def construction_3_3(k,n,m,i):
r"""
Return an `OA(k,nm+i)`.
This is Wilson's construction with `i` truncated columns of size 1 and such
that a block `B_0` of the incomplete OA intersects all truncated columns. As
a consequence, all other blocks intersect only `0` or `1` of the last `i`
columns. This allow to consider the block `B_0` only up to its first `k`
coordinates and then use a `OA(k,i)` instead of a `OA(k,m+i) - i.OA(k,1)`.
This is construction 3.3 from [AC07]_.
INPUT:
- ``k,n,m,i`` (integers) such that the following designs are available :
`OA(k,n),OA(k,m),OA(k,m+1),OA(k,r)`.
.. SEEALSO::
:func:`find_construction_3_3`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays_recursive import find_construction_3_3
sage: from sage.combinat.designs.orthogonal_arrays_recursive import construction_3_3
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: k=11;n=177
sage: is_orthogonal_array(construction_3_3(*find_construction_3_3(k,n)[1]),k,n,2)
True
"""
from orthogonal_arrays import wilson_construction, OA_relabel, incomplete_orthogonal_array
# Builds an OA(k+i,n) containing a block [0]*(k+i)
OA = incomplete_orthogonal_array(k+i,n,(1,))
OA = [[(x+1)%n for x in B] for B in OA]
# Truncated version
OA = [B[:k]+[0 if x == 0 else None for x in B[k:]] for B in OA]
OA = wilson_construction(OA,k,n,m,i,[1]*i,check=False)[:-i]
matrix = [range(m)+range(n*m,n*m+i)]*k
OA.extend(OA_relabel(orthogonal_array(k,m+i),k,m+i,matrix=matrix))
assert is_orthogonal_array(OA,k,n*m+i)
return OA
def simple_wilson_construction(k,r,m,u):
r"""
Return an `OA(k,rm + \sum u_i)` from Wilson construction.
INPUT:
- ``k,r,m`` -- integers
- ``u`` -- list of positive integers
.. TODO::
As soon as wilson construction accepts an empty master design we should
remove this intermediate functions.
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays_recursive import simple_wilson_construction
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: OA = simple_wilson_construction(6,7,12,())
sage: is_orthogonal_array(OA,6,84)
True
sage: OA = simple_wilson_construction(4,5,7,(3,))
sage: is_orthogonal_array(OA,4,38)
True
sage: OA = simple_wilson_construction(5,7,7,(4,5))
sage: is_orthogonal_array(OA,5,58)
True
"""
from sage.combinat.designs.orthogonal_arrays import wilson_construction, OA_relabel
n = r*m + sum(u)
n_trunc = len(u)
OA = orthogonal_array(k+n_trunc,r,check=False)
matrix = [range(r)]*k
for uu in u:
matrix.append(range(uu)+[None]*(r-uu))
OA = OA_relabel(OA,k+n_trunc,r,matrix=matrix)
return wilson_construction(OA,k,r,m,n_trunc,u,False)
def construction_3_6(k,n,m,i):
r"""
Return a `OA(k,nm+i)`
This is Wilson's construction with `r` columns of order `1`, in which each
block intersects at most two truncated columns. Such a design exists when
`n` is a prime power and is returned by :func:`OA_and_oval`.
INPUT:
- ``k,n,m,i`` (integers) -- `n` must be a prime power. The following designs
must be available: `OA(k+r,q),OA(k,m),OA(k,m+1),OA(k,m+2)`.
This is construction 3.6 from [AC07]_.
.. SEEALSO::
- :func:`construction_3_6`
- :func:`OA_and_oval`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays_recursive import find_