , 2), (10, 8), (12,None)],

[(0 ,None), (7 ,None), (1 , 2), (8 , 12), (2 , 4), (9 , 12), (3 , 0), (10, 10), (4 , 14), (11, 11), (5 , 14), (12, 9), (6 , 8), (10, 3), (10, 6)],

[(0 ,None), (8 ,None), (3 ,None), (11, 4), (6 , 12), (1 , 12), (9 , 14), (4 , 2), (12, 9), (7 , 9), (2 ,None), (10, 1), (5 , 14), (12, 5), (10, 8)],

[(0 ,None), (9 ,None), (5 , 9), (1 , 7), (10, 6), (6 , 3), (2 , 6), (11, 10), (7 , 11), (3 , 13), (12, 2), (8 , 0), (4 , 13), (3 , 3), (12, 14)],

[(0 ,None), (10,None), (7 , 7), (4 , 1), (1 , 8), (11, 1), (8 , 11), (5 , 4), (2 , 11), (12, 8), (9 , 12), (6 , 4), (3 , 0), (9 , 4), (3 , 8)],

[(0 ,None), (11,None), (9 , 3), (7 , 11), (5 , 14), (3 , 10), (1 , 10), (12, 0), (10, 2), (8 , 2), (6 , 6), (4 , 2), (2 , 12), (4 , 8), (9 , 10)],

[(0 ,None), (12,None), (11, 4), (10, 9), (9 , 2), (8 ,None), (7 , 9), (6 , 12), (5 , 5), (4 ,None), (3 , 7), (2 , 10), (1 , 13), (1 , 6), (4 , 0)],

[(0 ,None), (5 , 3), (7 , 5), (6 , 5), (2 , 14), (8 , 5), (11, 1), (11, 6), (8 , 13), (2 , 13), (6 , 9), (7 ,None), (5 , 10), (0 , 5), (2 , 8)],

[(0 ,None), (6 , 3), (9 , 4), (9 , 13), (6 , 4), (0 , 5), (4 , 6), (5 , 2), (3 ,None), (11, 14), (3 , 3), (5 , 7), (4 , 1), (2 , 8), (0 , 2)],

[(0 ,None), (7 , 3), (11, 5), (12, 12), (10,None), (5 , 5), (10, 7), (12, 9), (11, 9), (7 , 7), (0 , 0), (3 , 12), (3 , 11), (8 , 13), (2 , 14)],

[(0 ,None), (8 , 3), (0 , 8), (2 , 6), (1 ,None), (10, 9), (3 , 12), (6 , 8), (6 , 4), (3 , 9), (10, 2), (1 , 11), (2 , 7), (5 , 2), (8 , 2)],

[(0 ,None), (9 , 3), (2 , 3), (5 , 3), (5 , 8), (2 , 0), (9 , 1), (0 , 3), (1 , 14), (12, 3), (7 , 6), (12, 4), (1 , 3), (6 , 10), (5 , 7)],

[(0 ,None), (10, 3), (4 , 2), (8 , 0), (9 , 8), (7 , 1), (2 , 5), (7 ,None), (9 , 2), (8 , 4), (4 , 14), (10, 13), (0 , 10), (11, 7), (6 , 10)],

[(0 ,None), (11, 3), (6 , 9), (11, 14), (0 , 10), (12, 13), (8 , 6), (1 , 8), (4 , 7), (4 , 0), (1 , 14), (8 , 2), (12, 8), (7 , 10), (11, 7)], # 6,10 became 6,9

[(0 ,None), (12, 3), (8 , 12), (1 , 9), (4 , 6), (4 , 13), (1 , 6), (8 , 1), (12, 4), (0 , 7), (11, 5), (6 , 6), (11, 14), (7 , 3), (7 , 5)],

[(0 ,None), (0 , 3), (10, 10), (4 , 2), (8 , 1), (9 ,None), (7 , 2), (2 , 10), (7 , 13), (9 , 5), (8 , 14), (4 , 7), (10, 11), (11, 13), (7 , 0)],

[(0 ,None), (1 , 3), (12, 11), (7 , 12), (12, 13), (1 , 2), (0 , 9), (9 , 6), (2 , 13), (5 , 4), (5 , 13), (2 , 4), (9 , 12), (6 , 5), (11, 1)],

[(0 ,None), (2 , 3), (1 , 8), (10,None), (3 , 13), (6 ,None), (6 , 1), (3 , 0), (10, 4), (1 , 14), (2 , 0), (0 , 3), (8 , 13), (5 , 1), (6 , 7)], # 2,None became 2,0

[(0 ,None), (3 , 3), (3 , 14), (0 , 1), (7 , 14), (11, 4), (12, 9), (10, 1), (5 , 9), (10,None), (12, 13), (11,None), (7 , 7), (8 , 6), (5 , 0)],

[(0 ,None), (4 , 3), (5 , 10), (3 , 8), (11, 8), (3 , 0), (5 , 7), (4 , 12), (0 , 13), (6 ,None), (9 , 11), (9 , 5), (6 , 0), (2 , 5), (8 , 8)],

]

Y = [None, 0, 1, 2, 12, 9, 13, 11, 7, 4, 8, 5, 14, 6, 3]

return OA_n_times_2_pow_c_from_matrix(16,4,FiniteField(13),list(zip(*A)),Y,check=False)

def OA_15_224():

r"""

Returns an OA(15,224)

Published by Julian R. Abel in [AbelThesis]_ (uses the fact that `224=2^5

\times 7` is a product of a power of `2` and a prime number).

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_n_times_2_pow_c_from_matrix`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_15_224

sage: OA = OA_15_224() # not tested -- too long

sage: is_orthogonal_array(OA,15,224,2) # not tested -- too long

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(15,224)

True

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

A = [

[(0,None), (0,None), (0,None), (0,None), (0,None), (0,None), (0,None), (0,None), (1,None), (4,None), (2,None), (2,None), (4,None), (1,None)],

[(0,None), (1,None), (2, 9), (3, 23), (4, 29), (5, 4), (6, 30), (1, 26), (0,None), (1, 11), (4, 2), (2, 28), (2,None), (4, 13)],

[(0,None), (2,None), (4, 8), (6,None), (1, 29), (3, 21), (5, 4), (4, 5), (1, 4), (0, 14), (1, 5), (4, 6), (2, 0), (2, 2)],

[(0,None), (3,None), (6, 8), (2, 12), (5, 4), (1, 1), (4, 2), (2, 1), (4, 18), (1, 27), (0, 5), (1,None), (4, 1), (2,None)],

[(0,None), (4,None), (1, 9), (5, 2), (2, 29), (6, 17), (3, 0), (2, 12), (2, 5), (4, 22), (1, 0), (0, 29), (1, 19), (4,None)],

[(0,None), (5,None), (3, 26), (1, 0), (6, 29), (4, 16), (2, 11), (4, 21), (2, 28), (2, 16), (4, 0), (1, 3), (0, 11), (1, 2)],

[(0,None), (6,None), (5, 3), (4, 19), (3, 24), (2, 20), (1, 28), (1, 12), (4, 23), (2, 0), (2, 5), (4, 29), (1, 0), (0, 2)],

[(0,None), (4, 4), (2, 14), (1, 23), (1, 22), (2, 17), (4, 17), (0, 25), (6, 21), (3, 11), (5, 2), (5, 27), (3, 5), (6, 2)],

[(0,None), (5, 4), (4, 3), (4, 0), (5, 20), (0, 4), (3, 8), (6, 28), (0, 16), (6, 1), (3, 22), (5, 0), (5, 0), (3, 2)],

[(0,None), (6, 4), (6,None), (0, 18), (2, 0), (5, 20), (2, 4), (3, 11), (6, 15), (0, 18), (6, 5), (3, 0), (5,None), (5, 2)],

[(0,None), (0, 4), (1, 15), (3, 29), (6, 20), (3, 24), (1, 13), (5, 30), (3, 2), (6,None), (0, 10), (6, 3), (3, 0), (5,None)],

[(0,None), (1, 4), (3, 4), (6, 12), (3, 28), (1, 27), (0, 6), (5, 7), (5, 29), (3, 0), (6, 0), (0, 0), (6, 0), (3,None)], # 6,19 became 6,12

[(0,None), (2, 4), (5, 11), (2, 5), (0, 21), (6, 11), (6, 24), (3, 24), (5, 11), (5, 30), (3,None), (6,None), (0,None), (6, 1)],

[(0,None), (3, 4), (0, 11), (5, 11), (4, 22), (4, 2), (5, 23), (6, 22), (3, 27), (5, 1), (5, 0), (3,None), (6,None), (0,None)]

]

Y = [None, 0, 1, 2, 27, 22, 11, 4, 26, 25, 29, 24, 7, 20]

return OA_n_times_2_pow_c_from_matrix(15,5,FiniteField(7),list(zip(*A)),Y,check=False)

def OA_11_254():

r"""

Return an OA(11,254)

This constructions appears in [Greig99]_.

From a cyclic `PG(2,19)` whose base blocks contains 7,9, and 4 points in the

congruence classes mod 3, build a `(254,{11,13,16})-PBD` by ignoring the

points of a congruence class. There exist `OA(12,11),OA(12,13),OA(12,16)`,

which gives the `OA(11,254)`.

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_from_PBD`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_11_254

sage: OA = OA_11_254()

sage: is_orthogonal_array(OA,11,254,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(11,254)

True

"""

# Base block of a PG(2,19)

B = (0,1,19,28,96,118,151,153,176,202,240,254,290,296,300,307,337,361,366,369)

BIBD = [[(x+i)%381 for x in B] for i in range(381)]

# We only keep points congruent to 0,1 mod 3 and relabel the PBD. The result is

# a (254,{11,13,16})-PBD

BIBD = [[2*(x//3)+x%3 for x in B if x%3<2] for B in BIBD]

return OA_from_PBD(11,254,BIBD,check=False)

def OA_20_352():

r"""

Returns an OA(20,352)

Published by Julian R. Abel in [AbelThesis]_ (uses the fact that `352=2^5

\times 11` is the product of a power of `2` and a prime number).

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_n_times_2_pow_c_from_matrix`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_20_352

sage: OA = OA_20_352() # not tested (~25s)

sage: is_orthogonal_array(OA,20,352,2) # not tested (~25s)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(20,352)

True

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

# Column 8, line 6 : 4,25 became 4,27

# line 17: 3,0 became 3,None

# Column 14,line 1 : 4,1 became 4,0

# Column 18,line 18: 0,0 became 0,None

A = [

[(0,None),(0, None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(0,None),(1,None),(4,None),(9,None),(5, None),(3,None),(3,None),(5,None)],

[(0,None),(1, None),(2, 13),(3, 2),(4, 0),(5, 8),(6, 30),(7, 0),(8, 13),(9, 26),(10, 10),(1, 29),(0, 9),(1, 11),(4, 0),(9, 23),(5, 7),(3, 25),(3, 29)],

[(0,None),(2, None),(4, 29),(6, 6),(8, 3),(10, 18),(1, 21),(3, 24),(5, 4),(7, 7),(9, 29),(4, 22),(1, 2),(0, 27),(1, 10),(4, 13),(9, 22),(5, 6),(3, 20)],

[(0,None),(3, None),(6, 25),(9, 21),(1, 23),(4, 25),(7, 12),(10, 16),(2, 26),(5, 27),(8, 19),(9, 27),(4, 6),(1, 5),(0, 6),(1, 15),(4, 10),(9, 2),(5, 14)],

[(0,None),(4, None),(8, 3),(1, 23),(5, 17),(9, 7),(2, 7),(6, 25),(10, 27),(3, 30),(7, 5),(5, 23),(9, 24),(4, 16),(1, 12),(0, 8),(1, 12),(4, 17),(9, 28)],

[(0,None),(5, None),(10, 10),(4, 27),(9, 4),(3, 24),(8, 21),(2, 3),(7, 22),(1, 21),(6, 24),(3, 28),(5, 3),(9, 26),(4, 29),(1, 9),(0, 19),(1, 2),(4, 0)],

[(0,None),(6, None),(1, 11),(7, 9),(2, 14),(8, 15),(3, 11),(9, 7),(4, 27),(10, 13),(5, 4),(3, 18),(3, 0),(5, 5),(9, 2),(4, 7),(1, 30),(0, 10),(1,None)],

[(0,None),(7, None),(3, 25),(10, 7),(6, 29),(2, 4),(9, 10),(5, 22),(1, 25),(8, 18),(4, 11),(5, 21),(3, 29),(3, 14),(5, 12),(9, 25),(4, 2),(1, 13),(0, 19)],

[(0,None),(8, None),(5, 27),(2, 30),(10, 24),(7, 4),(4, 6),(1, 4),(9, 5),(6, 27),(3, 0),(9, 2),(5, 20),(3, 10),(3, 13),(5, 2),(9, 5),(4, 21),(1, 12)],

[(0,None),(9, None),(7, 21),(5, 0),(3, 9),(1, 13),(10, 17),(8, 1),(6, 15),(4, 30),(2, 28),(4, 3),(9, 28),(5, 0),(3,None),(3, 2),(5, 23),(9, 10),(4, 15)],

[(0,None),(10,None),(9, 29),(8, 8),(7, 6),(6, 6),(5, 18),(4, 20),(3, 22),(2, 7),(1, 13),(1, 24),(4, 13),(9, 14),(5, 29),(3, 27),(3, 16),(5, 12),(9, 4)],

[(0,None),(6, 4),(2, 17),(10, 16),(8, 26),(7, 17),(7, 21),(8, 9),(10, 2),(2, 25),(6, 27),(0, 20),(10, 8),(7, 12),(2, 26),(6, 22),(8, 8),(8, 16),(6, 13)],

[(0,None),(7, 4),(4, 1),(2, 0),(1, 8),(1, 18),(2, 10),(4, 9),(7, 2),(0, 11),(5, 27),(10, 27),(0, 16),(10, 19),(7, 0),(2, 2),(6, 26),(8, 30),(8, 6)],

[(0,None),(8, 4),(6, 19),(5, 24),(5, 16),(6, 20),(8,None),(0, 17),(4, 5),(9, 23),(4, 27),(7, 22),(10, 25),(0, 23),(10, 11),(7, 10),(2, 16),(6, 28),(8, 3)],

[(0,None),(9, 4),(8, 14),(8, 30),(9, 16),(0, 0),(3, 25),(7, 30),(1, 27),(7, 4),(3, 10),(2, 5),(7, 3),(10, 11),(0, 21),(10,None),(7, 7),(2, 19),(6, 24)],

[(0,None),(10, 4),(10, 30),(0, 12),(2, 9),(5, 9),(9, 0),(3, 14),(9, 17),(5, 17),(2, 18),(6, 10),(2, 0),(7, 16),(10, 23),(0, 1),(10, 26),(7, 18),(2, 9)],

[(0,None),(0, 4),(1, 13),(3, 28),(6, 25),(10, 28),(4, 16),(10, 17),(6, 23),(3, 7),(1, 22),(8, 22),(6, 27),(2, 29),(7, 5),(10, 14),(0, 12),(10, 14),(7, 6)],

[(0,None),(1, 4),(3, 6),(6, 4),(10, 13),(4, 12),(10, 15),(6, 27),(3,None),(1, 26),(0, 3),(8, 21),(8, 26),(6, 13),(2, 27),(7, 11),(10, 5),(0, 3),(10, 3)],

[(0,None),(2, 4),(5, 12),(9, 27),(3, 7),(9, 21),(5,None),(2, 22),(0, 28),(10, 30),(10, 25),(6, 12),(8, 6),(8, 30),(6, 28),(2, 6),(7, 26),(10, 3),(0,None)],

[(0,None),(3, 4),(7, 22),(1, 7),(7, 8),(3, 12),(0, 27),(9, 1),(8, 17),(8, 4),(9, 12),(2, 16),(6, 23),(8, 14),(8, 2),(6, 26),(2, 14),(7, 22),(10, 30)],

[(0,None),(4, 4),(9, 21),(4, 25),(0, 9),(8, 23),(6, 5),(5, 20),(5, 13),(6, 19),(8, 0),(7, 30),(2, 29),(6, 24),(8, 18),(8, 10),(6, 9),(2, 20),(7, 4)],

[(0,None),(5, 4),(0, 25),(7, 4),(4, 20),(2, 3),(1,None),(1, 21),(2,None),(4, 26),(7, 1),(10, 23),(7, 20),(2, 3),(6, 5),(8, 19),(8, 9),(6, 23),(2, 7)],

]

Y = [None, 0, 1, 2, 18, 5, 11, 4, 13, 26, 25, 29, 24, 7, 20, 19, 9, 12, 15]

return OA_n_times_2_pow_c_from_matrix(20,5,FiniteField(11),list(zip(*A)),Y,check=False)

def OA_20_416():

r"""

Returns an OA(20,416)

Published by Julian R. Abel in [AbelThesis]_ (uses the fact that `416=2^5

\times 13` is the product of a power of `2` and a prime number).

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_n_times_2_pow_c_from_matrix`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_20_416

sage: OA = OA_20_416() # not tested (~35s)

sage: is_orthogonal_array(OA,20,416,2) # not tested

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(20,416)

True

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

Z = None

A=[

[(0,Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (0 , Z), (1 , Z), (4 , Z), (9 , Z), (3 , Z), (12, Z)],

[(0,Z), (1 , Z), (2 ,18), (3 , 2), (4 ,20), (5 ,22), (6 ,11), (7 ,19), (8 , 0), (9 ,26), (10, Z), (11, 5), (12,27), (1 ,17), (0 ,30), (1 ,22), (4 ,29), (9 , 6), (3 ,19)],

[(0,Z), (2 , 4), (4 ,21), (6 ,10), (8 ,24), (10,13), (12, 7), (1 ,11), (3 ,29), (5 ,12), (7 ,21), (9 , 2), (11,11), (4 , 5), (1 ,11), (0 ,23), (1 ,13), (4 , 6), (9 ,15)],

[(0,Z), (3 , 4), (6 ,17), (9 ,20), (12,26), (2 , 2), (5 ,12), (8 ,29), (11, 1), (1 , Z), (4 ,15), (7 ,16), (10,27), (9 , 2), (4 , 7), (1 , 5), (0 ,23), (1 ,24), (4 , 8)],

[(0,Z), (4 , 4), (8 ,29), (12, 8), (3 , 3), (7 , 8), (11, 2), (2 ,17), (6 , 4), (10, 2), (1 ,21), (5 ,29), (9 ,20), (3 , 2), (9 , 1), (4 ,14), (1 ,21), (0 ,24), (1 ,28)],

[(0,Z), (5 , 4), (10,22), (2 ,18), (7 , 6), (12, 2), (4 ,18), (9 ,27), (1 ,15), (6 , Z), (11,20), (3 ,15), (8 , 9), (12, 9), (3 , 3), (9 ,13), (4 , 4), (1 , 7), (0 ,14)],

[(0,Z), (6 , Z), (12,23), (5 ,13), (11,11), (4 ,10), (10, 0), (3 , 4), (9 ,16), (2 ,28), (8 ,27), (1 , 1), (7 ,23), (10,17), (12, 9), (3 ,20), (9 ,16), (4 ,17), (1 ,26)],

[(0,Z), (7 , Z), (1 , 3), (8 ,13), (2 , 8), (9 , 9), (3 , 0), (10,26), (4 , 5), (11, 6), (5 ,22), (12, 1), (6 ,17), (10,10), (10, 5), (12,15), (3 ,25), (9 , Z), (4 , 4)],

[(0,Z), (8 , 4), (3 ,10), (11, 3), (6 ,17), (1 ,21), (9 ,18), (4 , 5), (12,27), (7 ,20), (2 ,16), (10,25), (5 ,22), (12,21), (10,25), (10,12), (12,28), (3 ,19), (9 ,29)],

[(0,Z), (9 , 4), (5 , 6), (1 ,16), (10, 4), (6 ,24), (2 ,14), (11,11), (7 , 2), (3 , 9), (12,30), (8 ,28), (4 , 2), (3 , 7), (12, 6), (10,17), (10, 2), (12,13), (3 ,26)],

[(0,Z), (10, 4), (7 ,11), (4 ,18), (1 ,23), (11,21), (8 ,28), (5 ,21), (2 ,29), (12,20), (9 , 0), (6 , 8), (3 , 6), (9 , 7), (3 ,12), (12, 5), (10, 1), (10,21), (12, 5)],

[(0,Z), (11, 4), (9 ,22), (7 ,11), (5 ,17), (3 , Z), (1 ,17), (12,25), (10,14), (8 ,18), (6 , 2), (4 ,17), (2 ,25), (4 ,29), (9 , 6), (3 , 2), (12, 8), (10,13), (10,14)],

[(0,Z), (12, Z), (11, 7), (10,26), (9 ,24), (8 , 4), (7 ,25), (6 , Z), (5 ,13), (4 , 9), (3 , 5), (2 ,19), (1 ,10), (1 ,26), (4 ,14), (9 , 7), (3 ,11), (12, 9), (10,20)],

[(0,Z), (5 , Z), (7 , 7), (6 ,27), (2 , 5), (8 , 1), (11,23), (11, Z), (8 ,23), (2 ,21), (6 ,20), (7 , 5), (5 , 6), (0 , 2), (2 ,12), (8 ,15), (5 ,22), (6 ,25), (11,10)],

[(0,Z), (6 , 4), (9 ,24), (9 ,18), (6 ,26), (0 ,26), (4 ,17), (5 ,24), (3 , 5), (11, 9), (3 ,15), (5 ,23), (4 ,22), (2 ,26), (0 , 8), (2 ,21), (8 ,25), (5 ,15), (6 , 8)],

[(0,Z), (7 , 4), (11,11), (12, 9), (10,10), (5 , 6), (10, 1), (12,24), (11, 6), (7 ,26), (0 , 8), (3 ,10), (3 ,29), (8 , 3), (2 ,24), (0 ,22), (2 ,13), (8 , 2), (5 , 0)],

[(0,Z), (8 , Z), (0 ,27), (2 , 0), (1 ,25), (10,21), (3 ,10), (6 ,20), (6 ,14), (3 , 1), (10, 3), (1 ,15), (2 ,14), (5 ,12), (8 ,11), (2 ,28), (0 ,15), (2 ,13), (8 ,22)],

[(0,Z), (9 , Z), (2 ,13), (5 ,11), (5 , 6), (2 ,24), (9 , 9), (0 ,14), (1 ,30), (12, 1), (7 ,15), (12,15), (1 , 5), (6 ,23), (5 , 9), (8 , 3), (2 ,27), (0 ,28), (2 ,12)],

[(0,Z), (10, Z), (4 ,18), (8 ,23), (9 ,27), (7 , 4), (2 , 2), (7 , Z), (9 ,10), (8 , 8), (4 , 0), (10,12), (0 ,21), (11,28), (6 ,15), (5 ,23), (8 , 5), (2 ,28), (0 , 7)],

[(0,Z), (11, Z), (6 , 7), (11,27), (0 , 0), (12,17), (8 ,11), (1 ,12), (4 ,22), (4 ,15), (1 ,16), (8 , 0), (12, 6), (7 ,16), (11,30), (6 ,21), (5 ,14), (8 ,17), (2 ,26)],

[(0,Z), (12, 4), (8 ,28), (1 ,22), (4 , 2), (4 ,15), (1 , 6), (8 ,12), (12,19), (0 ,21), (11, 2), (6 , 4), (11,19), (7 ,30), (7 ,11), (11,12), (6 ,20), (5 , 3), (8 , 7)],

[(0,Z), (0 , 4), (10,21), (4 , 4), (8 , 1), (9 , 6), (7 ,30), (2 , 4), (7 , 8), (9 ,30), (8 , 3), (4 ,22), (10, 3), (11,25), (7 , 1), (7 ,24), (11,20), (6 ,30), (5 , 4)],

[(0,Z), (1 , 4), (12,21), (7 , 3), (12, 2), (1 , 1), (0 , 6), (9 ,14), (2 ,19), (5 , 6), (5 ,12), (2 , 9), (9 , 9), (6 ,19), (11, Z), (7 , 4), (7 , 6), (11,29), (6 ,15)],

[(0,Z), (2 , Z), (1 ,22), (10, Z), (3 , 5), (6 ,30), (6 ,26), (3 , 1), (10,12), (1 ,16), (2 ,28), (0 ,20), (8 ,11), (5 ,29), (6 , 7), (11,21), (7 ,14), (7 , 8), (11,11)],

[(0,Z), (3 , Z), (3 , 4), (0 ,18), (7 , 2), (11,16), (12,28), (10, 4), (5 ,28), (10, 0), (12, 4), (11,10), (7 ,11), (8 ,17), (5 , 6), (6 ,16), (11, 4), (7 ,22), (7 ,28)],

[(0,Z), (4 , Z), (5 ,22), (3 ,18), (11, Z), (3 ,15), (5 , 1), (4 ,26), (0 ,10), (6 , 8), (9 , 9), (9 ,29), (6 , Z), (2 ,23), (8 ,28), (5 ,30), (6 , 8), (11,24), (7 ,16)]

]

Y = [None, 0, 1, 2, 18, 5, 11, 4, 13, 26, 25, 29, 24, 7, 20, 19, 9, 12, 15]

return OA_n_times_2_pow_c_from_matrix(20,5,FiniteField(13),list(zip(*A)),Y,check=False)

def OA_20_544():

r"""

Returns an OA(20,544)

Published by Julian R. Abel in [AbelThesis]_ (uses the fact that

`544=2^5 \times 17` is the product of a power of `2` and a prime number).

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_n_times_2_pow_c_from_matrix`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_20_544

sage: OA = OA_20_544() # not tested (too long ~1mn)

sage: is_orthogonal_array(OA,20,544,2) # not tested

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(20,544)

True

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

Z = None

A=[

[(0,Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(0 , Z),(1 , Z)],

[(0,Z),(1 , 4),(2 , 7),(3 ,30),(4 ,17),(5 , 2),(6 ,22),(7 ,23),(8 ,28),(9 , 2),(10,27),(11,26),(12,13),(13,25),(14,18),(15,15),(16,18),(1 ,14),(0 , 1)],

[(0,Z),(2 , 4),(4 ,20),(6 ,29),(8 ,27),(10, 7),(12,20),(14,19),(16,26),(1 ,28),(3 , Z),(5 ,27),(7 , Z),(9 ,11),(11, Z),(13,17),(15, 1),(4 ,14),(1 ,14)],

[(0,Z),(3 , Z),(6 ,14),(9 ,26),(12,17),(15,15),(1 ,26),(4 ,24),(7 ,27),(10,13),(13,10),(16, 7),(2 , 1),(5 , Z),(8 , 1),(11,15),(14,18),(9 ,21),(4 , 6)],

[(0,Z),(4 , 4),(8 , Z),(12, 2),(16,23),(3 ,19),(7 ,26),(11, 7),(15,26),(2 , 3),(6 ,11),(10,16),(14,23),(1 ,30),(5 , 1),(9 ,30),(13,19),(16,10),(9 , 4)],

[(0,Z),(5 , Z),(10,17),(15,19),(3 ,13),(8 , 4),(13,21),(1 , 9),(6 , 7),(11, 4),(16,24),(4 , 6),(9 ,11),(14, Z),(2 , 6),(7 ,14),(12,10),(8 ,12),(16, 1)],

[(0,Z),(6 , Z),(12, 1),(1 ,23),(7 ,21),(13,10),(2 , 0),(8 ,15),(14,19),(3 ,30),(9 ,21),(15,17),(4 ,25),(10,20),(16,15),(5 ,16),(11,15),(2 ,22),(8 ,29)], # 2,Z -> 2,0

[(0,Z),(7 , Z),(14,30),(4 ,26),(11,24),(1 ,22),(8 ,22),(15,27),(5 ,23),(12,13),(2 ,18),(9 ,22),(16, 6),(6 ,27),(13,19),(3 , 1),(10,16),(15, 9),(2 , 5)],

[(0,Z),(8 , 4),(16, 5),(7 ,18),(15,11),(6 , 1),(14,21),(5 ,28),(13,19),(4 , 7),(12,19),(3 ,15),(11,13),(2 ,23),(10, 1),(1 ,23),(9 ,19),(13,27),(15,25)], # 13,9 -> 15,25

[(0,Z),(9 , Z),(1 , 3),(10, 4),(2 ,29),(11,13),(3 ,27),(12,11),(4 ,30),(13, 9),(5 ,18),(14,17),(6 ,18),(15,10),(7 ,11),(16,28),(8 ,26),(13,12),(13, 9)],

[(0,Z),(10, Z),(3 ,18),(13,21),(6 , 8),(16, 1),(9 ,11),(2 ,11),(12,12),(5 ,20),(15,21),(8 ,12),(1 , 5),(11,28),(4 ,16),(14,16),(7 ,21),(15, 0),(13,20)],

[(0,Z),(11, 4),(5 ,25),(16, 2),(10,18),(4 , 6),(15,20),(9 ,29),(3 ,13),(14,24),(8 ,18),(2 ,22),(13, 1),(7 , 8),(1 ,21),(12,16),(6 ,23),(2 ,10),(15,26)],

[(0,Z),(12, 4),(7 ,11),(2 , 4),(14,25),(9 , 0),(4 , 5),(16,21),(11,18),(6 ,18),(1 ,22),(13,27),(8 ,23),(3 ,20),(15,18),(10, 7),(5 ,10),(8 ,11),(2 ,18)],

[(0,Z),(13, Z),(9 ,21),(5 ,17),(1 ,26),(14,30),(10,11),(6 , 1),(2 , 8),(15, 9),(11, 5),(7 ,29),(3 ,17),(16, 3),(12, 3),(8 ,30),(4 , 3),(16, 5),(8 ,21)],

[(0,Z),(14, Z),(11,20),(8 ,24),(5 , Z),(2 , 2),(16,24),(13,12),(10,21),(7 ,26),(4 ,29),(1 , 1),(15, 1),(12,19),(9 , 8),(6 ,26),(3 ,10),(9 ,20),(16,21)],

[(0,Z),(15, Z),(13,21),(11,10),(9 , 7),(7 ,21),(5 ,11),(3 ,19),(1 ,29),(16,13),(14, 9),(12, 9),(10, 8),(8 ,16),(6 ,15),(4 ,14),(2 ,29),(4 ,16),(9 , 9)],

[(0,Z),(16, 4),(15,19),(14,21),(13, 0),(12,13),(11,28),(10,21),(9 , 5),(8 ,18),(7 , 2),(6 , Z),(5 ,20),(4 ,26),(3 , 8),(2 , 9),(1 ,23),(1 ,19),(4 ,23)], # 13,Z -> 13,0

[(0,Z),(3 , 4),(12,11),(10,17),(14,14),(7 , 1),(6 ,27),(11,25),(5 , 2),(5 ,24),(11,15),(6 , 8),(7 ,28),(14,21),(10, 4),(12,20),(3 ,26),(0 , 5),(3 ,12)],

[(0,Z),(4 , Z),(14,17),(13,26),(1 ,12),(12,12),(12,23),(1 ,13),(13, 7),(14,10),(4 ,28),(0 ,11),(2 , 7),(10,15),(7 , Z),(10, 1),(2 , 6),(3 ,24),(0 ,18)],

[(0,Z),(5 , 4),(16,24),(16, 1),(5 ,27),(0 ,14),(1 ,11),(8 ,13),(4 ,25),(6 ,25),(14,14),(11, 6),(14, 4),(6 ,24),(4 , 4),(8 ,28),(1 ,14),(12,22),(3 ,11)],

[(0,Z),(6 , 4),(1 ,10),(2 , 6),(9 ,12),(5 , 3),(7 ,11),(15,30),(12,21),(15,26),(7 , 3),(5 ,12),(9 , 0),(2 ,25),(1 , 2),(6 , 0),(0 ,13),(10,13),(12,14)],

[(0,Z),(7 , 4),(3 ,24),(5 ,25),(13,20),(10,19),(13,16),(5 , 4),(3 ,23),(7 ,20),(0 , 8),(16, 4),(4 ,19),(15, 0),(15,10),(4 ,11),(16, 7),(14,11),(10, 6)],

[(0,Z),(8 , Z),(5 , 1),(8 ,21),(0 , 1),(15,17),(2 ,26),(12, 2),(11, 6),(16, 2),(10,15),(10,13),(16,16),(11,12),(12,22),(2 ,11),(15,22),(7 ,30),(14,22)], # 8,9 -> 8,21

[(0,Z),(9 , 4),(7 ,20),(11,24),(4 , 7),(3 ,11),(8 ,21),(2 ,23),(2 , 2),(8 ,12),(3 , 8),(4 ,13),(11,17),(7 , 4),(9 , 3),(0 ,18),(14,12),(6 ,26),(7 ,28)],

[(0,Z),(10, 4),(9 ,22),(14,23),(8 , 5),(8 , 8),(14,12),(9 , 6),(10,20),(0 ,11),(13,23),(15,26),(6 ,12),(3 ,15),(6 , Z),(15,18),(13, 1),(11,22),(6 ,24)],

[(0,Z),(11, Z),(11,11),(0 ,28),(12,16),(13,18),(3 , 3),(16,22),(1 , 9),(9 , Z),(6 ,21),(9 , 6),(1 , 0),(16, 1),(3 , 2),(13,28),(12, 6),(5 ,18),(11, 9)],

[(0,Z),(12, Z),(13, 5),(3 ,14),(16,22),(1 , 5),(9 , 1),(6 , Z),(9 , 3),(1 , 9),(16,21),(3 ,18),(13,17),(12,29),(0 ,13),(11, 4),(11,18),(5 ,21),(5 , 6)],

[(0,Z),(13, 4),(15,27),(6 ,26),(3 ,20),(6 ,29),(15,11),(13,18),(0 , 4),(10, 5),(9 ,16),(14,26),(8 ,20),(8 , 8),(14,11),(9 ,10),(10, 9),(11,17),(5 ,21)],

[(0,Z),(14, 4),(0 ,29),(9 , 8),(7 , 2),(11,18),(4 ,22),(3 ,22),(8 ,13),(2 ,23),(2 ,21),(8 , 9),(3 ,30),(4 ,21),(11, 5),(7 ,25),(9 , Z),(6 , 0),(11,17)],

[(0,Z),(15, 4),(2 ,27),(12,27),(11,28),(16, 0),(10, 6),(10,12),(16,11),(11,15),(12, 2),(2 ,10),(15,19),(0 ,11),(8 ,10),(5 , 6),(8 , 5),(7 , 7),(6 ,16)],

[(0,Z),(16, Z),(4 ,23),(15, 4),(15,30),(4 ,27),(16,12),(0 , 8),(7 , 9),(3 , 6),(5 ,26),(13,28),(10,12),(13,14),(5 ,30),(3 ,27),(7 , 6),(14,15),(7 ,18)],

[(0,Z),(0 , 4),(6 ,13),(1 ,14),(2 , 2),(9 ,11),(5 , 5),(7 ,13),(15,24),(12,16),(15,20),(7 ,24),(5 ,19),(9 ,25),(2 ,26),(1 ,20),(6 ,28),(10, 5),(14,11)],

[(0,Z),(1 , Z),(8 ,25),(4 , 5),(6 , 6),(14, 6),(11,11),(14,22),(6 , 2),(4 , 2),(8 ,14),(1 ,13),(0 , 3),(5 , 6),(16,21),(16,11),(5 , 8),(12,15),(10,20)], # 12,14->12,15

[(0,Z),(2 , Z),(10,19),(7 ,29),(10,22),(2 ,23),(0 ,15),(4 ,19),(14, 6),(13,14),(1 , 5),(12,24),(12, 8),(1 , 4),(13, 1),(14,21),(4 ,17),(3 , 3),(12,27)],

]

Y = [None, 0, 1, 2, 18, 5, 11, 4, 13, 26, 25, 29, 24, 7, 20, 19, 9, 12, 15]

return OA_n_times_2_pow_c_from_matrix(20,5,FiniteField(17),list(zip(*A)),Y,check=False)

def OA_17_560():

r"""

Returns an OA(17,560)

This OA is built in Corollary 2.2 of [Thwarts]_.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_17_560

sage: OA = OA_17_560()

sage: is_orthogonal_array(OA,17,560,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(17,560)

True

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF

alpha = 5

beta = 4

p = 2

k = 17

m = 16

n = p**alpha

G = GF(p**alpha,prefix='x')

G_set = sorted(G) # sorted by lexicographic order, G[1] = 1

G_to_int = {v:i for i,v in enumerate(G_set)}

# Builds an OA(n+1,n) whose last n-1 columns are

# \forall x \in G and x!=0, C_x(i,j) = i+x*j

# (only the necessary columns are built)

OA = [[G_to_int[i+x*j] for i in G_set for j in G_set] for x in G_set[k+1:0:-1]]

OA.append([j for i in range(n) for j in range(n)])

OA.append([i for i in range(n) for j in range(n)])

# The additive group F_{p^beta} appears in F_{p^alpha} as all polynomials

# with degree < beta

# We remove all elements except those from F_{p^alpha} in the last three

# columns

elements_of_subgroup = set([x for x in G_set if x.polynomial().degree() < beta])

relabel = {G_to_int[v]:i for i,v in enumerate(elements_of_subgroup)}

for x in range(p**alpha):

if x not in relabel:

relabel[x] = None

for C in OA[-3:]:

for i,x in enumerate(C):

C[i] = relabel[x]

OA = list(zip(*OA))

return wilson_construction(OA,k,n,m,[p**beta]*3,check=False)

def OA_11_640():

r"""

Returns an OA(11,640)

Published by Julian R. Abel in [AbelThesis]_ (uses the fact that `640=2^7

\times 5` is the product of a power of `2` and a prime number).

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_n_times_2_pow_c_from_matrix`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_11_640

sage: OA = OA_11_640() # not tested (too long)

sage: is_orthogonal_array(OA,11,640,2) # not tested (too long)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(11,640)

True

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

A = [

[(0,None), (0,None), (0,None), (0,None), (0,None), (0,None), (1,None), (4,None), (4,None), (1,None)],

[(0,None), (1,None), (2, 7), (3, 55), (4, 54), (1, 87), (0, 124), (1, 123), (4, 83), (4, 61)], # 0,25 became 0,124

[(0,None), (2,None), (4, 14), (1, 63), (3, 6), (4, 87), (1, 16), (0, 47), (1, 29), (4, 16)],

[(0,None), (3,None), (1, 1), (4, 15), (2, 5), (4, 32), (4, 30), (1, 3), (0, 12), (1, 14)],

[(0,None), (4,None), (3, 28), (2, 62), (1, 64), (1, 55), (4, 63), (4, 4), (1, 0), (0, 0)],

[(0,None), (2, 6), (3, 8), (3, 7), (2, 12), (0, 1), (2, 6), (3, 97), (3, 45), (2, 0)],

[(0,None), (3, 6), (0, 63), (1, 5), (1, 6), (2, 97), (0, 28), (2, 63), (3, 0), (3, 2)],

[(0,None), (4, 6), (2, 4), (4, 65), (0, 6), (3, 68), (2, 1), (0, 14), (2, 1), (3, 0)],

[(0,None), (0, 6), (4, 9), (2,None), (4, 29), (3, 15), (3, 0), (2, 1), (0, 7), (2, 4)],

[(0,None), (1, 6), (1, 14), (0, 14), (3, 4), (2, 0), (3,None), (3, 4), (2, 0), (0,None)]

]

Y = [None, 0, 1, 2, 121, 66, 77, 78, 41, 100]

return OA_n_times_2_pow_c_from_matrix(11,7,FiniteField(5),list(zip(*A)),Y,check=False)

def OA_10_796():

r"""

Returns an OA(10,796)

Construction shared by Julian R. Abel, from [AC07]_:

Truncate one block of a `TD(17,47)` to size `13`, then add an extra

point. Form a block on each group plus the extra point: we obtain a

`(796, \{13,16,17,47,48\})`-PBD in which only the extra point lies in

more than one block of size `48` (and each other point lies in exactly 1

such block).

For each block `B` (of size `k` say) not containing the extra point,

construct a `TD(10, k) - k.TD(k,1)` on `I(10) X B`. For each block `B`

(of size `k=47` or `48`) containing the extra point, construct a

`TD(10,k) - TD(k,1)` on `I(10) X B`, the size `1` hole being on `I(10) X

P` where `P` is the extra point. Finally form `1` extra block of size

`10` on `I(10) X P`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_10_796

sage: OA = OA_10_796()

sage: is_orthogonal_array(OA,10,796,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(10,796)

True

"""

from sage.combinat.designs.orthogonal_arrays import OA_relabel

from sage.combinat.designs.orthogonal_arrays import OA_from_PBD

from .orthogonal_arrays import incomplete_orthogonal_array

OA = orthogonal_array(17,47)

OA = OA_relabel(OA,17,47,blocks=[OA[0]]) # making sure [46]*17 is a block

PBD = [[i*47+x for i,x in enumerate(B) if (x<46 or i<13)] for B in OA]

extra_point = 10000

PBD.extend([list(range(i*47,(i+1)*47-int(i>=13)))+[extra_point] for i in range(17)]) # Adding the columns

rel = {v:i for i,v in enumerate(set(range(17*47)).difference([(i+1)*47-1 for i in range(13,17)]))}

rel[extra_point] = len(rel)

PBD = [[rel[x] for x in B] for B in PBD]

assert set(map(len,PBD)) == set([13, 16, 17, 47, 48])

extra_point = rel[extra_point]

others = []

OA = []

span = set()

iOA = {47: incomplete_orthogonal_array(10,47,(1,)),

48: incomplete_orthogonal_array(10,48,(1,))}

for B in PBD:

if len(B) >= 47:

B.sort(key=lambda x:int(x==extra_point))

OA.extend([[B[i] for i in BB] for BB in iOA[len(B)]])

span.update(B[:-1])

else:

others.append(B)

OA.extend(OA_from_PBD(10,796,others,check=False))

OA = OA[:-796] # removes the [x]*k

for x in set(range(796)).difference(span):

OA.append([x]*10)

return OA

def OA_10_469():

r"""

Return an OA(10,469)

This construction appears in [Brouwer80]_. It is based on the same technique

used in

:func:`~sage.combinat.designs.orthogonal_arrays_build_recursive.brouwer_separable_design`.

Julian R. Abel's instructions:

Brouwer notes that a cyclic `PG(2,37)` (or `(1407,38,1)`-BIBD) can be

obtained with a base block containing `13,9,` and `16` points in each

residue class mod 3. Thus, by reducing the `PG(2,37)` to its points

congruent to `0 \pmod 3` one obtains a `(469,\{9,13,16\})`-PBD which

consists in 3 symmetric designs, i.e. 469 blocks of size 9, 469 blocks

of size 13, and 469 blocks of size 16.

For each block size `s`, one can build a matrix with size `s\times 469`

in which each block is a row, and such that each point of the PBD

appears once per column. By multiplying a row of an `OA(9,s)-s.OA(9,1)`

with the rows of the matrix one obtains a parallel class of a resolvable

`OA(9,469)`.

Add to this the parallel class of all blocks `(0,0,...),(1,1,...),...`

to obtain a resolvable `OA(9,469)` equivalent to an `OA(10,469)`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_10_469

sage: OA = OA_10_469()

sage: is_orthogonal_array(OA,10,469,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(10,469)

True

"""

from .orthogonal_arrays_build_recursive import _reorder_matrix

from .orthogonal_arrays import incomplete_orthogonal_array

OA = []

# A cyclic (1407,38,1)-BIBD

B = (0,1,27,44,63,69,102,149,237,249,395,436,510,515,525,533,547,592,665,

731,824,837,848,932,1002,1051,1055,1089,1105,1145,1165,1196,1217,1226,

1274,1281,1309,1405)

BIBD = [[(x+i)%1407 for x in B] for i in range(1407)]

# Only keep points v congruent to 0 mod 3 and relabel

PBD = [[x//3 for x in B if x%3==0] for B in BIBD]

# Split the block according to their size

blocks = {9:[],13:[],16:[]}

for B in PBD:

blocks[len(B)].append(B)

# Product of each symmetric design with the OA

for b_size,symmetric_design in iteritems(blocks):

matrix = _reorder_matrix(symmetric_design)

OA.extend([[B[xx] for xx in R]

for R in incomplete_orthogonal_array(9,b_size,[1]*b_size)

for B in matrix])

# Last parallel class

OA.extend([[i]*9 for i in range(469)])

for i,R in enumerate(OA):

R.append(i//469)

return OA

def OA_520_plus_x(x):

r"""

Return an `OA(10+x,520+x)`.

The construction shared by Julian R. Abel works for :func:`OA(10,520)

<OA_10_520>`, :func:`OA(12,522) <OA_12_522>`, and :func:`OA(14,524)

<OA_14_524>`.

Let `n=520+x` and `k=10+x`. Build a `TD(17,31)`. Remove `8-x` points

contained in a common block, add a new point `p` and create a block

`g_i\cup \{p\}` for every (possibly truncated) group `g_i`. The result

is a `(520+x,{9+x,16,17,31,32})-PBD`. Note that all blocks of size `\geq

30` only intersect on `p`, and that the unique block `B_9` of size `9`

intersects all blocks of size `32` on one point. Now:

* Build an `OA(k,16)-16.OA(k,16)` for each block of size 16

* Build an `OA(k,17)-17.OA(k,17)` for each block of size 17

* Build an `OA(k,31)-OA(k,1)` for each block of size 31 (with the hole on

`p`).

* Build an `OA(k,32)-2.OA(k,1)` for each block `B` of size 32 (with the

holes on `p` and `B\cap B_9`).

* Build an `OA(k,9)` on `B_9`.

Only a row `[p,p,...]` is missing from the `OA(10+x,520+x)`

This construction is used in :func:`OA(10,520) <OA_10_520>`,

:func:`OA(12,522) <OA_12_522>`, and :func:`OA(14,524) <OA_14_524>`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_520_plus_x

sage: OA = OA_520_plus_x(0) # not tested (already tested in OA_10_520)

sage: is_orthogonal_array(OA,10,520,2) # not tested (already tested in OA_10_520)

True

"""

from .orthogonal_arrays import incomplete_orthogonal_array

k = 9+x+1

# The OA(17,31) with a block [30,30,...]

OA = incomplete_orthogonal_array(17,31,[1])

OA.append([30]*17)

# We truncate [30,30,...] to its first 9+x coordinates, and add sets

# corresponding to each (possibly truncated) group extended with a new

# point. The result is a (520+x,{9+x,16,17,31,32})-PBD.

new_point = 31*17

PBD = [[i*31+xx for i,xx in enumerate(B) if i<9+x or xx<30] for B in OA] # truncated blocks

PBD.extend([list(range(i*31,i*31+30+bool(i<9+x)))+[new_point] for i in range(17)]) # extended (+truncated) groups

relabel = {v:i for i,v in enumerate(sorted(set().union(*PBD)))}

PBD = [[relabel[xx] for xx in B] for B in PBD]

subdesigns = {

9+x: orthogonal_array(k,9+x),

16 : incomplete_orthogonal_array(k,16,[1]*16),

17 : incomplete_orthogonal_array(k,17,[1]*17),

31 : incomplete_orthogonal_array(k,31,[1]),

32 : incomplete_orthogonal_array(k,32,[1]*2),

}

OA = []

for B in PBD:

OA.extend([[B[xx] for xx in R]

for R in subdesigns[len(B)]])

OA.append([relabel[new_point]]*k)

return OA

def OA_10_520():

r"""

Return an OA(10,520).

This design is built by the slightly more general construction

:func:`OA_520_plus_x`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_10_520

sage: OA = OA_10_520()

sage: is_orthogonal_array(OA,10,520,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(10,520)

True

"""

return OA_520_plus_x(0)

def OA_12_522():

r"""

Return an OA(12,522)

This design is built by the slightly more general construction

:func:`OA_520_plus_x`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_12_522

sage: OA = OA_12_522()

sage: is_orthogonal_array(OA,12,522,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(12,522)

True

"""

return OA_520_plus_x(2)

def OA_14_524():

r"""

Return an OA(14,524)

This design is built by the slightly more general construction

:func:`OA_520_plus_x`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_14_524

sage: OA = OA_14_524()

sage: is_orthogonal_array(OA,14,524,2)

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(14,524)

True

"""

return OA_520_plus_x(4)

def OA_15_896():

r"""

Returns an OA(15,896)

Uses the fact that `896 = 2^7 \times 7` is the product of a power of `2` and

a prime number.

.. SEEALSO::

:func:`sage.combinat.designs.orthogonal_arrays.OA_n_times_2_pow_c_from_matrix`

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_15_896

sage: OA = OA_15_896() # not tested -- too long (~2min)

sage: is_orthogonal_array(OA,15,896,2) # not tested -- too long

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(15,896)

True

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

A = [

[(0,None), (0,None), (0,None), (0,None), (0,None), (0,None), (0,None), (0,None), (1,None), (4,None), (2,None), (2,None), (4,None), (1,None)],

[(0,None), (1,None), (2, 17), (3, 20), (4, 49), (5, 4), (6, 59), (1, 15), (0, 114), (1, 76), (4, 106), (2, 87), (2, 118), (4, 49)], # 4,120 became the leftmost 4,49

[(0,None), (2,None), (4, 2), (6, 98), (1, 53), (3, 97), (5, 123), (4, 3), (1, 32), (0, 10), (1, 45), (4, 3), (2, 1), (2, 14)],

[(0,None), (3,None), (6, 16), (2, 86), (5, 102), (1, 64), (4, 69), (2, 11), (4, 55), (1, 90), (0, 115), (1, 15), (4, 7), (2, 0)],

[(0,None), (4,None), (1, 4), (5, 110), (2, 51), (6, 118), (3, 8), (2, 81), (2, 79), (4, 98), (1, 2), (0, 3), (1, 7), (4,None)],

[(0,None), (5,None), (3, 66), (1, 70), (6, 102), (4, 119), (2, 20), (4, 86), (2, 59), (2, 15), (4, 63), (1, 126), (0, 1), (1, 0)],

[(0,None), (6,None), (5, 94), (4, 48), (3, 90), (2, 2), (1, 13), (1, 53), (4, 117), (2, 21), (2, 2), (4, 1), (1, 0), (0, 0)],

[(0,None), (4, 6), (2, 21), (1, 112), (1, 36), (2, 14), (4, 60), (0, 1), (6, 64), (3, 0), (5, 31), (5, 3), (3, 3), (6, 14)],

[(0,None), (5, 6), (4, 61), (4,None), (5, 108), (0, 91), (3, 10), (6, 15), (0,None), (6, 15), (3, 7), (5, 0), (5, 1), (3, 0)],

[(0,None), (6, 6), (6, 107), (0, 88), (2, 12), (5, 44), (2, 31), (3, 64), (6, 0), (0,None), (6, 2), (3, 3), (5,None), (5, 0)],

[(0,None), (0, 6), (1, 52), (3, 115), (6, 30), (3, 78), (1, 64), (5, 63), (3, 5), (6,None), (0,None), (6, 3), (3, 1), (5,None)],

[(0,None), (1, 6), (3, 117), (6, 19), (3, 9), (1, 31), (0, 56), (5, 0), (5, 63), (3,None), (6,None), (0,None), (6, 7), (3,None)],

[(0,None), (2, 6), (5, 116), (2, 3), (0, 0), (6,None), (6, 1), (3, 0), (5, 0), (5, 2), (3,None), (6,None), (0,None), (6, 0)],

[(0,None), (3, 6), (0, 0), (5, 0), (4, 1), (4,None), (5,None), (6, 0), (3, 2), (5, 0), (5,None), (3,None), (6,None), (0,None)] # 0,0 became the rightmost 0,None

]

Y = [None, 0,1,2,121,66,77,78,41,100,74,118,108,43]

return OA_n_times_2_pow_c_from_matrix(15,7,FiniteField(7),list(zip(*A)),Y,check=False)

def OA_9_1078():

r"""

Returns an OA(9,1078)

This is obtained through the generalized Brouwer-van Rees

construction. Indeed, `1078 = 89.11 + (99=9.11)` and there exists an

`OA(9,100) - OA(9,11)`.

.. NOTE::

This function should be removed once

:func:`~sage.combinat.designs.orthogonal_arrays_find_recursive.find_brouwer_van_rees_with_one_truncated_column`

can handle all incomplete orthogonal arrays obtained through

:func:`~sage.combinat.designs.orthogonal_arrays.incomplete_orthogonal_array`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_9_1078

sage: OA = OA_9_1078() # not tested -- ~3s

sage: is_orthogonal_array(OA,9,1078,2) # not tested -- ~3s

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(9,1078)

True

"""

return wilson_construction(None,9,11,89,[[(11,9)]])

def OA_25_1262():

r"""

Returns an OA(25,1262)

The construction is given in [Greig99]_. In Julian R. Abel's words:

Start with a cyclic `PG(2,43)` or `(1893,44,1)`-BIBD whose base block

contains respectively `12,13` and `19` point in the residue classes mod

3. In the resulting BIBD, remove one of the three classes: the result is

a `(1262, \{25, 31,32\})`-PBD, from which the `OA(25,1262)` is obtained.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_25_1262

sage: OA = OA_25_1262() # not tested -- too long

sage: is_orthogonal_array(OA,25,1262,2) # not tested -- too long

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(25,1262)

True

"""

from sage.combinat.designs.orthogonal_arrays import OA_from_PBD

B = (0, 68, 78, 106, 227, 296, 304, 330, 354, 411, 624, 631, 636, 732, 747,

772, 794, 846, 869, 939, 948, 1011, 1015, 1031, 1135, 1171, 1188, 1206,

1217, 1219, 1220, 1261, 1306, 1349, 1370, 1400, 1461, 1480, 1517, 1714,

1768, 1827, 1833, 1866)

BIBD = [[(x+i)%1893 for x in B] for i in range(1893)] # a (1893,44,1)-BIBD

PBD = [[x for x in B if (x%3)<2] for B in BIBD] # We only keep the x with x%3=0,1

PBD = [[2*(x//3)+(x%3) for x in B] for B in PBD] # The (1262, {25, 31,32})-PBD

return OA_from_PBD(25,1262,PBD,check=False)

def OA_9_1612():

r"""

Returns an OA(9,1612)

This is obtained through the generalized Brouwer-van Rees

construction. Indeed, `1612 = 89.17 + (99=9.11)` and there exists an

`OA(9,100) - OA(9,11)`.

.. NOTE::

This function should be removed once

:func:`~sage.combinat.designs.orthogonal_arrays_find_recursive.find_brouwer_van_rees_with_one_truncated_column`

can handle all incomplete orthogonal arrays obtained through

:func:`~sage.combinat.designs.orthogonal_arrays.incomplete_orthogonal_array`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_9_1612

sage: OA = OA_9_1612() # not tested -- ~6s

sage: is_orthogonal_array(OA,9,1612,2) # not tested -- ~6s

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(9,1612)

True

"""

return wilson_construction(None,9,17,89,[[(11,9)]])

def OA_10_1620():

r"""

Returns an OA(10,1620)

This is obtained through the generalized Brouwer-van Rees

construction. Indeed, `1620 = 144.11+(36=4.9)` and there exists an

`OA(10,153) - OA(10,9)`.

.. NOTE::

This function should be removed once

:func:`~sage.combinat.designs.orthogonal_arrays_find_recursive.find_brouwer_van_rees_with_one_truncated_column`

can handle all incomplete orthogonal arrays obtained through

:func:`~sage.combinat.designs.orthogonal_arrays.incomplete_orthogonal_array`.

EXAMPLES::

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

sage: from sage.combinat.designs.database import OA_10_1620

sage: OA = OA_10_1620() # not tested -- ~7s

sage: is_orthogonal_array(OA,10,1620,2) # not tested -- ~7s

True

The design is available from the general constructor::

sage: designs.orthogonal_arrays.is_available(10,1620)

True

"""

return wilson_construction(None,10,11,144,[[(9,4)]])

# Index of the OA constructions

# Associates to n the pair (k,f) where f() is a function that returns an OA(k,n)

# This dictionary is used by designs.orthogonal_array(k,n).

OA_constructions = {

18 : (7 , OA_7_18),

40 : (9 , OA_9_40),

66 : (7 , OA_7_66),

68 : (7 , OA_7_68),

69 : (8 , OA_8_69),

74 : (7 , OA_7_74),

76 : (8 , OA_8_76),

80 : (11 , OA_11_80),

112 : (15 , OA_15_112),

120 : (9 , OA_9_120),

135 : (9 , OA_9_135),

160 : (11 , OA_11_160),

176 : (16 , OA_16_176),

185 : (11 , OA_11_185),

205 : (10 , OA_10_205),

208 : (16 , OA_16_208),

224 : (15 , OA_15_224),

254 : (11 , OA_11_254),

352 : (20 , OA_20_352),

416 : (20 , OA_20_416),

469 : (10 , OA_10_469),

520 : (10 , OA_10_520),

522 : (12 , OA_12_522),

524 : (14 , OA_14_524),

544 : (20 , OA_20_544),

560 : (17 , OA_17_560),

640 : (11 , OA_11_640),

796 : (10 , OA_10_796),

896 : (15 , OA_15_896),

1078 : (9 , OA_9_1078),

1262 : (25, OA_25_1262),

1612 : (9 , OA_9_1612),

1620 : (10, OA_10_1620),

}

# Add this data to the module's doc

LIST_OF_OA_CONSTRUCTIONS = ", ".join(":func:`OA({},{}) <OA_{}_{}>`".format(k,n,k,n)

for n,(k,_) in OA_constructions.items())

def QDM_19_6_1_1_1():

r"""

Return a `(19,6;1,1;1)`-quasi-difference matrix.

Used to build an `OA(6,20)`

Given in the Handbook III.3.49 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_19_6_1_1_1

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_19_6_1_1_1()

sage: is_quasi_difference_matrix(M,G,6,1,1,1)

True

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

M=[[None, 7, 13, 1, 16, 9, 2],

[ 0, 1, 15, 7, 17, 6, 14],

[ 0, 11, 10, 11, 5, 4, 3],

[ 7,None, 13, 16, 1, 2, 9],

[ 1, 0, 15, 17, 7, 14, 6],

[ 11, 0, 10, 5, 11, 3, 4]]

Mb=[]

for R in zip(*M):

a,b,c,d,e,f = R

Mb.append([a,b,c,d,e,f])

Mb.append([b,c,a,f,d,e])

Mb.append([c,a,b,e,f,d])

return AdditiveCyclic(19), Mb

def QDM_21_5_1_1_1():

r"""

Return a `(21,5;1,1;1)`-quasi-difference matrix.

Used to build an `OA(5,22)`

Given in the Handbook III.3.51 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_21_5_1_1_1

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_21_5_1_1_1()

sage: is_quasi_difference_matrix(M,G,5,1,1,1)

True

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(21)

M = [

[ 1, 13, 18, 3, 16, 19,None],

[ 16, 19, 1, 13, 18, 3, 0],

[ 18, 3, 16, 19, 1, 13, 0],

[ 6, 15, 6, 15, 6, 15, 0],

[ 12, 9, 19, 16, 5, 2, 0],

]

Mb=[[0,7,14,None,0],

[0,14,7,0,None]]

for R in zip(*M):

a,b,c,d,e = [G(x) if x is not None else None for x in R]

Mb.append([a,b,c,d,e])

Mb.append([16*c,

None if a is None else 16*a,

16*b,

16*d+7,

16*e+14])

Mb.append([4*b,

4*c,

None if a is None else 4*a,

4*d+14,

4*e+7])

return G, Mb

def QDM_21_6_1_1_5():

r"""

Return a `(21,6;1,1;5)`-quasi-difference matrix.

Used to build an `OA(6,26)`

Given in the Handbook III.3.53 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_21_6_1_1_5

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_21_6_1_1_5()

sage: is_quasi_difference_matrix(M,G,6,1,1,5)

True

"""

M = [

[None,None,None,None,None],

[ 0, 0, 0, 0, 0],

[ 1, 6, 7, 8, 14],

[ 3, 11, 20, 18, 10],

[ 6, 10, 14, 1, 5],

[ 4, 19, 5, 12, 2],

]

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(21)

Mb=[[0,0,0,0,0,0]]

for R in zip(*M):

a,b,c,d,e,f = R

Mb.append([a,b,c,d,e,f])

Mb.append([b,c,d,e,f,a])

Mb.append([c,d,e,f,a,b])

Mb.append([d,e,f,a,b,c])

Mb.append([e,f,a,b,c,d])

Mb.append([f,a,b,c,d,e])

return G, Mb

def QDM_25_6_1_1_5():

r"""

Return a `(25,6;1,1;5)`-quasi-difference matrix.

Used to build an `OA(6,30)`

Given in the Handbook III.3.55 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_25_6_1_1_5

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_25_6_1_1_5()

sage: is_quasi_difference_matrix(M,G,6,1,1,5)

True

"""

M = [

[(0,0),None,(0,0),(0,0),(0,0),(0,0),(0,0)],

[(0,0),(0,0),None,(0,4),(0,2),(0,3),(0,1)],

[(0,0),(3,1),(3,0),None,(4,0),(1,0),(2,0)],

[(0,0),(3,0),(0,2),(1,2),None,(0,1),(0,3)],

[(0,0),(3,3),(1,2),(4,2),(2,0),None,(0,4)],

[(0,0),(4,2),(2,4),(0,3),(2,3),(3,2),None]

]

from sage.groups.additive_abelian.additive_abelian_group import AdditiveAbelianGroup

from sage.modules.free_module_element import free_module_element as vector

G = AdditiveAbelianGroup([5,5])

M = [[None if x is None else G(vector(x)) for x in L] for L in M]

Mb=[]

for R in zip(*M):

a,b,c,d,e,f = R

for i in range(5):

Mb.append([

None if a is None else a+G(vector((i,i))),

None if b is None else b+G(vector((2*i,i))),

None if c is None else c+G(vector((i,0))),

None if d is None else d+G(vector((4*i,0))),

None if e is None else e+G(vector((3*i,4*i))),

None if f is None else f+G(vector((4*i,4*i)))])

return G, Mb

def QDM_33_6_1_1_1():

r"""

Return a `(33,6;1,1;1)`-quasi-difference matrix.

Used to build an `OA(6,34)`

Given in the Handbook III.3.57 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_33_6_1_1_1

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_33_6_1_1_1()

sage: is_quasi_difference_matrix(M,G,6,1,1,1)

True

"""

M = [

[None, 0, 0, 0, 0, 0],

[ 30, 17, 10, 25, 23, 8],

[ 22, 4, 32, 29, 28, 22],

[ 25, 10, 20, 15, 21, 16],

[ 0, 12, 15, 16, 32, 23],

[ 6, 11, 18, 14, 9, 20]

]

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(33)

Mb = [[ 0, 0, 0, 0, 0, 0],

[ 1, 4, 16, 31, 25, 11],

[ 3, 12, 15, 27, 9, 11],

[10, 7, 28, 13, 19, 0],

[ 5, 20, 14, 23, 26,None]]

times4 = lambda x : None if x is None else 4*x

for R in zip(*M):

a,b,c,d,e,f = [None if x is None else G(x) for x in R]

for i in range(5):

Mb.append([a,b,c,d,e,f])

a,b,c,d,e,f = map(times4,[e,a,b,c,d,f])

return G, Mb

def QDM_37_6_1_1_1():

r"""

Return a `(37,6;1,1;1)`-quasi-difference matrix.

Used to build an `OA(6,38)`

Given in the Handbook III.3.60 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_37_6_1_1_1

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_37_6_1_1_1()

sage: is_quasi_difference_matrix(M,G,6,1,1,1)

True

"""

M = [

[None, 10, 1, 2, 6, 3, 22, 5, 7, 9, 14, 18, 28],

[ 0, 1, 10, 20, 23, 30, 35, 13, 33, 16, 29, 32, 21],

[ 0, 26, 26, 15, 8, 4, 17, 19, 34, 12, 31, 24, 25],

[ 10,None, 10, 6, 2, 22, 3, 7, 5, 14, 9, 28, 18],

[ 1, 0, 26, 23, 20, 35, 30, 33, 13, 29, 16, 21, 32],

[ 26, 0, 1, 8, 15, 17, 4, 34, 19, 31, 12, 25, 24]

]

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(37)

Mb=[]

for R in zip(*M):

a,b,c,d,e,f = R

Mb.append([a,b,c,d,e,f])

Mb.append([b,c,a,f,d,e])

Mb.append([c,a,b,e,f,d])

return G, Mb

def QDM_35_7_1_1_7():

r"""

Return a `(35,7;1,1;7)`-quasi-difference matrix.

Used to build an `OA(7,42)`

As explained in the Handbook III.3.63 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_35_7_1_1_7

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_35_7_1_1_7()

sage: is_quasi_difference_matrix(M,G,7,1,1,7)

True

"""

M = [

[None,None,None,None,None,None,None],

[ 0, 0, 0, 0, 0, 0, 0],

[ 18, -18, 11, -11, 5, -5, 4],

[ 26, -26, 10, -10, 30, -30, 23],

[ 20, -20, 3, -3, 33, -33, 23],

[ 5, -5, 25, -25, 24, -24, 4],

[ 17, -17, 4, -4, 22, -22, 0]

]

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(35)

Mb=[]

for R in zip(*M):

for i in range(7):

Mb.append(cyclic_shift(R,i))

return G, Mb

def QDM_45_7_1_1_9():

r"""

Return a `(45,7;1,1;9)`-quasi-difference matrix.

Used to build an `OA(7,54)`

As explained in the Handbook III.3.71 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_45_7_1_1_9

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_45_7_1_1_9()

sage: is_quasi_difference_matrix(M,G,7,1,1,9)

True

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(45)

M = [

[None,None,None,None,None,None,None,None,None],

[ 0, 0, 0, 0, 0, 0, 0, 0, 0],

[ 1, 27, 16, 7, -1, -27, -16, -7, 3],

[ 24, 40, 1, 35, -24, -40, -1, -35, 7],

[ 10, 30, 22, 44, -10, -30, -22, -44, 7],

[ 5, 18, 14, 33, -5, -18, -14, -33, 3],

[ 30, 16, 33, 27, -30, -16, -33, -27, 0],

]

Mb=[]

for R in zip(*M):

for c in range(7):

Mb.append(cyclic_shift(R,c))

return G, Mb

def QDM_54_7_1_1_8():

r"""

Return a `(54,7;1,1;8)`-quasi-difference matrix.

Used to build an `OA(7,62)`

As explained in the Handbook III.3.74 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_54_7_1_1_8

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_54_7_1_1_8()

sage: is_quasi_difference_matrix(M,G,7,1,1,8)

True

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(54)

M = [

[ 0 ,None,None,None, 0 ,None ,None ,None,None,None],

[17 , 0 , 0 , 0 , -17 , 0 , 0 , 0 , 1 , 11 ],

[29 , 28 , 35 , 23 , -29 , -28 , -35 , -23, 3 , 19 ],

[36 , 50 , 5 , 33 , -36 , -50 , -5 , -33, 7 , 33 ],

[31 , 2 , 43 , 30 , -31 , - 2 , -43 , -30, 34 , 33 ],

[16 , 47 , 44 , 51 , -16 , -47 , -44 , -51, 30 , 19 ],

[41 , 11 , 1 , 17 , -41 , -11 , - 1 , -17, 28 , 11 ]

]

Mb=[]

for R in zip(*M):

for c in range(7):

Mb.append(cyclic_shift(R,c))

return G, Mb

def QDM_57_9_1_1_8():

r"""

Return a `(57,9;1,1;8)`-quasi-difference matrix.

Used to build an `OA(9,65)`

Construction shared by Julian R. Abel

EXAMPLES::

sage: from sage.combinat.designs.database import QDM_57_9_1_1_8

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: G,M = QDM_57_9_1_1_8()

sage: is_quasi_difference_matrix(M,G,9,1,1,8)

True

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as G

B = [None,1, 6, 7, 9, 19, 38, 42, 49] # Base block of a (57,8,1)-BIBD

OA = orthogonal_array(9,9,2)

M = [R for R in OA if any(R[0] != x for x in R)]

M = [[B[x] for x in R] for R in M] # replacing [0,..,8] by the elements of B

M.append([0]*9)

return G(57), M

# Quasi-difference matrices

# The syntax of the dictionary is

# QDM = {

# (n+u,lmbda): { # QDM with mu<=lmbda=1 yields a OA(k,n+u)-OA(k,u)

# (n,lmbda,mu,u): (k,qdm_constructor),

# }

QDM = {}

for ((n,k,lmbda,mu,u),f) in [((19,6,1,1,1), QDM_19_6_1_1_1),

((21,5,1,1,1), QDM_21_5_1_1_1),

((21,6,1,1,5), QDM_21_6_1_1_5),

((25,6,1,1,5), QDM_25_6_1_1_5),

((33,6,1,1,1), QDM_33_6_1_1_1),

((37,6,1,1,1), QDM_37_6_1_1_1),

((35,7,1,1,7), QDM_35_7_1_1_7),

((45,7,1,1,9), QDM_45_7_1_1_9),

((54,7,1,1,8), QDM_54_7_1_1_8),

((57,9,1,1,8), QDM_57_9_1_1_8)]:

if not (n+u,lmbda) in QDM:

QDM[n+u,lmbda] = {}

QDM[n+u,lmbda][n,lmbda,mu,u] = (k,f)

# Create the list of QDM matrices for the doc

LIST_OF_QDM = ", ".join("`({},{};{},{};{})`".format(n,k,lmbda,mu,u)

for n,k,lmbda,mu,u in

sorted((n,k,lmbda,mu,u) for entry in QDM.values()

for (n,lmbda,mu,u),(k,_) in sorted(entry.items())))

_ref_Handbook = """Handbook of Combinatorial Designs (2ed),

C. Colbourn, J. Dinitz, 2010 CRC Press"""

_ref_Brouwer_vanRees = """A. Brouwer and J. van Rees, More mutually orthogonal Latin squares,

Discrete Mathematics 1982, vol 39, num 3, pp 263-281"""

_ref_Colbourn = """Charles J. Colbourn, Some direct constructions for incomplete transversal designs,

Journal of Statistical Planning and Inference, vol 56, num 1, pp 93-104"""

_ref_Abel_v_12_t = """J.R. Abel, Some V(12,t) vectors and designs from difference and quasi-difference matrices,

Australasian Journal of Combinatorics 2008, vol 40 pp 69-85"""

_ref_Abel_v_11_t = """J.R. Abel, Some new matrix-minus-diagonal V(11,t) vectors,

Journal of Combinatorial Designs 2003, vol 11, num 4, pp 304-306"""

Vmt_vectors = {

(3 ,2 ) : ((0,1,3,6), _ref_Handbook),

(3 ,4 ) : ((0,1,3,9), _ref_Handbook),

(3 ,10) : ((0,1,4,13), _ref_Handbook),

(3 ,12) : ((0,1,3,10), _ref_Handbook),

(3 ,20) : ((0,1,3,13), _ref_Handbook),

(3 ,6 ) : ((0,1,3,7), _ref_Handbook),

(3 ,26) : ((0,1,3,8), _ref_Handbook),

(3 ,32) : ((0,1,3,9), _ref_Handbook),

(3 ,6 ) : ((0,1,3,7), _ref_Handbook),

(3 ,14) : ((0,1,4,13), _ref_Handbook),

(3 ,24) : ((0,1,3,15), _ref_Handbook),

(3 ,34) : ((0,1,3,7), _ref_Handbook),

(4 ,3 ) : ((0,1,3,7,2), _ref_Handbook),

(4 ,7 ) : ((0,1,3,7,19), _ref_Handbook),

(4 ,9 ) : ((0,1,3,2,8), _ref_Brouwer_vanRees),

(4 ,13) : ((0,1,3,7,19), _ref_Handbook),

(4 ,15) : ((0,1,3,7,5), _ref_Handbook),

(4 ,25) : ((0,1,3,2,31), _ref_Handbook),

(5 ,6 ) : ((0,1,3,7,30,17), _ref_Handbook),

(5 ,8 ) : ((0,1,3,22,14,18), _ref_Handbook),

(5 ,12) : ((0,1,3,7,23,50), _ref_Handbook),

(5 ,14) : ((0,1,3,9,25,54), _ref_Handbook),

(5 ,20) : ((0,1,3,10,43,91), _ref_Handbook),

(5 ,26) : ((0,1,3,6,48,15), _ref_Handbook),

(6 ,5 ) : ((0,1,7,30,12,21,15), _ref_Handbook),

(6 ,7 ) : ((0,1,3,16,35,26,36), _ref_Colbourn),

(6 ,11) : ((0,1,3,14,7,24,27), _ref_Handbook),

(6 ,13) : ((0,1,3,7,55,47,34), _ref_Handbook),

(6 ,17) : ((0,1,3,2,14,99,29), _ref_Handbook),

(6 ,21) : ((0,1,4,13,66,93,45), _ref_Handbook),

(7 ,6 ) : ((0,1,12,27,37,16,30,35), _ref_Handbook),

(7 ,10) : ((0,1,3,45,9,50,28,16), _ref_Handbook),

(7 ,16) : ((0,1,3,7,82,72,93,39), _ref_Handbook),

(7 ,18) : ((0,1,3,6,97,114,99,26), _ref_Handbook),

(8 ,9 ) : ((0,1,20,70,23,59,3,8,19), _ref_Colbourn),

(8 ,11) : ((0,1,6,56,22,35,47,23,60), _ref_Colbourn),

(8 ,17) : ((0,1,3,2,133,126,47,109,74), _ref_Colbourn),

(8 ,29) : ((0,1,4,11,94,60,85,16,198), _ref_Colbourn),

(8 ,57) : ((0,1,3,2,12,333,363,154,340), _ref_Brouwer_vanRees),

(9 ,12) : ((0,1,4,19,56,22,83,95,52,96), _ref_Handbook),

(9 ,14) : ((0,1,11,25,37,8,100,23,95,42), _ref_Handbook),

(9 ,18) : ((0,1,3,7,36,30,158,94,52,70), _ref_Handbook),

(9 ,20) : ((0,1,3,19,145,70,173,159,18,85), _ref_Handbook),

(9 ,22) : ((0,1,3,31,99,190,174,46,87,127), _ref_Handbook),

(9 ,30) : ((0,1,3,8,197,68,119,13,215,105), _ref_Handbook),

(9 ,34) : ((0,1,3,13,140,81,74,131,303,238), _ref_Handbook),

(9 ,42) : ((0,1,3,6,66,258,186,346,104,152), _ref_Handbook),

(9 ,44) : ((0,1,4,11,144,103,216,77,160,363), _ref_Handbook),

(10,13) : ((0,1,5,10,22,6,14,9,53,129,84), _ref_Colbourn),

(10,15) : ((0,1,45,146,51,97,70,137,85,133,18), _ref_Handbook),

(10,19) : ((0,1,3,96,143,156,182,142,4,189,25), _ref_Colbourn),

(10,21) : ((0,1,6,188,205,39,101,113,30,32,42), _ref_Handbook),

(10,25) : ((0,1,3,85,140,178,195,22,48,179,188), _ref_Colbourn),

(10,27) : ((0,1,3,82,109,241,36,112,141,263,126), _ref_Colbourn),

(10,31) : ((0,1,3,57,128,247,289,239,70,271,96), _ref_Colbourn),

(10,33) : ((0,1,3,67,319,44,249,146,302,282,90), _ref_Handbook),

(10,43) : ((0,1,6,29,170,207,385,290,375,32,336), _ref_Colbourn),

(10,49) : ((0,1,3,8,406,72,335,197,324,383,395), _ref_Handbook),

(10,81) : ((0,1,3,2,27,438,615,708,168,410,656), _ref_Colbourn),

(10,97) : ((0,1,3,6,11,274,772,340,707,157,556), _ref_Colbourn),

(10,103) : ((0,1,3,2,7,744,342,797,468,46,561), _ref_Colbourn),

(10,181) : ((0,1,3,8,5,68,514,16,1168,225,929), _ref_Colbourn),

(10,187) : ((0,1,3,7,2,325,1138,730,1013,534,366), _ref_Colbourn),

(10,259) : ((0,1,3,7,2,15,324,1956,1353,2041,1616), _ref_Colbourn),

(10,273) : ((0,1,3,6,11,28,2573,38,1215,1299,2468), _ref_Colbourn),

(10,319) : ((0,1,3,7,2,43,239,1335,1586,2724,63), _ref_Colbourn),

(10,391) : ((0,1,3,2,5,32,555,3450,1242,1823,3833), _ref_Colbourn),

(10,409) : ((0,1,3,2,5,11,505,3202,1502,2521,3023), _ref_Colbourn),

(11,30 ) : ((0,1,58,61,235,82,160,120,260,161,204,174), _ref_Abel_v_11_t),

(11,32 ) : ((0,1,90,6,158,125,293,76,250,123,341,79), _ref_Abel_v_11_t),

(11,36 ) : ((0,1,3,57,250,77,196,255,371,107,305,260), _ref_Abel_v_11_t),

(11,38 ) : ((0,1,43,27,179,37,345,70,17,255,238,147), _ref_Abel_v_11_t),

(11,42 ) : ((0,1,3,12,87,104,392,328,346,314,23,359), _ref_Abel_v_11_t),

(11,56 ) : ((0,1,26,50,76,246,255,146,513,271,123,555), _ref_Abel_v_11_t),

(11,60 ) : ((0,1,5,46,324,206,537,621,304,307,529,547), _ref_Abel_v_11_t),

(11,62 ) : ((0,1,11,31,395,251,605,55,336,321,6,213), _ref_Abel_v_11_t),

(11,66 ) : ((0,1,4,32,15,586,669,112,240,496,490,210), _ref_Abel_v_11_t),

(11,78 ) : ((0,1,4,31,97,264,277,746,816,808,298,741), _ref_Abel_v_11_t),

(11,80 ) : ((0,1,3,73,68,71,569,409,127,110,554,432), _ref_Abel_v_11_t),

(11,86 ) : ((0,1,13,32,17,236,380,340,849,855,189,774), _ref_Abel_v_11_t),

(11,90 ) : ((0,1,6,19,193,213,529,661,52,952,638,605), _ref_Abel_v_11_t),

(11,92 ) : ((0,1,4,80,177,182,508,581,511,664,25,425), _ref_Abel_v_11_t),

(11,102) : ((0,1,9,34,747,766,884,887,812,12,255,475), _ref_Abel_v_11_t),

(11,116) : ((0,1,3,16,692,7,36,183,201,846,661,759), _ref_Abel_v_11_t),

(11,120) : ((0,1,4,29,531,536,732,1167,65,1033,508,1255), _ref_Abel_v_11_t),

(11,128) : ((0,1,6,53,50,492,599,1230,430,131,1063,677), _ref_Abel_v_11_t),

(11,132) : ((0,1,4,81,626,632,694,1352,744,60,105,821), _ref_Abel_v_11_t),

(11,146) : ((0,1,7,18,92,176,193,1088,114,515,791,548), _ref_Abel_v_11_t),

(11,162) : ((0,1,8,28,314,323,401,1569,1197,1455,1269,382), _ref_Abel_v_11_t),

(11,170) : ((0,1,8,41,1573,1585,1686,1750,358,1732,271,340), _ref_Abel_v_11_t),

(11,182) : ((0,1,5,23,675,682,732,1800,1821,1485,763,1913), _ref_Abel_v_11_t),

(11,188) : ((0,1,5,29,1454,1463,1493,1838,903,98,1692,1846), _ref_Abel_v_11_t),

(11,192) : ((0,1,4,9,1842,1851,1876,2035,139,979,1027,350), _ref_Abel_v_11_t),

(11,198) : ((0,1,3,52,250,255,278,347,418,856,1298,780), _ref_Abel_v_11_t),

(11,206) : ((0,1,6,99,1465,1469,1501,1530,869,2074,1786,674), _ref_Abel_v_11_t),

(11,210) : ((0,1,8,39,2228,2244,2274,2293,188,2181,537,867), _ref_Abel_v_11_t),

(11,212) : ((0,1,9,32,2219,2241,2310,2319,1253,352,920,365), _ref_Abel_v_11_t),

(11,216) : ((0,1,5,15,1606,1611,1627,2101,211,1821,1564,1688), _ref_Abel_v_11_t),

(11,218) : ((0,1,8,23,1347,1352,1358,1846,1479,2157,1910,292), _ref_Abel_v_11_t),

(11,230) : ((0,1,6,33,2387,2394,2488,2518,1893,728,246,65), _ref_Abel_v_11_t),

(11,242) : ((0,1,8,57,378,392,404,637,1708,567,1356,1903), _ref_Abel_v_11_t),

(11,246) : ((0,1,7,97,389,400,413,1253,1625,1071,1756,1440), _ref_Abel_v_11_t),

(11,248) : ((0,1,6,67,2112,2118,2142,2181,365,1315,2336,1283), _ref_Abel_v_11_t),

(11,260) : ((0,1,5,20,1158,1165,1171,1609,449,1990,1546,1222), _ref_Abel_v_11_t),

(11,266) : ((0,1,4,45,2132,2136,2164,2354,2407,2194,1459,394), _ref_Abel_v_11_t),

(11,270) : ((0,1,9,31,2085,2089,2100,2348,57,748,1440,2254), _ref_Abel_v_11_t),

(11,276) : ((0,1,5,42,1905,1910,1925,2382,618,594,2820,322), _ref_Abel_v_11_t),

(11,288) : ((0,1,7,21,2651,2656,2694,2953,190,545,311,3063), _ref_Abel_v_11_t),

(11,290) : ((0,1,5,95,1487,1492,1512,1523,1599,939,2724,971), _ref_Abel_v_11_t),

(11,296) : ((0,1,7,68,856,860,868,2884,2872,2339,2965,1715), _ref_Abel_v_11_t),

(11,300) : ((0,1,9,24,2221,2232,2246,2349,2196,3173,2190,1661), _ref_Abel_v_11_t),

(11,302) : ((0,1,8,24,1273,1277,1290,1750,2662,733,511,1147), _ref_Abel_v_11_t),

(11,308) : ((0,1,4,29,1159,1168,1174,2322,2963,1778,3071,2317), _ref_Abel_v_11_t),

(11,312) : ((0,1,4,43,121,128,136,1266,2919,603,3199,2590), _ref_Abel_v_11_t),

(11,318) : ((0,1,8,36,2701,2712,2733,2995,3281,2830,1262,2203), _ref_Abel_v_11_t),

(11,330) : ((0,1,9,22,2312,2316,2326,2517,1311,488,1406,267), _ref_Abel_v_11_t),

(11,336) : ((0,1,3,69,117,126,133,456,1399,579,3469,1157), _ref_Abel_v_11_t),

(11,338) : ((0,1,9,52,1012,1017,1027,1511,3139,243,2560,139), _ref_Abel_v_11_t),

(11,350) : ((0,1,5,37,2650,2655,2666,3213,3709,86,3456,1383), _ref_Abel_v_11_t),

(11,356) : ((0,1,6,23,2647,2651,2657,2942,2733,1481,301,831), _ref_Abel_v_11_t),

(11,366) : ((0,1,6,28,1144,1151,1160,1349,392,1114,1006,1906), _ref_Abel_v_11_t),

(11,368) : ((0,1,9,47,1259,1263,1269,1319,1029,2121,2206,3959), _ref_Abel_v_11_t),

(11,372) : ((0,1,7,89,1015,1022,1035,1280,361,3425,1101,2744), _ref_Abel_v_11_t),

(11,378) : ((0,1,3,35,551,558,570,750,481,464,118,2491), _ref_Abel_v_11_t),

(11,396) : ((0,1,9,58,1938,1942,1956,2251,434,768,582,1489), _ref_Abel_v_11_t),

(11,402) : ((0,1,8,49,4331,4336,4350,4399,4169,1114,3877,3795), _ref_Abel_v_11_t),

(11,420) : ((0,1,9,23,207,214,220,359,1273,1500,1817,1048), _ref_Abel_v_11_t),

(11,422) : ((0,1,7,27,86,97,125,246,3796,3663,2211,2422), _ref_Abel_v_11_t),

(11,450) : ((0,1,7,31,4808,4812,4826,4931,1333,4783,1152,162), _ref_Abel_v_11_t),

(11,452) : ((0,1,5,58,4530,4536,4544,4568,3644,1121,561,1732), _ref_Abel_v_11_t),

(12,33 ) : ((0,1,117,331,131,309,321,386,204,276,278,40,118), _ref_Abel_v_12_t),

(12,35 ) : ((0,1,110,361,349,226,98,68,80,234,347,198,321), _ref_Abel_v_12_t),

(12,45 ) : ((0,1,128,372,85,361,484,394,242,41,412,388,480), _ref_Abel_v_12_t),

(12,51 ) : ((0,1,216,516,92,426,559,292,568,184,387,460,162), _ref_Abel_v_12_t),

(12,55 ) : ((0,1,354,581,101,391,639,534,523,252,338,379,77), _ref_Abel_v_12_t),

(12,59 ) : ((0,1,287,561,431,482,527,513,234,518,366,673,670), _ref_Abel_v_12_t),

(12,61 ) : ((0,1,289,562,361,385,125,613,219,637,686,732,185), _ref_Abel_v_12_t),

(12,63 ) : ((0,1,216,562,384,653,218,584,188,704,11,29,122), _ref_Abel_v_12_t),

(12,69 ) : ((0,1,527,449,471,497,677,20,778,88,366,721,753), _ref_Abel_v_12_t),

(12,71 ) : ((0,1,645,446,813,543,413,7,55,177,468,503,646), _ref_Abel_v_12_t),

(12,73 ) : ((0,1,607,719,837,496,240,645,184,829,451,830,770), _ref_Abel_v_12_t),

(12,83 ) : ((0,1,627,898,836,939,742,42,847,531,173,607,361), _ref_Abel_v_12_t),

(12,85 ) : ((0,1,778,1000,913,819,961,456,507,186,509,495,300), _ref_Abel_v_12_t),

(12,89 ) : ((0,1,602,894,827,661,350,647,304,47,430,533,550), _ref_Abel_v_12_t),

(12,91 ) : ((0,1,777,1054,855,892,792,134,224,740,240,898,631), _ref_Abel_v_12_t),

(12,93 ) : ((0,1,601,1004,872,557,599,819,381,248,270,1091,49), _ref_Abel_v_12_t),

(12,101) : ((0,1,787,1049,818,1064,288,346,464,958,1188,340,1192), _ref_Abel_v_12_t),

(12,103) : ((0,1,770,1027,806,1082,515,436,1096,1060,57,1135,1144), _ref_Abel_v_12_t),

(12,115) : ((0,1,747,1179,873,484,969,692,679,153,1237,1110,616), _ref_Abel_v_12_t),

(12,119) : ((0,1,701,1225,834,515,367,727,1349,407,891,1189,153), _ref_Abel_v_12_t),

(12,121) : ((0,1,713,1265,848,421,998,69,874,1126,693,467,1164), _ref_Abel_v_12_t),

(12,129) : ((0,1,623,1170,824,450,1099,418,948,177,207,797,59), _ref_Abel_v_12_t),

(12,133) : ((0,1,648,1157,822,371,407,180,1120,898,342,548,117), _ref_Abel_v_12_t),

(12,135) : ((0,1,712,1253,844,623,943,992,191,845,299,1381,611), _ref_Abel_v_12_t),

(12,139) : ((0,1,627,1216,711,489,642,904,733,1246,96,1617,12), _ref_Abel_v_12_t),

(12,141) : ((0,1,447,522,967,763,1035,344,93,561,1137,523,828), _ref_Abel_v_12_t),

(12,145) : ((0,1,426,582,937,534,1538,1606,1148,1436,191,1406,823), _ref_Abel_v_12_t),

(12,149) : ((0,1,420,509,957,593,835,1031,1502,319,1552,1047,993), _ref_Abel_v_12_t),

(12,155) : ((0,1,300,482,962,638,1207,1682,885,211,1838,1244,531), _ref_Abel_v_12_t),

(12,161) : ((0,1,455,318,952,400,470,584,1368,292,678,1138,383), _ref_Abel_v_12_t),

(12,169) : ((0,1,425,326,951,1211,1881,1063,1631,1363,1554,665,1600), _ref_Abel_v_12_t),

(12,171) : ((0,1,432,319,933,688,549,63,2002,1702,653,1081,1813), _ref_Abel_v_12_t),

(12,185) : ((0,1,404,324,935,605,366,360,178,221,533,1940,30), _ref_Abel_v_12_t),

(12,189) : ((0,1,303,329,957,866,2180,1899,597,2209,1186,994,1301), _ref_Abel_v_12_t),

(12,191) : ((0,1,491,527,939,377,1685,1735,1967,1176,391,2192,681), _ref_Abel_v_12_t),

(12,195) : ((0,1,331,313,934,384,2105,479,1546,86,184,1127,1822), _ref_Abel_v_12_t),

(12,199) : ((0,1,377,524,946,560,316,1591,2036,273,1841,2091,713), _ref_Abel_v_12_t),

(12,203) : ((0,1,324,312,933,341,547,68,39,1008,561,1372,1300), _ref_Abel_v_12_t),

(12,213) : ((0,1,343,312,933,378,229,60,1179,1781,1960,66,536), _ref_Abel_v_12_t),

(12,223) : ((0,1,463,316,933,413,970,1083,2322,491,1226,1809,560), _ref_Abel_v_12_t),

(12,229) : ((0,1,338,312,933,380,401,2398,612,1279,1514,268,528), _ref_Abel_v_12_t),

(12,233) : ((0,1,405,314,934,398,1053,310,2254,2250,2652,1300,1079), _ref_Abel_v_12_t),

(12,243) : ((0,1,486,314,933,375,697,151,1964,1623,1590,1756,1152), _ref_Abel_v_12_t),

(12,253) : ((0,1,322,312,933,395,1047,12,176,1859,881,1220,2465), _ref_Abel_v_12_t),

(12,255) : ((0,1,463,316,938,345,360,2537,2648,2270,789,2959,2796), _ref_Abel_v_12_t),

(12,259) : ((0,1,486,314,933,350,575,1962,2347,750,3054,2719,1841), _ref_Abel_v_12_t),

(12,265) : ((0,1,333,312,933,343,759,1754,2650,1633,2479,2718,1164), _ref_Abel_v_12_t),

(12,269) : ((0,1,432,312,938,345,567,2441,966,1935,470,2105,3043), _ref_Abel_v_12_t),

(12,271) : ((0,1,463,313,933,356,453,2869,793,748,2116,3126,2839), _ref_Abel_v_12_t),

(12,275) : ((0,1,477,313,943,358,474,2312,1258,52,1452,2370,260), _ref_Abel_v_12_t),

(12,281) : ((0,1,483,313,933,387,418,961,1586,766,2937,275,2569), _ref_Abel_v_12_t),

(12,289) : ((0,1,474,313,943,367,963,3147,2157,238,12,1610,2189), _ref_Abel_v_12_t),

(12,293) : ((0,1,423,335,945,397,235,2878,1793,2484,2440,503,1609), _ref_Abel_v_12_t),

(12,295) : ((0,1,428,337,931,406,360,1978,68,375,721,2390,2465), _ref_Abel_v_12_t),

(12,301) : ((0,1,436,351,924,367,1196,265,2527,720,664,105,250), _ref_Abel_v_12_t),

(12,303) : ((0,1,487,572,946,462,2646,2616,1249,3143,21,2537,2128), _ref_Abel_v_12_t),

(12,309) : ((0,1,417,327,944,341,1924,1975,2308,1234,1658,1829,1606), _ref_Abel_v_12_t),

(12,311) : ((0,1,435,557,937,371,267,428,1289,3355,2948,3030,861), _ref_Abel_v_12_t),

(12,321) : ((0,1,319,325,952,364,674,2128,643,393,1025,619,868), _ref_Abel_v_12_t),

(12,323) : ((0,1,445,344,920,365,567,3483,3364,1240,344,2683,3070), _ref_Abel_v_12_t),

(12,335) : ((0,1,478,557,969,462,1587,1457,2552,2575,2420,168,924), _ref_Abel_v_12_t),

(12,341) : ((0,1,498,362,954,440,584,421,3867,3964,404,664,2233), _ref_Abel_v_12_t),

(12,355) : ((0,1,415,329,927,512,615,2336,127,2245,2250,2272,1888), _ref_Abel_v_12_t),

(12,363) : ((0,1,541,368,971,370,297,555,148,4195,1197,1527,211), _ref_Abel_v_12_t),

(12,379) : ((0,1,424,545,948,415,378,1181,2984,3458,3288,3888,74), _ref_Abel_v_12_t),

(12,383) : ((0,1,477,534,964,441,246,972,2504,3957,3101,4366,2168), _ref_Abel_v_12_t),

(12,385) : ((0,1,543,334,943,531,793,1852,538,4231,4492,580,3816), _ref_Abel_v_12_t),

(12,399) : ((0,1,487,571,964,391,300,4515,2211,3063,2771,2586,1056), _ref_Abel_v_12_t),

(12,401) : ((0,1,442,543,964,514,567,763,3816,3621,2124,1092,1456), _ref_Abel_v_12_t),

(12,405) : ((0,1,433,552,963,385,684,63,4243,3494,3500,560,4611), _ref_Abel_v_12_t),

(12,409) : ((0,1,426,541,954,411,708,1875,2058,2443,1913,2924,3673), _ref_Abel_v_12_t),

(12,411) : ((0,1,430,558,963,397,372,492,2502,3948,18,1191,3761), _ref_Abel_v_12_t),

(12,413) : ((0,1,436,546,977,467,242,3695,682,483,3026,461,1334), _ref_Abel_v_12_t),

}

# Translate all V(m,t) into (mt+1,m+2;1,0;t)-QDM constructors

for (m,t),(vec,source) in iteritems(Vmt_vectors):

n,k,lmbda,mu,u = (m*t+1,m+2,1,0,t)

if not (n+u,lmbda) in QDM:

QDM[n+u,lmbda] = {}

QDM[n+u,lmbda][n,lmbda,mu,u] = (k,lambda m=m,t=t,vec=vec:QDM_from_Vmt(m,t,vec))

# Create the list of V(m,t) vectors for the doc

_all_m = sorted(set(m for m,_ in Vmt_vectors.keys()))

LIST_OF_VMT_VECTORS = "\n".join(" - `m={}` and `t=` ".format(m) +

", ".join("`{}`".format(t) for _,t in sorted(Vmt_vectors.keys()) if _ == m)

for m in _all_m)

r""""

Tests for the Vmt vectors

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_quasi_difference_matrix

sage: from sage.combinat.designs.orthogonal_arrays import QDM_from_Vmt

sage: from sage.combinat.designs.database import Vmt_vectors

sage: for (m,t),(vec,source) in sorted(Vmt_vectors.items()):

....: G,M = QDM_from_Vmt(m,t,vec)

....: if m*t < 600:

....: assert is_quasi_difference_matrix(M,G,m+2,1,1,t,verbose=1),(m,t)

....: assert len(source)>10

"""

DF = {

##############

# lambda = 1 #

##############

( 15, 3, 1):

{(15,): [[0,1,4],[0,2,9],[0,5,10]]},

( 21, 3, 1):

{(21,): [[0,1,3],[0,4,12],[0,5,11],[0,7,14]]},

( 21, 5, 1):

{(21,): [[0,1,4,14,16]]},

( 25, 3, 1):

{(25,): [[0,1,3],[0,4,11],[0,5,13],[0,6,15]]},

( 25, 4, 1):

{(5,5): [[(0,0),(0,1),(1,0),(2,2)],[(0,0),(0,2),(2,0),(4,4)]]},

( 27, 3, 1):

{(27,): [[0,1,3],[0,4,11],[0,5,15],[0,6,14],[0,9,18]]},

( 33, 3, 1):

{(33,): [[0,1,3],[0,4,10],[0,5,18],[0,7,19],[0,8,17],[0,11,22]]},

( 37, 4, 1):

{(37,): [[0,1,3,24],[0,4,26,32],[0,10,18,30]]},

( 39, 3, 1):

{(39,): [[0,1,3],[0,4,18],[0,5,27],[0,6,16],[0,7,15],[0,9,20],[0,13,26]]},

( 40, 4, 1):

{(40,): [[0,1,4,13],[0,2,7,24],[0,6,14,25],[0,10,20,30]]},

( 45, 3, 1):

{(45,): [[0,1,3],[0,4,10],[0,5,28],[0,7,34],[0,8,32],[0,9,29],[0,12,26],[0,15,30]]},

( 45, 5, 1):

{(3,3,5): [[(0,1,0),(0,2,0),(1,0,2),(2,0,2),(0,0,1)],

[(2,1,0),(1,2,0),(2,2,2),(1,1,2),(0,0,1)],

[(0,0,0),(0,0,1),(0,0,2),(0,0,3),(0,0,4)]]},

( 49, 3, 1):

{(49,): [[0,1,3],[0,4,9],[0,6,17],[0,7,23],[0,8,30],[0,10,31],[0,12,36],[0,14,34]]},

( 49, 4, 1):

{(49,): [[0,1,3,8,],[0,4,18,29],[0,6,21,33],[0,9,19,32]]},

( 51, 3, 1):

{(51,): [[0,1,3],[0,4,9],[0,6,25],[0,7,35],

[0,8,22],[0,10,21],[0,12,27],[0,13,31],[0,17,34]]},

( 52, 4, 1):

{(52,): [[0,1,3,7,],[0,5,19,35],[0,8,20,31],[0,9,24,34],[0,13,26,39]]},

( 55, 3, 1):

{(55,): [[0,1,3],[0,4,9],[0,6,16],[0,7,32],[0,8,29],

[0,11,42],[0,12,27],[0,14,36],[0,17,37]]},

( 57, 3, 1):

{(57,): [[0,1,3],[0,4,9],[0,6,13],[0,8,26],[0,10,33],

[0,11,32],[0,12,40],[0,14,41],[0,15,35],[0,19,38]]},

( 63, 3, 1):

{(63,): [[0,1,3],[0,4,9],[0,6,13],[0,8,25],[0,10,41],[0,11,44],

[0,12,36],[0,14,37],[0,15,43],[0,16,34],[0,21,42]]},

( 64, 4, 1):

{(64,): [[0,1,3,7,],[0,5,18,47],[0,8,33,44],

[0,9,19,43],[0,12,26,49],[0,16,32,48]]},

( 65, 5, 1):

{(65,): [[0,1,3,31,45],[0,4,10,19,57],[0,5,16,41,48],[0,13,26,39,52]]},

( 69, 3, 1):

{(69,): [[0,1,3],[0,4,9],[0,6,13],[0,8,24],[0,10,38],[0,11,47],[0,12,32],

[0,14,40],[0,15,50],[0,17,42],[0,18,39],[0,23,46]]},

( 75, 3, 1):

{(75,): [[0,1,67],[0,2,47],[0,3,41],[0,4,69],[0,5,68],[0,11,55],[0,13,61],

[0,15,33],[0,16,52],[0,17,43],[0,19,40],[0,22,51],[0,25,50]]},

( 76, 4, 1):

{(76,): [[0,1,7,22],[0,2,11,45],[0,3,59,71],[0,4,32,50],

[0,10,37,51],[0,13,36,60],[0,19,38,57]]},

( 81, 3, 1):

{(81,): [[0,1,39],[0,2,58],[0,3,34],[0,4,21],[0,5,67],[0,6,15],[0,7,36],

[0,8,59],[0,10,63],[0,11,37],[0,12,61],[0,13,48],[0,16,40],[0,27,54]]},

( 81, 5, 1):

{(81,): [[0,1,5,12,26],[0,2,10,40,64],[0,3,18,47,53],[0,9,32,48,68]]},

( 85, 4, 1):

{(85,): [[0,2,41,42],[0,17,32,38],[0,18,27,37],[0,13,29,36],

[0,11,31,35],[0,12,26,34,],[0,5,30,33]]},

( 91, 6, 1):

{(91,): [[0,1,3,7,25,38], [0,16,21,36,48,62], [0,30,40,63,74,82]]},

( 91, 7, 1): # from the La Jolla covering repository, attributed to Jan de Heer and Steve Muir

{(91,): [[8, 9, 14, 25, 58, 81, 85], [5, 33, 35, 42, 45, 67, 88], [4, 17, 30, 43, 56, 69, 82]]},

(121, 5, 1):

{(121,): [[0,14,26,51,60],[0,15,31,55,59],[0,10,23,52,58],

[0,3,36,56,57],[0,7,18,45,50],[0,8,30,47,49]]},

(121, 6, 1):

{(11,11): [[(0,0),(0,3),(0,4),(1,1),(1,7),(4,6)],

[(0,0),(0,2),(2,5),(4,7),(6,4),(8,0)],

[(0,0),(1,5),(2,0),(4,1),(6,0),(7,2)],

[(0,0),(1,0),(3,9),(4,8),(6,1),(9,5)]]},

(141, 5, 1):

{(141,): [[0,33,60,92,97],[0,3,45,88,110],[0,18,39,68,139],[0,12,67,75,113],

[0,1,15,84,94],[0,7,11,24,30],[0,36,90,116,125]]},

(161, 5, 1):

{(161,): [[0,19,34,73,80],[0,16,44,71,79],[0,12,33,74,78],[0,13,30,72,77],

[0,11,36,67,76],[0,18,32,69,75],[0,10,48,68,70],[0,3,29,52,53]]},

(175, 7, 1):

{(7,5,5): [[(0,0,0),(1,0,0),(2,0,0),(3,0,0),(4,0,0),(5,0,0),(6,0,0)],

[(0,0,0),(1,1,3),(1,4,2),(2,2,2),(2,3,3),(4,2,0),(4,3,0)],

[(0,0,0),(1,3,4),(1,2,1),(2,2,3),(2,3,2),(4,0,2),(4,0,3)],

[(0,0,0),(1,1,2),(1,4,3),(2,1,1),(2,4,4),(4,0,1),(4,0,4)],

[(0,0,0),(1,3,1),(1,2,4),(2,4,1),(2,1,4),(4,1,0),(4,4,0)]]},

(201, 5, 1):

{(201,): [[0,1,45,98,100],[0,3,32,65,89],[0,4,54,70,75],[0,6,49,69,91],[0,7,58,81,95],

[0,8,34,72,90],[0,9,36,77,96],[0,10,35,83,94],[0,12,40,79,92],[0,15,46,76,93]]},

(217, 7, 1):

{(217,): [[0,1,37,67,88,92,149],[0,15,18,65,78,121,137],[0,8,53,79,85,102,107],

[0,11,86,100,120,144,190],[0,29,64,165,198,205,207],[0,31,62,93,124,155,186]]},

(221, 5, 1):

{(221,): [[0,1,24,61,116],[0,3,46,65,113],[0,4,73,89,130],[0,5,77,122,124],

[0,6,39,50,118],[0,7,66,81,94],[0,8,38,64,139],[0,9,29,80,107],

[0,10,35,93,135],[0,12,34,52,88],[0,14,31,63,84]]},

(259, 7, 1): # the following one is lemma 2.2 in Abel "Some new BIBDs with block size 7"

{(7,37): [[(0,0),(1,0),(2,0),(3,0),(4,0),(5,0),(6,0)],

[(0,0),(0,1),(0,6),(1,4),(2,19),(3,25),(6,26)],

[(0,0),(0,10),(0,23),(2,3),(4,5),(5,1),(6,28)],

[(0,0),(0,8),(0,26),(1,13),(3,10),(4,30),(5,21)],

[(0,0),(0,4),(1,25),(1,34),(2,33),(2,35),(4,10)],

[(0,0),(0,3),(1,26),(2,7),(2,28),(4,17),(4,34)],

[(0,0),(0,30),(1,7),(1,22),(2,1),(4,21),(4,33)]]},

##############

# lambda = 2 #

##############

( 16, 3, 2):

{(16,): [[0,1,2],[0,2,8],[0,3,7],[0,4,7],[0,5,10]]},

( 28, 3, 2):

{(28,): [[0,1,12],[0,2,11],[0,2,12],[0,3,7],[0,3,13],

[0,4,9],[0,5,13],[0,6,7],[0,6,14]]},

( 40, 3, 2):

{(40,): [[0,1,4],[0,1,16],[0,2,7],[0,2,9],[0,3,17],[0,4,17],[0,5,19],

[0,6,16],[0,6,18],[0,8,18],[0,8,19],[0,9,20],[0,12,25]]},

( 19, 4, 2):

{(19,): [[0,1,3,12],[0,1,5,13],[0,4,6,9]]},

( 21, 4, 3):

{(21,): [[0,2,3,7],[0,3,5,9],[0,1,7,11],[0,2,8,11],[0,1,9,14]]},

( 22, 4, 2):

{(22,): [[0,4,16,17],[0,12,14,21],[0,14,16,19],[0,4,11,15]]},

( 31, 4, 2):

{(31,): [[0,1,8,11],[0,1,13,17],[0,2,11,14],[0,5,7,13],[0,5,9,15]]},

( 34, 4, 2):

{(34,): [[0,1,22,24],[0,1,19,25],[0,2,6,29],[0,4,7,20],[0,5,8,20],[0,8,17,25]]},

( 43, 4, 2):

{(43,): [[0,1,6,36],[0,3,18,22],[0,9,11,23],[0,10,12,26],[0,26,27,33],

[0,13,35,38],[0,19,28,39,]]},

( 46, 4, 2):

{(46,): [[0,2,7,10],[0,4,19,32],[0,10,34,35],[0,5,8,24],[0,26,30,39],

[0,17,26,32],[0,28,34,45],[0,2,23,25]]},

(31, 5, 2):

{(31,): [[0,1,3,7,15],[0,3,9,14,21],[0,4,5,13,15,]]},

( 35, 5, 2):

{(35,): [[0,2,8,12,13],[0,3,18,22,27],[0,17,23,32,33],

[0,7,14,21,28],[0,7,14,21,28]]},

( 51, 5, 2):

{(51,): [[0,1,14,31,35],[0,1,9,23,33],[0,11,16,18,42],

[0,7,13,36,39],[0,4,10,12,15]]},

( 71, 5, 2):

{(71,): [[1,5,25,54,57],[3,4,15,20,29],[9,12,16,45,60],[27,36,38,48,64],

[2,10,37,43,50],[6,8,30,40,58],[18,19,24,32,49]]},

( 46, 6, 2):

{(46,): [[0,1,3,11,31,35],[0,1,4,10,23,29],[0,2,7,15,32,41]]},

( 61, 6, 2):

{(61,): [[12,15,28,34,35,59],[1,13,18,47,51,53],

[8,10,11,21,29,43],[16,20,25,32,40,50]]},

( 43, 7, 2):

{(43,): [[0,1,11,19,31,38,40],[0,2,10,16,25,38,42]]},

( 64, 7, 2):

{(64,): [[0,1,2,4,7,28,52],[0,4,9,21,31,39,53],[0,6,15,23,34,41,54]]},

( 75, 5, 2):

{(5,15): [[(0,0),(1,10),(1,8),(4,1),(4,2)],

[(0,0),(2,5),(2,10),(3,7),(3,13)],

[(0,0),(1,10),(1,2),(4,4),(4,8)],

[(0,0),(2,5),(2,10),(3,14),(3,11)],

[(0,0),(1,4),(1,5),(4,1),(4,8)],

[(0,0),(1,1),(1,5),(4,4),(4,2)],

[(0,0),(2,7),(2,13),(3,1),(3,4)],

[(0,0),(1,0),(2,0),(3,0),(4,0)],

[(0,0),(1,0),(2,0),(3,0),(4,0)]]},

( 85, 7, 2):

{(85,): [[0,1,11,20,32,35,39],[0,2,6,16,29,50,65],

[0,3,9,27,55,72,80],[0,5,7,30,47,48,59]]},

( 85, 8, 2):

{(85,): [[24,31,39,50,67,68,70,82],[20,49,51,55,56,60,72,81],

[9,19,29,37,43,56,59,81]]},

(153, 9, 2):

{(3,3,17): [[(0,0,0),(0,1,0),(0,2,0),(1,0,0),(1,1,0),(1,2,0),(2,0,0),(2,1,0),(2,2,0)],

[(0,0,0),(0,1,0),(0,2,0),(1,0,0),(1,1,0),(1,2,0),(2,0,0),(2,1,0),(2,2,0)],

[(0,0,0),(0,1,1),(0,1,16),(0,2,4),(0,2,13),(1,0,3),(1,0,14),(2,0,5),(2,0,12)],

[(0,0,0),(0,1,2),(0,1,15),(0,2,8),(0,2,9),(1,0,6),(1,0,11),(2,0,10),(2,0,7)],

[(0,0,0),(0,1,3),(0,1,14),(0,2,12),(0,2,5),(1,0,9),(1,0,8),(2,0,15),(2,0,2)],

[(0,0,0),(0,1,6),(0,1,11),(0,2,7),(0,2,10),(1,0,1),(1,0,16),(2,0,13),(2,0,4)]]},

(181,10, 2):

{(181,): [[1,7,40,42,51,59,113,125,135,151],

[19,22,31,35,36,64,74,133,154,156],

[10,15,34,47,58,65,83,87,161,164],

[12,18,32,52,77,78,142,157,165,172]]},

##############

# lambda = 3 #

##############

( 21, 6, 3):

{(21,): [[0,2,10,15,19,20],[0,3,7,9,10,16]]},

( 41, 6, 3):

{(41,): [[0,1,10,16,18,37],[0,6,14,17,19,26],

[0,2,20,32,33,36],[0,11,12,28,34,38]]},

( 51, 6, 3):

{(51,): [[15,17,18,27,34,48],[3,17,30,34,42,45],[9,17,24,33,34,39],

[3,25,41,43,44,48],[3,5,25,29,43,48]]},

( 61, 6, 3):

{(61,): [[0,1,9,20,58,34],[0,2,7,18,40,55],[0,4,14,19,36,49],

[0,8,11,28,37,38],[0,13,15,16,22,56],[0,26,30,32,44,51]]},

( 29, 7, 3):

{(29,): [[1,7,16,20,23,24,25],[2,3,11,14,17,19,21]]},

( 43, 7, 3):

{(43,): [[1,4,11,16,21,35,41],[3,5,12,19,20,33,37],[9,13,14,15,17,25,36]]},

( 57, 7, 3):

{(57,): [[0,1,11,12,15,35,53],[0,7,17,20,27,29,48],

[0,5,18,26,32,49,51],[0,2,6,9,14,41,42]]},

( 61,10, 3):

{(61,): [[1,4,18,20,32,35,36,41,42,54],[11,13,14,21,23,28,34,39,43,47]]},

( 71, 7, 3):

{(71,): [[1,20,30,32,37,45,48],[2,3,19,25,40,60,64],[4,6,9,38,49,50,57],

[5,8,12,18,27,29,43],[10,15,16,24,36,54,58]]},

( 85, 7, 3):

{(85,): [[0,7,23,27,28,31,71],[0,12,22,41,61,74,79],

[0,6,11,13,38,42,77],[0,1,7,16,19,27,49],

[0,9,26,39,54,56,71],[0,2,3,12,37,53,63]]},

( 97, 9, 3):

{(97,): [[1,2,25,35,46,58,61,70,90],[3,4,8,38,43,50,69,86,87],

[6,12,16,32,53,55,57,75,82],[9,18,24,26,31,34,37,48,64]]},

( 49, 9, 3):

{(49,): [[0,1,3,5,9,14,19,25,37],[0,2,12,13,16,19,34,41,42]]},

(121,10, 3):

{(11,11): [[(0,1),(0,3),(0,4),(0,5),(0,9),(1,8),(3,2),(4,10),(5,7),(9,6)],

[(1,2),(3,6),(4,8),(5,10),(9,7),(10,2),(8,6),(7,8),(6,10),(2,7)],

[(1,7),(3,10),(4,6),(5,2),(9,8),(1,4),(3,1),(4,5),(5,9),(9,3)],

[(10,10),(8,8),(7,7),(6,6),(2,2),(1,0),(3,0),(4,0),(5,0),(9,0)]]},

###############

# lambda = 4 #

###############

( 22, 7, 4):

{(22,): [[0,2,6,8,9,10,13],[0,3,5,6,12,13,17]]},

( 29, 8, 4):

{(29,): [[0,1,7,16,20,23,24,25],[0,2,3,11,14,17,19,21]]},

( 71, 8, 4):

{(71,): [[0,1,20,30,32,37,45,48],[0,2,3,19,25,40,60,64],

[0,4,6,9,38,49,50,57],[0,5,8,12,18,27,29,43],

[0,10,15,16,24,36,54,58]]},

( 43, 8, 4):

{(43,): [[0,1,4,11,16,21,35,41],[0,3,5,12,19,20,33,37],

[0,9,13,14,15,17,25,36]]},

( 46,10, 4):

{(46,): [[3,7,13,16,23,24,25,28,30,42],[2,10,12,18,25,34,40,43,44,45]]},

( 55, 9, 4):

{(55,): [[0,4,21,25,26,42,45,53,54],[0,6,8,25,37,39,45,48,52],

[2,5,6,13,15,20,25,39,45]]},

( 67,12, 4):

{(67,): [[1,8,23,25,28,29,31,37,47,54,55,64],

[3,20,25,32,36,39,44,45,54,55,57,59]]},

##############

# lambda = 5 #

##############

( 13, 5, 5):

{(13,): [[0,1,2,4,8],[0,1,3,6,12],[0,2,5,6,10]]},

( 17, 5, 5):

{(17,): [[0,1,4,13,16],[0,3,5,12,14],[0,2,8,9,15],[0,6,7,10,11]]},

( 21, 6, 5):

{(21,): [[0,2,6,12,15,16],[0,3,6,7,11,19],

[0,7,15,16,17,18],[0,2,7,9,14,16]]},

( 22, 6, 5):

{(22,): [[0,1,2,5,10,13],[0,1,5,6,8,15],

[0,2,3,6,16,18],[0,2,6,11,13,17]]},

( 28, 6, 5):

{(28,): [[0,4,7,8,16,21],[5,7,8,9,14,20],[7,12,14,16,17,25],

[1,4,7,13,14,24],[2,4,8,16,18,22]]},

( 33, 5, 5):

{(33,): [[0,2,3,7,25],[0,3,13,14,29],[0,4,5,12,13],[0,2,12,16,26],

[0,3,12,20,31],[3,9,12,15,27],[0,8,13,14,31],[0,2,7,13,29]]},

( 33, 6, 5):

{(33,): [[0,3,12,17,18,28],[0,2,3,16,28,29],[0,16,20,26,28,30],

[0,2,3,12,16,27],[0,6,20,21,28,30],[0,4,11,15,22,26]]},

( 37,10, 5):

{(37,): [[0,1,7,9,10,12,16,26,33,34],[0,2,14,15,18,20,24,29,31,32]]},

( 39, 6,5):

{(39,): [[0,3,4,17,19,32],[0,1,5,12,30,36],[0,3,8,9,25,27],[0,7,10,12,17,21],

[0,16,18,19,27,35],[0,2,18,27,28,33],[0,6,13,19,26,32]]},

( 45,11, 5):

{(45,): [[1,3,7,10,22,25,30,35,37,38,44],[0,2,3,14,22,26,27,28,31,32,38]]},

( 46,10, 5):

{(46,): [[0,4,6,11,12,15,24,25,28,42],[0,2,5,7,8,9,14,24,34,35],

[0,2,12,32,40,23,25,35,9,17]]},

( 55,10, 5):

{(55,): [[0,5,11,15,20,22,25,33,44,45],[3,7,8,10,31,37,39,45,46,49],

[3,7,8,10,31,37,39,45,46,49]]},

( 67,11, 5):

{(67,): [[1,9,14,15,22,24,25,40,59,62,64],[2,13,18,28,30,44,48,50,51,57,61],

[4,21,26,29,33,35,36,47,55,56,60]]},

( 73,10, 5):

{(73,): [[0,1,2,4,8,16,32,37,55,64],[0,5,7,10,14,20,28,39,40,56],

[0,25,27,35,49,50,54,61,67,70],[0,11,15,21,22,30,42,44,47,60]]},

###############

# lambda >= 6 #

###############

( 11, 4,6):

{(11,): [[0,1,8,9],[0,2,5,7],[0,1,4,5],[0,2,3,5],[0,4,5,9]]},

( 15, 4,6):

{(15,): [[0,1,2,3],[0,2,4,6],[0,4,8,12],[0,8,1,9],

[3,6,9,12],[0,1,5,10],[0,2,5,10]]},

( 15, 5,6):

{(15,): [[0,1,2,3,6],[0,2,4,7,8],[0,2,4,9,10],

[0,3,6,10,11],[0,3,6,9,12]]},

( 21, 8,14):

{(21,): [[0,9,10,13,14,15,18,19],[0,1,4,7,9,15,16,18],[0,1,2,4,6,14,15,16],

[0,1,3,4,8,14,16,18],[0,1,4,9,11,12,14,16]]},

( 21, 10, 9):

{(21,): [[0,1,2,3,4,7,8,11,14,16],[0,6,7,9,11,12,15,16,17,19]]},

( 22, 8, 8):

{(22,): [[0,1,5,7,13,17,20,21],[0,2,7,11,13,14,16,17],[0,3,4,12,14,15,17,21]]},

( 22, 8,12):

{(22,): [[1,2,3,5,6,9,15,18], [1,2,3,5,8,9,10,15],

[1,3,4,9,13,18,19,21], [2,4,6,12,13,15,17,1],

[2,4,8,12,13,15,19,1], [2,4,8,16,13,15,19,5]]},

( 25, 7, 7):

{(5,5): [[(0,0),(0,1),(0,4),(1,1),(1,2),(4,3),(4,4)],

[(0,0),(1,0),(1,3),(2,3),(3,2),(4,0),(4,2)],

[(0,0),(0,2),(0,3),(2,2),(2,4),(3,1),(3,3)],

[(0,0),(1,4),(2,0),(2,1),(3,0),(3,4),(4,1)]]},

( 29, 8,6):

{(29,): [[0,5,10,11,12,13,16,20],[0,8,10,12,17,22,23,26],

[0,4,5,11,13,23,25,26]]},

( 34,12, 8):

{(34,): [[0,5,9,14,15,17,20,25,26,27,28,30],

[0,6,7,10,13,17,18,20,22,24,25,26]]},

( 34,12,10):

{(34,): [[0,2,3,4,8,9,11,13,14,24,27,30],

[0,2,6,7,8,11,13,14,22,25,26,32],

[0,2,10,18,22,32,17,19,27,1,5,15]]},

( 43,15,10):

{(43,): [[1,3,6,13,18,21,22,25,26,27,33,35,36,38,40],

[9,10,11,13,16,17,19,23,26,27,28,33,35,38,39]]},

( 45,12, 3):

{(3,3,5): [[(0,0,0),(0,0,1),(0,0,2),(0,2,1),(0,0,3),(0,1,1),

(1,0,0),(1,1,2),(1,2,3),(2,0,0),(2,1,3),(2,2,2)]]},

( 46,10, 6):

{(46,): [[0,2,11,13,21,22,30,33,34,40],[0,2,6,7,22,23,28,32,35,38],

[0,2,4,7,8,9,12,23,26,41]]},

( 49,21,10):

{(7,7): [[(0,1),(0,2),(0,4),(1,1),(1,2),(1,4),(2,1),(2,2),(2,4),(3,1),(3,2),

(3,4),(4,1),(4,2),(4,4),(5,1),(5,2),(5,4),(6,1),(6,2),(6,4)],

[(1,0),(1,1),(1,2),(1,4),(2,0),(2,1),(2,2),(2,4),(4,0),(4,1),(4,2),

(4,4),(3,3),(3,5),(3,6),(5,3),(5,5),(5,6),(6,3),(6,5),(6,6)]]},

( 53,13, 6):

{(53,): [[1,10,13,15,16,24,28,36,42,44,46,47,49],

[2,3,19,20,26,30,31,32,35,39,41,45,48]]},

( 53,14, 7):

{(53,): [[0,1,10,13,15,16,24,28,36,42,44,46,47,49],

[0,2,3,19,20,26,30,31,32,35,39,41,45,48]]},

( 61,15, 7):

{(61,): [[0,1,3,4,8,10,13,22,30,35,44,45,46,50,58],

[0,1,3,5,13,18,29,34,35,37,41,43,44,51,55]]},

( 67,12, 6):

{(67,): [[0,1,9,14,15,22,24,25,40,59,62,64],

[0,2,13,18,28,30,44,48,50,51,57,61],

[0,4,21,26,29,33,35,36,47,55,56,60]]},

# a 133-cyclic set from Ken Smith database

# see http://www.ccrwest.org/diffsets/diff_sets/DS_133_33_8_133.html

(133,33, 8):

{(133,): [[0,4,7,8,15,17,19,22,24,25,29,30,38,

47,49,50,55,58,61,62,71,73,76,77,78,

82,95,111,113,114,121,123,127]]},

# a 901-cyclic

# see http://www.ccrwest.org/diffsets/diff_sets/DS_901_225_56_901.html

(901,225,56):

{(901,): [[ 0, 1, 5, 9, 12, 13, 14, 16, 22, 25, 41, 43,

45, 47, 53, 59, 60, 65, 69, 70, 71, 79, 80, 81,

89, 92, 93,106,108,109,110,114,117,124,125,126,

133,139,144,147,152,156,159,167,168,169,173,174,

182,183,192,194,196,198,202,203,205,208,209,212,

214,215,219,222,223,224,225,226,229,231,232,233,

235,244,254,256,259,264,265,274,277,286,292,293,

295,296,300,307,308,313,318,319,325,326,345,350,

352,355,363,369,371,379,382,387,394,395,397,400,

401,402,405,407,419,422,423,424,433,445,447,460,

461,465,467,469,477,484,492,498,502,503,516,523,

526,529,530,531,533,536,540,543,545,550,559,564,

570,571,574,577,579,581,583,585,587,596,599,602,

611,617,618,620,621,622,625,630,634,636,639,641,

656,658,661,664,665,688,689,691,694,695,706,708,

711,713,720,721,724,729,735,737,742,746,752,760,

766,767,772,778,780,786,795,801,813,824,826,827,

828,835,837,840,843,845,848,849,852,853,859,862,

863,865,870,874,878,881,886,897,898]]}

}

# Create the list of DF for the documentation

_all_l = sorted(set(l for v,k,l in DF.keys()))

LIST_OF_DF = "\n".join(" - `\lambda={}`:\n ".format(l) +

", ".join("`({},{},{})`".format(v,k,l) for v,k,_ in sorted(DF.keys()) if _ == l)

for l in _all_l)

def DM_12_6_1():

r"""

Return a `(12,6,1)`-difference matrix as built in [Hanani75]_.

This design is Lemma 3.21 from [Hanani75]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_12_6_1

sage: G,M = DM_12_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(12,6)

REFERENCES:

.. [Hanani75] Haim Hanani,

Balanced incomplete block designs and related designs,

:doi:`10.1016/0012-365X(75)90040-0`,

Discrete Mathematics, Volume 11, Issue 3, 1975, Pages 255-369.

"""

from sage.groups.additive_abelian.additive_abelian_group import AdditiveAbelianGroup

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(2).cartesian_product(AdditiveCyclic(6))

M = [[(0,0),(0,0),(0,0),(0,0),(0,0),(0,0)],

[(0,0),(0,1),(1,0),(0,3),(1,2),(0,4)],

[(0,0),(0,2),(1,2),(1,0),(0,1),(1,5)],

[(0,0),(0,3),(0,2),(0,1),(1,5),(1,4)],

[(0,0),(0,4),(1,1),(1,3),(0,5),(0,2)],

[(0,0),(0,5),(0,1),(1,5),(1,3),(1,1)],

[(0,0),(1,0),(1,3),(0,2),(0,3),(1,2)],

[(0,0),(1,1),(1,5),(1,2),(1,4),(1,0)],

[(0,0),(1,2),(0,4),(0,5),(0,2),(1,3)],

[(0,0),(1,3),(1,4),(0,4),(1,1),(0,1)],

[(0,0),(1,4),(0,5),(1,1),(1,0),(0,3)],

[(0,0),(1,5),(0,3),(1,4),(0,4),(0,5)]]

return G,M

def DM_21_6_1():

r"""

Return a `(21,6,1)`-difference matrix.

As explained in the Handbook III.3.50 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_21_6_1

sage: G,M = DM_21_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(21,6)

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

M = [[ 8, 17, 20, 2],

[ 9, 16, 4, 15],

[ 11, 5, 10, 6],

[ 14, 1, 3, 13],

[ 18, 19, 12, 7]]

Mb = [[0,0,0,0,0,0]]

for a,b,c,d,e in zip(*M):

Mb.append([a,b,c,d,e,0])

Mb.append([b,c,d,e,a,0])

Mb.append([c,d,e,a,b,0])

Mb.append([d,e,a,b,c,0])

Mb.append([e,a,b,c,d,0])

return AdditiveCyclic(21), Mb

def DM_24_8_1():

r"""

Return a `(24,8,1)`-difference matrix.

As explained in the Handbook III.3.52 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_24_8_1

sage: G,M = DM_24_8_1()

sage: is_difference_matrix(M,G,8,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(24,8)

"""

M = ("0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 "+

"0000 0010 0100 0110 1000 1010 1100 1110 2000 2010 2100 2110 "+

"0000 0011 1001 2110 0111 2011 2111 1000 0100 1100 1101 2010 "+

"0000 1010 1011 2000 1101 2110 0001 0101 2100 2001 0111 1100 "+

"0000 0001 2010 1111 2111 2100 1101 0011 1010 2101 1000 0110 "+

"0000 1000 2001 1011 0100 1100 0110 2101 2111 0010 1111 2011 "+

"0000 1001 0111 2100 2000 0010 1110 2011 1100 1011 0101 2111 "+

"0000 1011 2101 0100 2110 1001 2000 0110 0101 1111 2011 1010 ")

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

from sage.categories.cartesian_product import cartesian_product

G = cartesian_product([AdditiveCyclic(_) for _ in [2, 2, 6]])

rlabel = {(x%2,x%3):x for x in range(6)}

M = [G([int(c),int(d),rlabel[int(b),int(a)]]) for a,b,c,d in M.split()]

M = [M[i*12:(i+1)*12] for i in range(8)]

Mb = []

for a,b,c,d,e,f,g,h in zip(*M):

Mb.append([a,b,c,d,e,f,g,h])

Mb.append([a + G([0,0,rlabel[0,0]]),

b + G([0,1,rlabel[0,0]]),

c + G([1,0,rlabel[0,0]]),

d + G([1,1,rlabel[0,0]]),

e + G([0,0,rlabel[1,0]]),

f + G([0,1,rlabel[1,0]]),

g + G([1,0,rlabel[1,0]]),

h + G([1,1,rlabel[1,0]])])

return G, Mb

def DM_28_6_1():

r"""

Return a `(28,6,1)`-difference matrix.

As explained in the Handbook III.3.54 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_28_6_1

sage: G,M = DM_28_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(28,6)

"""

z=2

M = [

[(0,0), (z+1,6),(1,1) ,(1,1) ,(1,3) ,(1,4) ,(0,0) ,(1,4), (z,5) ],

[(z,2), (0,0) ,(1,5) ,(z,1) ,(z,2) ,(z,6) ,(z+1,3),(0,0), (z,1) ],

[(z,3), (z+1,4),(0,0) ,(z+1,5),(z+1,2),(z+1,4),(z+1,2),(1,6), (0,0) ],

[(0,5), (z,6) ,(0,5) ,(0,6) ,(z,3) ,(0,0) ,(0,4) ,(1,5), (z+1,4)],

[(0,3), (0,3) ,(z+1,5),(0,0) ,(0,5) ,(z+1,6),(1,1) ,(0,1), (z,3) ],

[(1,3), (0,6) ,(0,6) ,(1,5) ,(0,0) ,(0,3) ,(z+1,6),(z,2), (0,2) ],

]

from sage.groups.additive_abelian.additive_abelian_group import AdditiveAbelianGroup

from sage.modules.free_module_element import free_module_element as vector

G = AdditiveAbelianGroup([2,2,7])

M = [[G(vector([x//2,x%2,y])) for x,y in L] for L in M]

Mb=[[0,0,0,0,0,0]]

for R in zip(*M):

a,b,c,d,e,f = R

Mb.append([a,b,c,d,e,f])

Mb.append([b,c,a,f,d,e])

Mb.append([c,a,b,e,f,d])

return G, Mb

def DM_33_6_1():

r"""

Return a `(33,6,1)`-difference matrix.

As explained in the Handbook III.3.56 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_33_6_1

sage: G,M = DM_33_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(33,6)

"""

M = [

[ 0, 0, 0, 0, 0, 0],

[ 15, 11, 22, 4, 17, 8],

[ 19, 7, 14, 32, 22, 18],

[ 22, 19, 8, 24, 21, 6],

[ 9, 12, 15, 7, 26, 14],

[ 14, 28, 23, 2, 19, 3]

]

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(33)

Mb = [[0, 0, 0, 0, 0, 0],

[1, 4,16,31,25,22],

[7,28,13,19,10, 0]]

for R in zip(*M):

a,b,c,d,e,f = R

for i in range(5):

Mb.append([a,b,c,d,e,f])

a,b,c,d,e,f = 4*e,4*a,4*b,4*c,4*d,4*f

return G, Mb

def DM_35_6_1():

r"""

Return a `(35,6,1)`-difference matrix.

As explained in the Handbook III.3.58 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_35_6_1

sage: G,M = DM_35_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(35,6)

"""

M = [

[ 0, 15, 30, 10, 25, 1, 16, 31, 11, 26, 2, 17, 32, 12, 6, 3, 18, 33, 27, 21, 4, 19, 13, 7, 22, 5, 34, 28, 8, 23, 20, 14, 29, 9, 24],

[ 0, 22, 16, 3, 4, 9, 10, 32, 26, 13, 18, 5, 27, 14, 15, 20, 7, 1, 23, 31, 29, 2, 24, 11, 19, 17, 25, 12, 6, 28, 33, 34, 21, 8, 30],

[ 0, 29, 2, 31, 18, 10, 32, 26, 34, 28, 27, 21, 15, 9, 17, 30, 3, 4, 5, 20, 12, 6, 14, 22, 16, 8, 23, 24, 25, 33, 11, 19, 13, 7, 1],

[ 0, 8, 9, 17, 11, 25, 19, 27, 28, 1, 15, 23, 31, 4, 26, 12, 6, 14, 29, 16, 2, 3, 18, 33, 34, 20, 7, 22, 30, 24, 10, 32, 5, 13, 21],

[ 0, 1, 23, 24, 32, 33, 6, 7, 29, 30, 10, 11, 12, 13, 28, 8, 9, 31, 4, 5, 27, 14, 15, 16, 3, 25, 26, 34, 21, 22, 2, 17, 18, 19, 20],

[0]*35

]

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(35)

return G, list(zip(*M))

def DM_36_9_1():

r"""

Return a `(36,9,1)`-difference matrix.

As explained in the Handbook III.3.59 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_36_9_1

sage: G,M = DM_36_9_1()

sage: is_difference_matrix(M,G,9,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(36,9)

"""

M = [

[(0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0), (0,0,0,0)],

[(0,0,0,0), (0,1,0,0), (1,0,0,0), (1,1,0,0), (0,0,0,1), (0,1,0,1), (1,0,0,1), (1,1,0,1), (0,0,0,2), (0,1,0,2), (1,0,0,2), (1,1,0,2)],

[(0,0,0,0), (1,1,1,2), (0,0,2,1), (0,0,1,2), (0,1,2,0), (0,1,0,2), (1,1,1,1), (0,1,1,1), (1,1,1,0), (1,0,2,2), (1,0,0,1), (1,0,1,0)],

[(0,0,0,0), (0,0,1,0), (1,0,1,0), (0,1,0,0), (1,1,0,0), (1,0,2,0), (1,0,0,0), (0,1,2,0), (1,1,2,0), (0,0,2,0), (1,1,1,0), (0,1,1,0)],

[(0,0,0,0), (0,1,2,0), (0,0,1,0), (1,1,1,0), (1,0,2,0), (1,0,1,0), (0,1,0,0), (0,0,2,0), (0,1,1,0), (1,1,0,0), (1,1,2,0), (1,0,0,0)],

[(0,0,0,0), (0,1,1,0), (0,1,2,0), (1,1,2,0), (1,1,0,2), (0,0,1,2), (1,1,2,2), (1,0,0,2), (1,0,0,1), (1,0,1,1), (0,0,2,1), (0,1,1,1)],

[(0,0,0,0), (1,0,1,0), (1,1,0,1), (1,0,1,2), (1,0,2,2), (0,0,2,1), (0,1,0,1), (0,1,0,0), (1,1,2,2), (0,1,1,0), (0,0,1,2), (1,1,2,1)],

[(0,0,0,0), (1,1,0,0), (0,1,1,0), (1,0,2,1), (0,1,0,2), (1,0,2,2), (0,0,2,2), (1,1,1,0), (1,0,1,1), (0,1,2,1), (1,1,1,1), (0,0,0,2)],

[(0,0,0,0), (1,0,0,0), (1,1,1,0), (0,1,1,2), (1,1,2,1), (0,1,1,1), (0,0,1,1), (1,0,2,0), (0,1,2,2), (1,1,0,2), (1,0,2,2), (0,0,0,1)]

]

from sage.groups.additive_abelian.additive_abelian_group import AdditiveAbelianGroup

from sage.modules.free_module_element import free_module_element as vector

G = AdditiveAbelianGroup([2,2,3,3])

M = [[G(vector(x)) for x in L] for L in M]

Mb=[]

for R in zip(*M):

a,b,c,d,e,f,g,h,i = R

for y in range(3):

Mb.append([

a+G(vector([0,0,0,0])),

b+G(vector([0,0,y,0])),

c+G(vector([0,0,2*y,0])),

d+G(vector([0,0,0,y])),

e+G(vector([0,0,0,2*y])),

f+G(vector([0,0,y,y])),

g+G(vector([0,0,2*y,2*y])),

h+G(vector([0,0,y,2*y])),

i+G(vector([0,0,2*y,y]))])

return G, Mb

def DM_39_6_1():

r"""

Return a `(39,6,1)`-difference matrix.

As explained in the Handbook III.3.61 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_39_6_1

sage: G,M = DM_39_6_1()

sage: is_difference_matrix(M,G,6,1)

True

The design is available from the general constructor::

sage: designs.difference_matrix(39,6,existence=True)

True

"""

M = [

[ 0, 0, 0, 0, 0, 0],

[ 4, 23, 13, 5, 12, 11],

[ 25, 11, 22, 34, 23, 6],

[ 13, 4, 20, 17, 15, 29],

[ 27, 21, 8, 16, 19, 26],

[ 16, 19, 34, 38, 26, 21]

]

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(39)

Mb=[[ 0, 0, 0, 0, 0, 0],

[ 1, 16, 22, 17, 38, 23],

[-1,-16,-22,-17,-38,-23]]

for R in zip(*M):

a,b,c,d,e,f = map(G,R)

for i in range(3):

Mb.append([ a, b, c, d, e, f])

Mb.append([-a,-b,-c,-d,-e,-f])

a,b,c,d,e,f = [16*x for x in [c,a,b,f,d,e]]

return G,Mb

def DM_44_6_1():

r"""

Return a `(44,6,1)`-difference matrix.

As explained in the Handbook III.3.64 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_44_6_1

sage: G,M = DM_44_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(44,6)

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

from sage.categories.cartesian_product import cartesian_product

G2 = AdditiveCyclic(2)

G11 = AdditiveCyclic(11)

G2211 = cartesian_product((G2,G2,G11))

M = [

[(0,0,0), (0,0,0), (0,0,0), (0,0,0), (0,0,0), (0,0,0), (0,0,0), (0,0,0)],

[(1,1,4), (0,1,4), (1,1,7), (1,0,6), (1,1,9), (0,1,2), (0,1,5), (0,1,1)],

[(1,0,6), (0,1,3), (1,0,0), (0,1,9), (1,1,1), (0,1,4), (1,1,9), (1,0,9)],

[(1,1,6), (1,1,9), (0,1,2), (1,1,0), (0,1,0), (1,1,5), (0,0,4), (0,0,9)],

[(1,0,9), (0,0,2), (0,0,1), (1,0,2), (0,0,7), (1,1,6), (1,1,0), (1,0,7)],

[(1,0,1), (1,0,6), (1,1,3), (0,1,5), (0,0,5), (0,1,3), (0,1,0), (1,1,0)]

]

M = [[G2211(x) for x in L] for L in M]

Mb=[]

for R in zip(*M):

for c in range(5):

(x1,y1,z1),(x2,y2,z2),(x3,y3,z3),(x4,y4,z4),(x5,y5,z5),(x6,y6,z6) = R

Mb.append(list(R))

R = [(x5,y5,5*z5),

(x1,y1,5*z1),

(x2,y2,5*z2),

(x3,y3,5*z3),

(x4,y4,5*z4),

(x6,y6,5*z6)]

for x,y,z in [(0,0,0), (1,0,1),(1,1,2),(0,0,8)]:

Mb.append([(x,y,z),

(x,y,5*z),

(x,y,3*z),

(x,y,4*z),

(x,y,9*z),

(0,0,0)])

return G2211, Mb

def DM_45_7_1():

r"""

Return a `(45,7,1)`-difference matrix.

As explained in the Handbook III.3.65 [DesignHandbook]_.

... whose description contained a very deadly typo, kindly fixed by Julian

R. Abel.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_45_7_1

sage: G,M = DM_45_7_1()

sage: is_difference_matrix(M,G,7,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(45,7)

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

from sage.categories.cartesian_product import cartesian_product

G533 = cartesian_product((FiniteField(5),FiniteField(3),FiniteField(3)))

M = [

[(0,0,0), (2,2,1), (3,1,1), (4,1,2), (4,0,1), (0,1,1), (0,2,1), (3,2,2)],

[(0,0,0), (1,2,1), (4,2,2), (1,2,0), (4,1,0), (3,1,1), (3,0,0), (2,1,2)],

[(0,0,0), (4,1,1), (2,2,1), (3,2,0), (1,2,0), (2,1,0), (1,0,0), (3,2,1)],

[(0,0,0), (0,1,0), (2,1,1), (4,0,0), (0,0,2), (4,2,2), (3,2,2), (1,2,2)],

[(0,0,0), (3,1,2), (2,1,0), (0,2,2), (4,2,1), (0,2,1), (2,0,1), (1,1,2)],

[(0,0,0), (2,1,1), (1,2,2), (3,0,1), (2,0,1), (1,0,0), (4,2,1), (1,1,0)],

[(0,0,0), (0,0,0), (0,0,0), (0,0,0), (0,0,0), (0,0,0), (0,0,0), (0,0,0)]

]

for i in range(6):

M[i].extend(M[5-i][1:8])

M[6].extend(M[6][1:8])

Mb=[]

for R in zip(*M):

(x1,y1,z1),(x2,y2,z2),(x3,y3,z3),(x4,y4,z4),(x5,y5,z5),(x6,y6,z6),(x7,y7,z7) = R

for i in range(3):

Mb.append([(x1, y1 , z1+i ),

(x2, y2+2*i, z2 ),

(x3, y3+i , z3+2*i),

(x4, y4+2*i, z4+i ),

(x5, y5+i , z5 ),

(x6, y6 , z6+2*i),

(x7, y7 , z7 )])

return G533, Mb

def DM_48_9_1():

r"""

Return a `(48,9,1)`-difference matrix.

As explained in the Handbook III.3.67 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_48_9_1

sage: G,M = DM_48_9_1()

sage: is_difference_matrix(M,G,9,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(48,9)

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

F16 = FiniteField(16,'x')

F3 = FiniteField(3)

F3F16 = F3.cartesian_product(F16)

w = F16.primitive_element()

assert w**4 == w+1

A = [

[ (0, 4), (2, 2), (2,2), (0,13), (0,4), (2,13), (0,1), (0,7), (1,7) , (2,2) , (0,6), (2,9)],

[ (2, 7), (0, 9), (2,7), (2,3) , (0,3), (0,9) , (1,12), (0,6), (0,12), (2,14), (2,7), (0,11)],

[ (2,12), (2,12), (0,14), (0,14), (2,8), (0,8) , (0,2), (1,2), (0,11), (0,1) , (2,4), (2,12)],

[ (1, 3), (0, 2), (0,10), (0,14), (0,9), (1,3) , (0,12), (2,13), (2,1) , (2,9) , (2,0), (1,7)],

[ (0, 0), (1, 8), (0,7), (1,8) , (0,4), (0,14), (2,6), (0,2), (2,3) , (1,12), (2,14), (2,5)],

[ (0,12), (0, 5), (1,13), (0,4) , (1,13), (0,9) , (2,8), (2,11), (0,7) , (2,10), (1,2), (2,4)],

[ (1,12), (2, 0), (1,14), (0,6) , (1,9), (0,14), (1,4), (0,5), (1,8) , (1,3) , (2,1), (1,1)],

[ (1, 4), (1, 2), (2,5), (0,4) , (0,11), (1,14), (1,13), (1,9), (0,10), (1,6) , (1,8), (2,6)],

[ (2,10), (1, 9), (1,7), (1,4) , (0,9), (0,1) , (0,0), (1,3), (1,14), (2,11), (1,11), (1,13)],

]

A = [[F3F16((F3(a),w**b)) for a,b in L] for L in A]

V = [12,2,7,0,5,10,3,8,13]

Mb = []

for L in zip(*A):

Mb.append(L)

for u in [0,1,4]:

Mb.append([e+F3F16((0,w**(x+u))) for (e,x) in zip(L,V)])

return F3F16, Mb

def DM_51_6_1():

r"""

Return a `(51,6,1)`-difference matrix.

As explained in the Handbook III.3.69 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_51_6_1

sage: G,M = DM_51_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(51,6)

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(51)

M = [

[ 5, 33, 29, 30, 1],

[ 8, 3, 47, 10, 13],

[ 14, 27, 6, 12, 28],

[ 9, 16, 44, 49, 11],

[ 34, 32, 36, 26, 20]

]

Mb=[[0,0,0,0,0]]

for R in zip(*M):

for i in range(5):

for RR in [list(R), [-x for x in R]]:

Mb.append(RR)

R = cyclic_shift(R,1)

for R in Mb:

R.append(0)

return G,Mb

def DM_52_6_1():

r"""

Return a `(52,6,1)`-difference matrix.

As explained in the Handbook III.3.70 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_52_6_1

sage: G,M = DM_52_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(52,6)

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

from sage.rings.finite_rings.finite_field_constructor import FiniteField

F4 = FiniteField(4,'z')

G13 = FiniteField(13)

G = F4.cartesian_product(G13)

z = F4('z')

assert z**2 == z+1

M = [

[ (0,0), (0,0), (0,0), (0,0), (0,0)],

[(z**2,10), (0,7), (1,10), (z,10),(z**2,3)],

[ (z,10), (z**2,2), (1,11), (z,2),(z**2,7)],

[ (z,8),(z**2,12), (0,10),(z**2,11),(z**2,6)],

[ (1,2), (0,2), (z**2,8), (z,3), (z,7)],

[ (1,6), (z,12), (0,7), (z**2,6), (z,2)]

]

M2 = [

[ (1,1),(z**2,11)],

[ (z,3), (1,7)],

[ (z**2,9), (z,8)],

[ (1,4), (z**2,3)],

[ (z,12), (1,9)],

[(z**2,10), (z,1)]

]

M = [[G(x) for x in L] for L in M]

M2= [[G(x) for x in L] for L in M2]

Mb=[[(0,0)]*6]

from itertools import product

p = lambda x,y : G(tuple([x*yy for yy in G(y)]))

def t1(i,R):

if i > 1:

return t1(1,t1(i-1,R))

((x1,y1),(x2,y2),(x3,y3),(x4,y4),(x5,y5),(x6,y6)) = R

return [(z*x3, 3*y3), (z*x1, 3*y1), (z*x2, 3*y2), (z*x6, 3*y6), (z*x4, 3*y4), (z*x5, 3*y5)]

def t2(i,R):

if i > 1:

return t2(1,t2(i-1,R))

((x1,y1),(x2,y2),(x3,y3),(x4,y4),(x5,y5),(x6,y6)) = R

return [( x3, y3), ( x1, y1), ( x2, y2), ( x5, y5), ( x6, y6), ( x4, y4)]

for R in zip(*M):

for c1,c2 in product([1,2,3],repeat=2):

Mb.append(t2(c2,t1(c1,R)))

for R in zip(*M2):

for c2 in [1,2,3]:

Mb.append(t2(c2,R))

return G, Mb

def DM_55_7_1():

r"""

Return a `(55,7,1)`-difference matrix.

As explained in the Handbook III.3.72 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_55_7_1

sage: G,M = DM_55_7_1()

sage: is_difference_matrix(M,G,7,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(55,7)

"""

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(55)

M = [

[ 1 , 7 , 14 , 19 , 28 , 33 , 40 , 46 , 50],

[ 2 , 13 , 25 , 38 , 52 , 12 , 20 , 32 , 45],

[ 39 , 6 , 8 , 26 , 24 , 51 , 11 , 34 , 37],

[ 54 , 48 , 41 , 36 , 27 , 22 , 15 , 9 , 5],

[ 53 , 42 , 30 , 17 , 3 , 43 , 35 , 23 , 10],

[ 16 , 49 , 47 , 29 , 31 , 4 , 44 , 21 , 18]

]

Mb=[[0,0,0,0,0,0,0]]

for R in zip(*M):

R = list(R)

for c in range(6):

Mb.append(cyclic_shift(R,c)+[0])

return G, Mb

def DM_56_8_1():

r"""

Return a `(56,8,1)`-difference matrix.

As explained in the Handbook III.3.73 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_56_8_1

sage: G,M = DM_56_8_1()

sage: is_difference_matrix(M,G,8,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(56,8)

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

F8 = FiniteField(8,'z')

F7 = FiniteField(7)

G = F8.cartesian_product(F7)

w = F8.primitive_element()

assert w**3 == w+1

M = [

[(0,0), (w**0,0), (w**1,0), (w**2,0), (w**3,0), (w**4,0), (w**5,0), (w**6,0)],

[(0,1), (w**1,6), (w**2,1), (w**3,1), (w**4,6), (w**5,1), (w**6,6), (w**0,6)],

[(0,4), (w**2,3), (w**3,4), (w**4,4), (w**5,3), (w**6,4), (w**0,3), (w**1,3)],

[(0,2), (w**3,5), (w**4,2), (w**5,2), (w**6,5), (w**0,2), (w**1,5), (w**2,5)],

[(0,2), (w**4,5), (w**5,2), (w**6,2), (w**0,5), (w**1,2), (w**2,5), (w**3,5)],

[(0,4), (w**5,3), (w**6,4), (w**0,4), (w**1,3), (w**2,4), (w**3,3), (w**4,3)],

[(0,1), (w**6,6), (w**0,1), (w**1,1), (w**2,6), (w**3,1), (w**4,6), (w**5,6)],

[(1,0), ( 1,0), ( 1,0), ( 1,0), ( 1,0), ( 1,0), ( 1,0), ( 1,0)]

]

Mb=[]

for R in zip(*M):

for _ in range(7):

Mb.append(R)

(x1,y1),(x2,y2),(x3,y3),(x4,y4),(x5,y5),(x6,y6),(x7,y7),(x8,y8) = R

R = [(w*x7,y7), (w*x1,y1), (w*x2,y2), (w*x3,y3), (w*x4,y4), (w*x5,y5), (w*x6,y6), (w*x8,y8)]

return G, Mb

def DM_57_8_1():

r"""

Return a `(57,8,1)`-difference matrix.

Given by Julian R. Abel.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_57_8_1

sage: G,M = DM_57_8_1()

sage: is_difference_matrix(M,G,8,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(57,8)

"""

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

M = [[B[x] for x in R] for R in M]

M.append([0]*8)

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(57)

return G, M

def DM_60_6_1():

r"""

Return a `(60,6,1)`-difference matrix.

As explained in [JulianAbel13]_.

REFERENCES:

.. [JulianAbel13] Existence of Five MOLS of Orders 18 and 60

R. Julian R. Abel

Journal of Combinatorial Designs

2013

http://onlinelibrary.wiley.com/doi/10.1002/jcd.21384/abstract

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_60_6_1

sage: G,M = DM_60_6_1()

sage: is_difference_matrix(M,G,6,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(60,6)

"""

M60 = [[(0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0)],

[(1, 10), (1, 6), (0, 17), (0, 7), (1, 5), (0, 9), (0, 3), (1, 13), (1, 17), (0, 13)],

[(1, 22), (1, 1), (1, 8), (0, 9), (1, 21), (1, 29), (1, 0), (0, 2), (0, 12), (1, 15)],

[(1, 24), (1, 1), (0, 14), (0, 0), (0, 16), (0, 18), (0, 8), (0, 28), (0, 17), (0, 7)],

[(0, 17), (0, 7), (0, 20), (0, 1), (1, 4), (0, 26), (0, 19), (0, 28), (1, 21), (0, 6)],

[(1, 14), (1, 9), (0, 10), (0, 27), (1, 20), (0, 11), (0, 13), (1, 12), (0, 28), (1, 18)]]

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

from sage.categories.cartesian_product import cartesian_product

G = cartesian_product((AdditiveCyclic(2),AdditiveCyclic(30)))

M60b=[]

onezero = G((1,0))

for R in zip(*M60):

a,b,c,d,e,f = map(G,R)

M60b.append([a,b,c,d,e,f])

M60b.append([c,a,b,e,f,d])

M60b.append([b,c,a,f,d,e])

M60b.append([-d,-e,-f,-a+onezero,-b+onezero,-c+onezero])

M60b.append([-e,-f,-d,-c+onezero,-a+onezero,-b+onezero])

M60b.append([-f,-d,-e,-b+onezero,-c+onezero,-a+onezero])

return G, M60b

def DM_75_8_1():

r"""

Return a `(75,8,1)`-difference matrix.

As explained in the Handbook III.3.75 [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_75_8_1

sage: G,M = DM_75_8_1()

sage: is_difference_matrix(M,G,8,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(75,8)

"""

from sage.rings.finite_rings.finite_field_constructor import FiniteField

from sage.categories.cartesian_product import cartesian_product

F3 = FiniteField(3)

F5 = FiniteField(5)

G = cartesian_product((F3,F5,F5))

M = [

[(2,0,0), (0,0,0), (0,0,0), (1,0,0), (0,0,0), (1,0,0), (1,0,0), (0,0,0)],

[(0,2,3), (1,4,4), (1,1,3), (1,0,4), (2,4,3), (0,0,3), (1,4,4), (0,0,0)],

[(1,3,2), (2,1,1), (1,4,0), (0,3,0), (1,0,4), (2,4,1), (0,1,2), (0,0,0)],

[(0,2,4), (1,3,1), (2,0,2), (0,0,1), (2,4,0), (1,2,2), (0,0,0), (0,0,0)],

[(1,1,2), (2,2,3), (0,3,1), (1,4,2), (2,1,0), (1,4,3), (2,4,4), (0,0,0)],

[(0,1,4), (0,4,4), (2,4,1), (1,3,0), (1,3,1), (2,0,0), (2,4,0), (0,0,0)],

[(0,4,4), (2,0,1), (2,3,3), (2,3,2), (0,0,2), (2,1,2), (1,4,2), (0,0,0)],

[(2,4,2), (2,4,1), (2,3,1), (1,2,2), (1,3,0), (0,0,2), (2,4,2), (0,0,0)]

]

for i in range(8):

M[i].extend(M[7-i][:7])

Mb=[]

for R in zip(*M):

for x in range(5):

V = [(0,0,x), (0,x,0), (0,x,2*x),(0,2*x,2*x), (0,3*x,3*x), (0,4*x,3*x), (0,4*x,0), (0,0,4*x)]

Mb.append([G(e)+G(ee) for e,ee in zip(R,V)])

return G, Mb

def DM_273_17_1():

r"""

Return a `(273,17,1)`-difference matrix.

Given by Julian R. Abel.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_273_17_1

sage: G,M = DM_273_17_1()

sage: is_difference_matrix(M,G,17,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(273,17)

"""

M = orthogonal_array(17,17)

M = [R for R in M if any(x!=R[0] for x in R)] # removing the 0..0, 1..1, ... rows.

B = (1,2,4,8,16,32,64,91,117,128,137,182,195,205,234,239,256) # (273,17,1) difference set

M = [[B[x] for x in R] for R in M]

M.append([0]*17)

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(273)

return G, M

def DM_993_32_1():

r"""

Return a `(993,32,1)`-difference matrix.

Given by Julian R. Abel.

EXAMPLES::

sage: from sage.combinat.designs.designs_pyx import is_difference_matrix

sage: from sage.combinat.designs.database import DM_993_32_1

sage: G,M = DM_993_32_1()

sage: is_difference_matrix(M,G,32,1)

True

Can be obtained from the constructor::

sage: _ = designs.difference_matrix(993,32)

"""

M = orthogonal_array(32,32)

M = [R for R in M if any(x!=R[0] for x in R)] # removing the 0..0, 1..1, ... rows.

B = (0,74,81,126,254,282,308,331,344,375,387,409,525,563, # (993,32,1) difference set

572,611,631,661,694,702,734,763,798,809,814,851,906,

908,909,923,927,933)

M = [[B[x] for x in R] for R in M]

M.append([0]*32)

from sage.rings.finite_rings.integer_mod_ring import IntegerModRing as AdditiveCyclic

G = AdditiveCyclic(993)

return G, M

DM = {

(12 ,1) : (6 ,DM_12_6_1),

(21 ,1) : (6 ,DM_21_6_1),

(24 ,1) : (8 ,DM_24_8_1),

(28 ,1) : (6 ,DM_28_6_1),

(33 ,1) : (6 ,DM_33_6_1),

(35 ,1) : (6 ,DM_35_6_1),

(36 ,1) : (9 ,DM_36_9_1),

(39 ,1) : (6 ,DM_39_6_1),

(44 ,1) : (6 ,DM_44_6_1),

(45 ,1) : (7 ,DM_45_7_1),

(48 ,1) : (9 ,DM_48_9_1),

(51 ,1) : (6 ,DM_51_6_1),

(52 ,1) : (6 ,DM_52_6_1),

(55 ,1) : (7 ,DM_55_7_1),

(56 ,1) : (8 ,DM_56_8_1),

(57 ,1) : (8 ,DM_57_8_1),

(60 ,1) : (6 ,DM_60_6_1),

(75 ,1) : (8 ,DM_75_8_1),

(273,1) : (17,DM_273_17_1),

(993,1) : (32,DM_993_32_1),

}

# Create the list of DM for the documentation

_all_l = sorted(set(l for v,l in DM.keys()))

LIST_OF_DM = "\n".join(" - `\lambda={}`:\n ".format(l)+

", ".join("`({},{},{})`".format(v,k,l) for (v,_),(k,__) in sorted(DM.items()) if _ == l)

for l in _all_l)

def RBIBD_120_8_1():

r"""

Return a resolvable `BIBD(120,8,1)`

This function output a list ``L`` of `17\times 15` blocks such that

``L[i*15:(i+1)*15]`` is a partition of `120`.

Construction shared by Julian R. Abel:

Seiden's method: Start with a cyclic `(273,17,1)-BIBD` and let `B` be an

hyperoval, i.e. a set of 18 points which intersects any block of the

BIBD in either 0 points (153 blocks) or 2 points (120 blocks). Dualise

this design and take these last 120 blocks as points in the design;

blocks in the design will correspond to the `273-18=255` non-hyperoval

points.

The design is also resolvable. In the original `PG(2,16)` take any

point `T` in the hyperoval and consider a block `B1` containing `T`.

The `15` points in `B1` that do not belong to the hyperoval correspond

to `15` blocks forming a parallel class in the dualised design. The

other `16` parallel classes come in a similar way, by using point `T`

and the other `16` blocks containing `T`.

.. SEEALSO::

:func:`OA_9_120`

EXAMPLES::

sage: from sage.combinat.designs.database import RBIBD_120_8_1

sage: from sage.combinat.designs.bibd import is_pairwise_balanced_design

sage: RBIBD = RBIBD_120_8_1()

sage: is_pairwise_balanced_design(RBIBD,120,[8])

True

It is indeed resolvable, and the parallel classes are given by 17 slices of

consecutive 15 blocks::

sage: for i in range(17):

....: assert len(set(sum(RBIBD[i*15:(i+1)*15],[]))) == 120

The BIBD is available from the constructor::

sage: _ = designs.balanced_incomplete_block_design(120,8)

"""

from .incidence_structures import IncidenceStructure

n=273

# Base block of a cyclic BIBD(273,16,1)

B = [1,2,4,8,16,32,64,91,117,128,137,182,195,205,234,239,256]

BIBD = [[(x+c)%n for x in B] for c in range(n)]

# A (precomputed) set that every block of the BIBD intersects on 0 or 2 points

hyperoval = [128, 192, 194, 4, 262, 140, 175, 48, 81, 180, 245, 271, 119, 212, 249, 189, 62, 255]

#for B in BIBD:

# len_trace = sum(x in hyperoval for x in B)

# assert len_trace == 0 or len_trace == 2

# Equivalence classes

p = hyperoval[0]

equiv = []

new_BIBD = []

for B in BIBD:

if any(x in hyperoval for x in B):

if p in B:

equiv.append([x for x in B if x not in hyperoval])

else:

new_BIBD.append([x for x in B])

BIBD = new_BIBD

r = {v:i for i,v in enumerate(x for x in range(n) if x not in hyperoval)}

BIBD = [[r[x] for x in B] for B in BIBD ]

equiv = [[r[x] for x in B] for B in equiv]

BIBD = IncidenceStructure(range(255),BIBD)

M = BIBD.incidence_matrix()

equiv = [[M.nonzero_positions_in_row(x) for x in S] for S in equiv]

return [B for S in equiv for B in S]

def BIBD_45_9_8(from_code=False):

r"""

Return a `(45,9,1)`-BIBD.

This BIBD is obtained from the codewords of minimal weight in the

:func:`~sage.coding.code_constructions.ExtendedQuadraticResidueCode` of

length 48. This construction appears in VII.11.2 from [DesignHandbook]_,

which cites [HT95]_.

INPUT:

- ``from_code`` (boolean) -- whether to build the design from hardcoded data

(default) or from the code object (much longer).

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_45_9_8

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: B = BalancedIncompleteBlockDesign(45, BIBD_45_9_8(),lambd=8); B

(45,9,8)-Balanced Incomplete Block Design

TESTS:

From the definition (takes around 12s)::

sage: B2 = Hypergraph(BIBD_45_9_8(from_code=True)) # not tested

sage: B2.is_isomorphic(B) # not tested

True

REFERENCE:

.. [HT95] \W. Huffman and V. Tonchev,

The existence of extremal self-dual `[50, 25, 10]` codes and

quasi-symmetric `2-(49, 9, 6)` designs,

Designs, Codes and Cryptography

September 1995, Volume 6, Issue 2, pp 97-106

"""

if from_code:

from sage.coding.code_constructions import ExtendedQuadraticResidueCode

from sage.rings.finite_rings.finite_field_constructor import FiniteField

C = ExtendedQuadraticResidueCode(47,FiniteField(2))

min_weight = [map(int,x)[3:] for x in C

if x.hamming_weight() == 12 and

x[0]==1 and x[1]==1 and x[2]==1]

return [[i for i,v in enumerate(x) if v] for x in min_weight]

from sage.rings.integer import Integer

B = ['acs1v', 'l8lsx', '4ga1vw', '6q9amr', 'nb3ui8', 'sgjocw', '11vsoy2', '28791ts', '30tm1z8', '38ktnwh',

'3saz8jk', '41qkwme', '4g3jxmt', '56qhwuc', '711w45k', '8nz2gx4', '903uha8', '957z8dc', '9wejz7k', 'fs905ic',

'ftzzh28', 'gb4g448', 'hvreal0', 'nqlhxu8', 'rmluazm', 'vlyqayx', 'w52detk', 'zisjk02', 'zw9811c', '10i7qfl1',

'13ibtse8', '1rbsbvvc', '1sdy0o5c', '1z14s09e', '2nbz5a80', '2uuhib2a', '2wkn4r9d', '3iaaat5w', '3iiwq53s',

'3j9ubv43', '3mpxpngz', '3qamndc0', '3saomh3t', '3uhhi5cw', '4334rx4x', '4dxy3xts', '4tn9w2z1', '4vlr2h00',

'59f1meqm', '59h6udc1', '5cep4nc0', '5ddcxsw2', '70msua7k', '70ofjm82', '70p8jig0', '721o664h', '72jutmfk',

'74jowaad', '78ihrfgo', '7meufihs', '7wv5mtxj', '84akgj0w', '8m9vyb60', '8s0c6p04', '8soi6m8g', '9kawy0ow',

'awnpg9a8', 'biu8xww0', 'e1lptwxx', 'e79x2we8', 'eh0t1q9y', 'eh65daci', 'ehxytwjk', 'extc1udk', 'f4toqhpg',

'fgeqg214', 'ftiem9lk', 'fw77kcnc', 'h5kt9cf4', 'hjwhwym8', 'hz8d60xs', 'jb6bp0g0', 'l22bzw1w', 'l3pj9hq8',

'lbj1fubp', 'lxal1lk2', 's27vq70q', 's2bb5mki', 's2w95y0w', 's3cek9og', 's4703jk4', 's67g5qf5', 's8kgdkat',

'sckruupw', 'se4vzkao', 'si57d0vl', 'sjhd20i8', 'sqne2mf6', 'sxtju9ds', 'ttd710kw', 'ttkayw5e', 'u96baslc',

'vtdhrbj5', 'y79i706c', 'zycu7tsa', '10uwf8sh4', '11boo6mmc', '12sxyeebs', '163xyccg3', '16cpesdfk',

'18q18bpc0', '1k4hvvgq4', '1k5f63ok4', '1k5olig3u', '1k6fsqalm', '1kacr2gi8', '1kcc6rzu1', '1kkpot632',

'1kwdghpts', '1l2644l68', '1l3yxmj9s', '1m04wgmyo', '1mtm16z5s', '1np6u1q0w', '1nuo1tbfk', '1oy4n1mo0',

'1r5lsxju0', '1sx57vdfq', '1v4j675ds', '1y5oldkzm', '1ydfr4jno', '1ylc38ah4', '1z14mw0td', '223vcx1xc',

'26xq9hn29', '2c7wa6r0w', '2cbc8qbcw', '2jn9ojll5', '2qjlkoz69', '2tr1zn5ds', '348vfurgh', '348vlaoc0',

'348ynt0qx', '34ahl37ds', '34b3cgc8y', '34ooa1ix0', '34r4ejl82', '35p5m8r28', '360i7uazl', '36289j761',

'3650mzlzg', '36aev2c00', '36noxmex2', '36vlw3k3k', '37rw4rghs', '37t554ikq', '387avhseb', '3b9o5lbwi',

'3ewmteale', '3ibz0r8n4', '3id5iv5ky', '3ihxwcvvc', '3k5k1k174', '3pau9ujnl', '3wf1e2dck', '43rfm4du8',

'47pqff6yo', '4e2i4y684', '4hio30v0o', '4odb0lr5s', '4odcmkvt0', '4p94elixc', '4p9zffz0k', '4qciqf9mp',

'4ywafln9c', '5hf4nw08w', '68ijggco4', '68jq73cxs', '68maap98g', '68prdfhqg', '68qm8divl', '691ibd2ps',

'69dbnd8ur', '69esd0djg', '69w6eo0sh', '6ad6zcetk', '6aonwwkjk', '6aozhe8zl', '6cvyitslw', '6dr7i6olg',

'6fibvzxtw', '6fmd4bv28', '6gmqtkr9e', '6j14n6n7k', '6miukvtc1', '6mjvifon4', '6mormb3fm', '6mr9hvhna',

'6q533lm6w', '6rsie7cbk', '6tjgpxic0', '70k7ao9m0', '7103zqlvk', '71i1x52bm', '7447g0dfw', '7sogja9z4',

'7up5z9m9u', '7w7esu6fm', '7zmqtlrpd', '81tsbnzsw', '8kofgi1he', '8mhi35nc1', '9cv1pjiaw', '9d6ef1dah',

'9dftsor9c', '9du8c1vcw', '9jr5vsnj4', 'a8b405mps', 'ajqhmxkj4', 'ax2xsvfic']

B = [Integer(x, base=36) for x in B]

return [[i for i in range(45) if x&(1<<i)]

for x in B]

def BIBD_66_6_1():

r"""

Return a (66,6,1)-BIBD.

This BIBD was obtained from La Jolla covering repository

(https://www.ccrwest.org/cover.html) where it is attributed to Colin Barker.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_66_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(66, BIBD_66_6_1())

(66,6,1)-Balanced Incomplete Block Design

"""

BIBD = [frozenset([(x+i*5)%65 if x<65 else x for x in b])

for i in range(65)

for b in

[[6, 38, 42, 46, 53, 62], [9, 11, 21, 49, 56, 60], [18, 31, 37, 44, 52, 60],

[0, 12, 29, 46, 51, 63], [0, 6, 21, 30, 43, 48], [4, 17, 22, 36, 47, 59],

[0, 1, 2, 3, 4, 65], [23, 39, 44, 53, 59, 63], [12, 22, 28, 48, 55, 60],

[19, 22, 25, 40, 49, 50], [4, 30, 37, 50, 58, 61]]]

return [list(t) for t in frozenset(BIBD)]

def BIBD_76_6_1():

r"""

Return a (76,6,1)-BIBD.

This BIBD was obtained from La Jolla covering repository

(https://www.ccrwest.org/cover.html) where it is attributed to Colin Barker.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_76_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(76, BIBD_76_6_1())

(76,6,1)-Balanced Incomplete Block Design

"""

BIBD = [frozenset([(x+i*4)%76 if x<76 else x for x in b])

for i in range(76)

for b in

[[3, 5, 21, 33, 72, 73], [4, 37, 57, 58, 64, 75], [7, 14, 44, 47, 59, 63],

[10, 20, 61, 63, 71, 72], [13, 26, 30, 39, 45, 67], [11, 21, 25, 30, 55, 58],

[2, 5, 34, 52, 54, 70], [6, 8, 29, 48, 70, 71], [10, 15, 36, 41, 44, 56],

[0, 6, 13, 27, 44, 72]]]

return [list(t) for t in frozenset(BIBD)]

def BIBD_96_6_1():

r"""

Return a (96,6,1)-BIBD.

This BIBD was obtained from La Jolla covering repository

(https://www.ccrwest.org/cover.html) where it is attributed to Colin Barker.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_96_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(96, BIBD_96_6_1())

(96,6,1)-Balanced Incomplete Block Design

"""

BIBD = [frozenset([(x+i*2)%96 if x<96 else x for x in b])

for i in range(96)

for b in

[[3, 13, 32, 47, 68, 87], [9, 36, 70, 75, 81, 88], [22, 52, 72, 76, 78, 79],

[15, 23, 41, 43, 46, 58], [7, 8, 21, 57, 66, 94], [8, 22, 30, 51, 55, 93],

[15, 31, 47, 63, 79, 95], [2, 18, 34, 50, 66, 82]]]

return [list(t) for t in frozenset(BIBD)]

def BIBD_106_6_1():

r"""

Return a (106,6,1)-BIBD.

This constructions appears in II.3.32 from [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_106_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(106, BIBD_106_6_1())

(106,6,1)-Balanced Incomplete Block Design

"""

bibd = [((0,0), ( 1,0), ( 3,0), (11,0), (38,0), ( 0,1)),

((0,0), (13,0), (30,0), (23,1), (35,1), (51,1)),

((0,0), ( 5,0), (19,0), (25,0), (36,1), (39,1)),

((0,0), ( 4,0), (28,1), (30,1), (37,1), (47,1)),

((0,0), ( 7,0), (29,0), ( 8,1), (16,1), (48,1)),

((0,0), ( 2,1), ( 7,1), (25,1), (29,1), (49,1)),

((0,0), ( 9,0), (21,0), (12,1), (13,1), (27,1))]

return [[((x+i)%53+y*53) for x,y in B] for i in range(53) for B in bibd]

def BIBD_111_6_1():

r"""

Return a (111,6,1)-BIBD.

This constructions appears in II.3.32 from [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_111_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(111, BIBD_111_6_1())

(111,6,1)-Balanced Incomplete Block Design

"""

from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet

from .incidence_structures import IncidenceStructure

bibd = [(( 0,0), ( 1,0), ( 3,0), ( 7,0), (17,0), ( 0,1)),

(( 0,0), ( 5,0), (19,1), (28,1), (10,2), (30,2)),

(( 5,0), (33,0), (13,1), (34,1), (19,2), ( 7,2)),

(( 9,0), (27,0), (16,1), (11,1), (12,2), (36,2)),

((10,0), (23,0), (26,1), ( 8,1), ( 1,2), ( 6,2)),

((13,0), (24,0), (19,1), (18,1), ( 5,2), (32,2)),

((26,0), (34,0), ( 1,1), ( 7,1), (10,2), (33,2))]

gens = lambda B: [frozenset(((x*10)%37,(y+1)%3) for x,y in B),

frozenset(((x+1) %37, y) for x,y in B)]

bibd = RecursivelyEnumeratedSet(map(frozenset,bibd), successors=gens)

return IncidenceStructure(bibd)._blocks

def BIBD_126_6_1():

r"""

Return a (126,6,1)-BIBD.

This constructions appears in VI.16.92 from [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_126_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(126, BIBD_126_6_1())

(126,6,1)-Balanced Incomplete Block Design

"""

from itertools import product

bibd = [[((x+xx)%5, (y+yy)%5, (z+zz)%5) for x,y,z in B]

for xx,yy,zz in product(range(5),repeat=3)

for B in

[[(0,0,1),(0,0,4),(1,2,2),(1,3,3),(4,2,1),(4,3,4)],

[(0,0,2),(0,0,3),(1,4,4),(1,1,1),(4,4,2),(4,1,3)],

[(0,4,3),(0,1,2),(2,2,0),(2,3,0),(3,3,2),(3,2,3)],

[(0,3,1),(0,2,4),(2,4,0),(2,1,0),(3,1,4),(3,4,1)]]]

bibd.extend([[(125,0,0), (0,x,y),(1,x,y),(2,x,y),(3,x,y),(4,x,y)]

for x,y in product(range(5),repeat=2)])

return [[x+y*5+z*25 for x,y,z in B]

for B in bibd]

def BIBD_136_6_1():

r"""

Return a (136,6,1)-BIBD.

This constructions appears in II.3.32 from [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_136_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(136, BIBD_136_6_1())

(136,6,1)-Balanced Incomplete Block Design

"""

from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet

from .incidence_structures import IncidenceStructure

inf=(None,None)

bibd = [((0,0), ( 3,0), (15,0), (35,0), ( 6,2), (10,2)),

((0,0), (22,0), (11,1), (30,1), ( 1,2), (18,2)),

((0,0), ( 5,0), (18,1), (41,1), (13,2), (42,2)),

((0,0), (11,0), (17,0), ( 4,2), ( 5,2), (28,2)),

((0,0), ( 1,0), ( 0,1), (16,1), ( 0,2), (31,2)),

( inf ,( 0,0), ( 9,0), (18,0), (27,0), (36,0))]

gens = lambda B: [frozenset(((x*16)%45,(y+1)%3) if (x,y)!=inf else inf for x,y in B),

frozenset(((x+1) %45,y) if (x,y)!=inf else inf for x,y in B)]

bibd = RecursivelyEnumeratedSet(map(frozenset,bibd), successors=gens)

return IncidenceStructure(bibd)._blocks

def BIBD_141_6_1():

r"""

Return a (141,6,1)-BIBD.

This constructions appears in II.3.32 from [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_141_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(141, BIBD_141_6_1())

(141,6,1)-Balanced Incomplete Block Design

"""

from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet

from .incidence_structures import IncidenceStructure

a = 'a'

inf = (None,None)

bibd = [((0,0), (16,0), (24,0), (24,1), (15,2), (25,2)),

((0,0), ( 3,0), (26,0), (13,1), (33,1), (34,a)),

((0,0), (13,0), (18,0), (15,1), ( 7,2), ( 0,a)),

((0,0), ( 2,0), (14,1), (23,1), (26,a), (32,a)),

((0,0), ( 4,0), (29,1), ( 6,2), ( 9,a), (20,a)),

((0,0), ( 1,0), (12,2), ( 2,a), ( 4,a), (19,a)),

( inf ,( 0,0), ( 7,0), (14,0), (21,0), (28,0)),

( inf ,( 0,a), ( 7,a), (14,a), (21,a), (28,a))]

gens = lambda B: [frozenset(((x*16)%35,(y+1)%3 if y!=a else a) if (x,y)!=inf else inf for x,y in B),

frozenset(((x+1) %35, y ) if (x,y)!=inf else inf for x,y in B)]

bibd = RecursivelyEnumeratedSet(map(frozenset,bibd), successors=gens)

return IncidenceStructure(bibd)._blocks

def BIBD_171_6_1():

r"""

Return a (171,6,1)-BIBD.

This constructions appears in II.3.32 from [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_171_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(171, BIBD_171_6_1())

(171,6,1)-Balanced Incomplete Block Design

"""

from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet

from .incidence_structures import IncidenceStructure

bibd = [(( 0,0), (19,0), (39,0), (41,0), (14,1), (38,2)),

(( 0,0), (21,0), (44,0), (48,0), (26,1), (11,2)),

(( 0,0), ( 1,0), (43,0), ( 8,2), (15,2), (44,2)),

(( 0,0), ( 3,0), (31,0), (23,1), (43,1), (36,2)),

(( 0,0), (40,0), (50,0), (11,1), (25,2), (34,2)),

(( 0,0), (12,0), ( 0,1), (27,1), ( 0,2), (18,2)),

((37,0), (42,0), (31,1), ( 9,1), (46,2), ( 6,2))]

gens = lambda B: [frozenset(((x*7) %57,(y+1)%3) for x,y in B),

frozenset(((x+1) %57, y) for x,y in B)]

bibd = RecursivelyEnumeratedSet(map(frozenset,bibd), successors=gens)

return IncidenceStructure(bibd)._blocks

def HigmanSimsDesign():

r"""

Return the Higman-Sims designs, which is a `(176, 50, 14)`-BIBD.

This design is built from a from the :func:`WittDesign

<sage.combinat.designs.block_design.WittDesign>` `W` on 24 points. We define

two points `a,b`, and consider:

- The collection `W_a` of all blocks of `W` containing `a` but not

containing `b`.

- The collection `W_b` of all blocks of `W` containing `b` but not

containing `a`.

The design is then obtained from the incidence structure produced by the

blocks `A\in W_a` and `B\in W_b` whose intersection has cardinality 2. This

construction, due to M.Smith, can be found in [KY04]_ or in 10.A.(v) of [BvL84]_.

EXAMPLES::

sage: H = designs.HigmanSimsDesign(); H # optional - gap_packages

Incidence structure with 176 points and 176 blocks

sage: H.is_t_design(return_parameters=1) # optional - gap_packages

(True, (2, 176, 50, 14))

Make sure that the automorphism group of this designs is isomorphic to the

automorphism group of the

:func:`~sage.graphs.generators.smallgraphs.HigmanSimsGraph`. Note that the

first of those permutation groups acts on 176 points, while the second acts

on 100::

sage: gH = H.automorphism_group() # optional - gap_packages

sage: gG = graphs.HigmanSimsGraph().automorphism_group() # optional - gap_packages

sage: gG.is_isomorphic(gG) # long time # optional - gap_packages

True

REFERENCE:

.. [KY04] \S. Klee and L. Yates,

Tight Subdesigns of the Higman-Sims Design,

Rose-Hulman Undergraduate Math. J 5.2 (2004).

https://www.rose-hulman.edu/mathjournal/archives/2004/vol5-n2/paper9/v5n2-9pd.pdf

"""

from sage.combinat.designs.block_design import WittDesign

from .incidence_structures import IncidenceStructure

W = WittDesign(24)

a,b = 0,1

Wa = [set(B) for B in W

if (a in B and

b not in B)]

Wb = [set(B) for B in W

if (b in B and

a not in B)]

H = [[i for i,A in enumerate(Wa) if len(A&B) != 2]

for B in Wb]

H = IncidenceStructure(H)

return H

def BIBD_196_6_1():

r"""

Return a (196,6,1)-BIBD.

This constructions appears in II.3.32 from [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_196_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(196, BIBD_196_6_1())

(196,6,1)-Balanced Incomplete Block Design

"""

from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet

from .incidence_structures import IncidenceStructure

a = 'a'

bibd = [((0,0), ( 2,0), (12,0), (45,0), ( 3,1), (11,a)),

((0,0), ( 3,0), ( 8,0), ( 5,1), (17,1), (39,a)),

((0,0), ( 9,0), (36,0), (24,1), (44,1), (37,a)),

((0,0), (15,0), (34,1), (41,1), (47,2), (18,a)),

((0,0), ( 7,0), (31,0), (13,1), (35,2), (41,a)),

((0,0), (14,0), (32,1), (10,2), (22,a), (44,a)),

((0,0), (23,0), (21,1), (39,1), (19,a), (25,a)),

((0,0), (33,1), ( 0,a), ( 5,a), (29,a), (47,a)),

((0,0), ( 1,0), ( 0,1), (30,1), ( 0,2), (18,2)),

((8,0), (19,0), (44,1), (31,1), (46,2), (48,2))]

gens = lambda B: [frozenset(((x*30)%49,(y+1)%3 if y!=a else a) for x,y in B),

frozenset(((x+1) %49, y) for x,y in B)]

bibd = RecursivelyEnumeratedSet(map(frozenset,bibd), successors=gens)

return IncidenceStructure(bibd)._blocks

def BIBD_201_6_1():

r"""

Return a (201,6,1)-BIBD.

This constructions appears in II.3.32 from [DesignHandbook]_.

EXAMPLES::

sage: from sage.combinat.designs.database import BIBD_201_6_1

sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign

sage: BalancedIncompleteBlockDesign(201, BIBD_201_6_1())

(201,6,1)-Balanced Incomplete Block Design

"""

from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet

from .incidence_structures import IncidenceStructure

bibd = [((0,0), ( 1,0), ( 4,2), ( 9,2), (34,2), (62,2)),

((0,1), ( 2,1), (15,1), ( 8,2), (27,2), (49,2)),

((0,0), ( 3,0), (22,0), (54,1), (13,2), (40,2)),

((0,0), (36,0), (40,0), (31,1), (34,1), ( 5,2)),

((0,0), (50,0), (55,0), ( 6,1), (24,1), (26,2)),

((0,0), ( 2,0), ( 3,1), (14,1), (35,1), (25,2)),

((3,1), (20,1), (44,1), (36,2), (39,2), (59,2)),

((0,0), ( 0,1), (30,1), (38,1), (66,1), ( 0,2))]

gens = lambda B: [frozenset(((x*29)%67,y) for x,y in B),

frozenset(((x+1) %67,y) for x,y in B)]

bibd = RecursivelyEnumeratedSet(map(frozenset,bibd), successors=gens)

return IncidenceStructure(bibd)._blocks

# Index of the BIBD constructions

# Associates to triple (v,k,lambda) a function that return a

# (n,k,lambda)-BIBD family.

# This dictionary is used by designs.BalancedIncompleteBlockDesign

BIBD_constructions = {

( 45,9,8): BIBD_45_9_8,

( 66,6,1): BIBD_66_6_1,

( 76,6,1): BIBD_76_6_1,

( 96,6,1): BIBD_96_6_1,

(120,8,1): RBIBD_120_8_1,

(106,6,1): BIBD_106_6_1,

(111,6,1): BIBD_111_6_1,

(126,6,1): BIBD_126_6_1,

(136,6,1): BIBD_136_6_1,

(141,6,1): BIBD_141_6_1,

(171,6,1): BIBD_171_6_1,

(176,50,14): HigmanSimsDesign,

(196,6,1): BIBD_196_6_1,

(201,6,1): BIBD_201_6_1,

}

# Create the list of DF for the documentation

_all_l = sorted(set(l for v,k,l in BIBD_constructions.keys()))

LIST_OF_BIBD = "\n".join(" - `\lambda={}`:\n ".format(l) +

", ".join("`({},{},{})`".format(v,k,l) for v,k,_ in sorted(BIBD_constructions) if _ == l)

for l in _all_l)

# Evenly Distributed Sets (EDS)

# For the definition see the documentation of the class

# EvenlyDistributedSetsBacktracker in the file evenly_distributed_sets.pyx

# EDS is a dictionary of dictionaries whose keys are the integers

# 4, 5,..., 10. For each k in {4,...,10} the keys of EDS[k] are the prime powers

# `q` so that `q = 1 modulo k(k-1)`.

# The value at position EDS[k][q] is one of:

# - ``(None, B)`` if `q` is prime and `B` is an evenly distributed set in Z/pZ

# - ``(poly, B)`` if `q=p^k` is a prime power (but not a prime). The

# polynomial ``poly`` is such that GF(p)[x] / (poly) is a finite field of

# cardinality q. The set `B` is then given in terms of the canonical

# generator `x`.

from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

from sage.rings.integer_ring import ZZ

R = PolynomialRing(ZZ,'a')

a = R.gen()

EDS={

4:{

13: (None, [0, 1, 11, 5]),

25: (a**2 + 4*a + 2, [0, 1, a, 3*a + 4]),

37: (None, [0, 1, 17, 30]),

49: (a**2 + 6*a + 3, [0, 1, a + 6, 4*a + 1]),

61: (None, [0, 1, 6, 37]),

73: (None, [0, 1, 5, 18]),

97: (None, [0, 1, 5, 24]),

109: (None, [0, 1, 6, 60]),

121: (a**2 + 7*a + 2, [0, 1, 2*a, 3*a + 7]),

157: (None, [0, 1, 20, 132]),

169: (a**2 + 12*a + 2, [0, 1, a + 12, a + 6]),

181: (None, [0, 1, 10, 87]),

193: (None, [0, 1, 5, 11]),

229: (None, [0, 1, 6, 13]),

241: (None, [0, 1, 11, 24]),

277: (None, [0, 1, 11, 228]),

289: (a**2 + 16*a + 3, [0, 1, a, 6*a + 13]),

313: (None, [0, 1, 10, 121]),

337: (None, [0, 1, 10, 21]),

349: (None, [0, 1, 7, 19]),

361: (a**2 + 18*a + 2, [0, 1, a + 3, 9*a + 5]),

373: (None, [0, 1, 5, 231]),

397: (None, [0, 1, 18, 11]),

409: (None, [0, 1, 21, 60]),

421: (None, [0, 1, 14, 31]),

433: (None, [0, 1, 10, 97]),

457: (None, [0, 1, 13, 195]),

529: (a**2 + 21*a + 5, [0, 1, a + 5, 3*a + 11]),

541: (None, [0, 1, 11, 45]),

577: (None, [0, 1, 5, 115]),

601: (None, [0, 1, 7, 69]),

613: (None, [0, 1, 6, 88]),

625: (a**4 + 4*a**2 + 4*a + 2, [0, 1, a + 3, 2*a**2 + a]),

661: (None, [0, 1, 6, 66]),

673: (None, [0, 1, 5, 46]),

709: (None, [0, 1, 17, 256]),

733: (None, [0, 1, 6, 49]),

757: (None, [0, 1, 5, 224]),

769: (None, [0, 1, 11, 79]),

829: (None, [0, 1, 19, 44]),

841: (a**2 + 24*a + 2, [0, 1, a + 8, 4*a + 27]),

853: (None, [0, 1, 6, 58]),

877: (None, [0, 1, 5, 46]),

937: (None, [0, 1, 5, 160]),

961: (a**2 + 29*a + 3, [0, 1, a + 16, 3*a + 8]),

997: (None, [0, 1, 7, 102]),

1009: (None, [0, 1, 11, 131]),

1021: (None, [0, 1, 19, 153]),

1033: (None, [0, 1, 5, 15]),

1069: (None, [0, 1, 6, 36]),

1093: (None, [0, 1, 15, 25]),

1117: (None, [0, 1, 6, 23]),

1129: (None, [0, 1, 11, 37]),

1153: (None, [0, 1, 5, 151]),

1201: (None, [0, 1, 17, 48]),

1213: (None, [0, 1, 20, 217]),

1237: (None, [0, 1, 7, 199]),

1249: (None, [0, 1, 7, 36]),

1297: (None, [0, 1, 10, 103]),

1321: (None, [0, 1, 7, 112]),

1369: (a**2 + 33*a + 2, [0, 1, a + 33, a + 9]),

1381: (None, [0, 1, 19, 84]),

1429: (None, [0, 1, 14, 116]),

1453: (None, [0, 1, 5, 377]),

1489: (None, [0, 1, 14, 44]),

1549: (None, [0, 1, 22, 89]),

1597: (None, [0, 1, 33, 228]),

1609: (None, [0, 1, 7, 95]),

1621: (None, [0, 1, 6, 165]),

1657: (None, [0, 1, 11, 121]),

1669: (None, [0, 1, 6, 155]),

1681: (a**2 + 38*a + 6, [0, 1, a, 6*a + 6]),

1693: (None, [0, 1, 5, 50]),

1741: (None, [0, 1, 19, 341]),

1753: (None, [0, 1, 7, 146]),

1777: (None, [0, 1, 10, 100]),

1789: (None, [0, 1, 6, 238]),

1801: (None, [0, 1, 11, 79]),

1849: (a**2 + 42*a + 3, [0, 1, a + 5, 2*a + 35]),

1861: (None, [0, 1, 18, 110]),

1873: (None, [0, 1, 10, 40]),

1933: (None, [0, 1, 14, 100]),

1993: (None, [0, 1, 5, 34]),

2017: (None, [0, 1, 10, 57]),

2029: (None, [0, 1, 6, 25]),

2053: (None, [0, 1, 14, 95]),

2089: (None, [0, 1, 7, 66]),

2113: (None, [0, 1, 7, 117]),

2137: (None, [0, 1, 10, 60]),

2161: (None, [0, 1, 31, 78]),

2197: (a**3 + 2*a + 11, [0, 1, 2*a + 9, 11*a + 3]),

2209: (a**2 + 45*a + 5, [0, 1, a + 5, 2*a + 12]),

2221: (None, [0, 1, 18, 201]),

2269: (None, [0, 1, 6, 99]),

2281: (None, [0, 1, 7, 212]),

2293: (None, [0, 1, 5, 116]),

2341: (None, [0, 1, 7, 99]),

2377: (None, [0, 1, 5, 214]),

2389: (None, [0, 1, 18, 29]),

2401: (a**4 + 5*a**2 + 4*a + 3, [0, 1, a, 2*a**2 + 6]),

2437: (None, [0, 1, 5, 45]),

2473: (None, [0, 1, 5, 298]),

2521: (None, [0, 1, 17, 150]),

2557: (None, [0, 1, 5, 68]),

2593: (None, [0, 1, 7, 255]),

2617: (None, [0, 1, 5, 11]),

2677: (None, [0, 1, 7, 57]),

2689: (None, [0, 1, 19, 115]),

2713: (None, [0, 1, 5, 139]),

2749: (None, [0, 1, 13, 243]),

2797: (None, [0, 1, 5, 95]),

2809: (a**2 + 49*a + 2, [0, 1, a, 3*a + 22])},

5: {

41: (None, [0, 1, 13, 38, 31]),

61: (None, [0, 1, 26, 11, 7]),

101: (None, [0, 1, 12, 43, 81]),

121: (a**2 + 7*a + 2, [0, 1, a, 9*a + 5, 3*a + 1]),

181: (None, [0, 1, 21, 47, 123]),

241: (None, [0, 1, 7, 51, 189]),

281: (None, [0, 1, 3, 143, 74]),

361: (a**2 + 18*a + 2, [0, 1, a, 2*a + 14, 18*a + 9]),

401: (None, [0, 1, 3, 128, 133]),

421: (None, [0, 1, 40, 132, 8]),

461: (None, [0, 1, 28, 53, 287]),

521: (None, [0, 1, 3, 9, 217]),

541: (None, [0, 1, 30, 124, 370]),

601: (None, [0, 1, 7, 10, 545]),

641: (None, [0, 1, 12, 79, 185]),

661: (None, [0, 1, 6, 36, 286]),

701: (None, [0, 1, 12, 97, 365]),

761: (None, [0, 1, 11, 4, 260]),

821: (None, [0, 1, 13, 62, 571]),

841: (a**2 + 24*a + 2, [0, 1, a, 2*a + 5, 5*a + 19]),

881: (None, [0, 1, 3, 9, 836]),

941: (None, [0, 1, 7, 49, 96]),

961: (a**2 + 29*a + 3, [0, 1, a, 3, 3*a]),

1021: (None, [0, 1, 30, 6, 171]),

1061: (None, [0, 1, 15, 51, 60]),

1181: (None, [0, 1, 7, 90, 87]),

1201: (None, [0, 1, 11, 14, 621]),

1301: (None, [0, 1, 7, 19, 138]),

1321: (None, [0, 1, 13, 5, 1168]),

1361: (None, [0, 1, 3, 9, 159]),

1381: (None, [0, 1, 26, 35, 547]),

1481: (None, [0, 1, 3, 9, 730]),

1601: (None, [0, 1, 3, 17, 1077]),

1621: (None, [0, 1, 14, 4, 1380]),

1681: (a**2 + 38*a + 6, [0, 1, a, a + 15, 40*a + 22]),

1721: (None, [0, 1, 3, 121, 687]),

1741: (None, [0, 1, 7, 29, 32]),

1801: (None, [0, 1, 11, 51, 142]),

1861: (None, [0, 1, 10, 62, 643]),

1901: (None, [0, 1, 12, 4, 477])

6: {

31: (None, [0, 1, 3, 12, 18, 8]),

151: (None, [0, 1, 69, 36, 57, 89]),

181: (None, [0, 1, 14, 4, 59, 139]),

211: (None, [0, 1, 24, 141, 128, 202]),

241: (None, [0, 1, 7, 151, 232, 136]),

271: (None, [0, 1, 6, 15, 81, 225]),

331: (None, [0, 1, 29, 113, 21, 69]),

361: (a**2 + 18*a + 2, [0, 1, a, 3*a + 2, 14*a, 10*a + 9]),

421: (None, [0, 1, 11, 4, 111, 394]),

541: (None, [0, 1, 5, 42, 157, 322]),

571: (None, [0, 1, 3, 52, 549, 137]),

601: (None, [0, 1, 6, 114, 490, 359]),

631: (None, [0, 1, 3, 73, 144, 466]),

661: (None, [0, 1, 6, 73, 182, 44]),

691: (None, [0, 1, 3, 9, 554, 425]),

751: (None, [0, 1, 3, 9, 314, 226]),

811: (None, [0, 1, 3, 9, 504, 341]),

841: (a**2 + 24*a + 2, [0, 1, a, 3*a + 11, 12*a + 24, 22*a + 10]),

961: (a**2 + 29*a + 3, [0, 1, 11, 28, 15*a + 25, 4*a + 3]),

991: (None, [0, 1, 6, 36, 234, 834]),

1021: (None, [0, 1, 30, 6, 476, 154]),

1051: (None, [0, 1, 7, 23, 324, 266]),

1171: (None, [0, 1, 37, 4, 1163, 302]),

1201: (None, [0, 1, 11, 5, 130, 146]),

1231: (None, [0, 1, 3, 9, 768, 476]),

1291: (None, [0, 1, 45, 79, 320, 390]),

1321: (None, [0, 1, 13, 33, 445, 894]),

1381: (None, [0, 1, 26, 56, 474, 839]),

1471: (None, [0, 1, 6, 36, 425, 676]),

1531: (None, [0, 1, 38, 8, 465, 1376]),

1621: (None, [0, 1, 5, 20, 117, 1486]),

1681: (a**2 + 38*a + 6, [0, 1, a, a + 5, 2*a + 28, 2*a + 34]),

1741: (None, [0, 1, 9, 4, 301, 420]),

1801: (None, [0, 1, 6, 4, 1263, 260]),

1831: (None, [0, 1, 3, 9, 452, 1532]),

1861: (None, [0, 1, 10, 4, 188, 1405]),

1951: (None, [0, 1, 3, 7, 27, 1032]),

7: {

169: (a**2 + 12*a + 2, [0, 1, a, 5*a + 3, 11*a + 10, 11*a + 6, 5*a + 6]),

337: (None, [0, 1, 10, 28, 80, 224, 129]),

379: (None, [0, 1, 9, 175, 287, 14, 271]),

421: (None, [0, 1, 26, 4, 191, 250, 298]),

463: (None, [0, 1, 3, 9, 310, 243, 415]),

547: (None, [0, 1, 25, 4, 430, 9, 210]),

631: (None, [0, 1, 3, 104, 303, 257, 447]),

673: (None, [0, 1, 5, 25, 405, 476, 131]),

757: (None, [0, 1, 6, 36, 232, 557, 274]),

841: (a**2 + 24*a + 2, [0, 1, a + 28, 2*a + 1, 7*a + 22, 25*a + 20, 11*a + 10]),

883: (None, [0, 1, 54, 4, 870, 638, 310]),

967: (None, [0, 1, 5, 22, 775, 577, 819]),

1009: (None, [0, 1, 5, 36, 911, 650, 412]),

1051: (None, [0, 1, 7, 49, 274, 1012, 213]),

1093: (None, [0, 1, 5, 25, 274, 214, 735]),

1303: (None, [0, 1, 30, 70, 1107, 39, 1271]),

1429: (None, [0, 1, 6, 15, 289, 975, 314]),

1471: (None, [0, 1, 6, 36, 216, 947, 568]),

1597: (None, [0, 1, 7, 38, 266, 223, 1316]),

1681: (a**2 + 38*a + 6, [0, 1, a, 2*a + 12, 7*a + 9, 35*a + 29, 33*a + 2]),

1723: (None, [0, 1, 3, 9, 1169, 420, 1651]),

1849: (a**2 + 42*a + 3, [0, 1, 13, 3, 39, 19, 5*a + 13]),

1933: (None, [0, 1, 5, 25, 319, 1607, 1782])

8: {

449: (None, [0, 1, 3, 332, 8, 104, 381, 61]),

617: (None, [0, 1, 3, 610, 397, 318, 465, 84]),

673: (None, [0, 1, 20, 355, 92, 491, 315, 478]),

729: (a**6 + 2*a**4 + a**2 + 2*a + 2, [0, 1, a,

a**2, 2*a**4 + a**3 + a**2 + a + 1,

a**4, a**3 + 2*a**2 + 2, 2*a**5 + a**4 + 2*a**2 + 2*a]),

841: (a**2 + 24*a + 2, [0, 1, a, 27, 27*a + 25, 5*a + 18,

11*a + 14, 14*a + 2]),

953: (None, [0, 1, 3, 36, 727, 636, 899, 448]),

1009: (None, [0, 1, 11, 20, 202, 283, 698, 629]),

1289: (None, [0, 1, 6, 133, 579, 793, 361, 658]),

1681: (a**2 + 38*a + 6, [0, 1, a, 3*a + 25, 5*a + 33, 34*a + 12,

23*a + 31, 38*a + 14]),

1849: (a**2 + 42*a + 3, [0, 1, a, a + 2, 4*a + 36, 5*a,

20*a + 22, 18*a + 5]),

9: {

73: (None, [0, 1, 5, 21, 59, 18, 12, 51, 49]),

433: (None, [0, 1, 5, 145, 347, 248, 57, 267, 110]),

937: (None, [0, 1, 5, 265, 828, 773, 328, 587, 866]),

1009: (None, [0, 1, 11, 251, 944, 497, 700, 99, 545]),

1153: (None, [0, 1, 5, 522, 1116, 495, 215, 859, 167]),

1297: (None, [0, 1, 10, 244, 30, 1111, 392, 1183, 123]),

1369: (a**2 + 33*a + 2, [0, 1, a, 8*a + 34, 36*a + 33, 2*a + 21, 20,

32*a + 15, 25*a + 20]),

1657: (None, [0, 1, 11, 121, 396, 269, 266, 873, 345]),

1801: (None, [0, 1, 11, 105, 603, 966, 746, 1585, 1298]),

1873: (None, [0, 1, 10, 32, 1837, 1823, 1040, 1826, 1496]),

10:{

1171: (None, [0, 1, 817, 856, 143, 881, 833, 82, 870, 564]),

1531: (None, [0, 1, 61, 1109, 417, 590, 1273, 11, 1445, 326]),

1621: (None, [0, 1, 52, 111, 779, 365, 1225, 378, 535, 1012]),

1801: (None, [0, 1, 6, 369, 80, 1717, 138, 1782, 1301, 82]),

}

LIST_OF_EDS = "\n".join(" - `k = {}`: {}".format(

k, ', '.join('`{}`'.format(q) for q in sorted(EDS[k]) if EDS[k][q] is not False))

for k in sorted(EDS))

__doc__ = __doc__.format(

LIST_OF_OA_CONSTRUCTIONS = LIST_OF_OA_CONSTRUCTIONS,

LIST_OF_MOLS_CONSTRUCTIONS = LIST_OF_MOLS_CONSTRUCTIONS,

LIST_OF_VMT_VECTORS = LIST_OF_VMT_VECTORS,

LIST_OF_BIBD = LIST_OF_BIBD,

LIST_OF_DF = LIST_OF_DF,

LIST_OF_DM = LIST_OF_DM,

LIST_OF_QDM = LIST_OF_QDM,

LIST_OF_EDS = LIST_OF_EDS)

del LIST_OF_OA_CONSTRUCTIONS, LIST_OF_MOLS_CONSTRUCTIONS, LIST_OF_VMT_VECTORS,LIST_OF_DF, LIST_OF_DM, LIST_OF_QDM, LIST_OF_EDS, LIST_OF_BIBD

del PolynomialRing, ZZ, a,

Coverage for local/lib/python2.7/site-packages/sage/combinat/designs/database.py : 79%

887 statements 705 run 182 missing 0 excluded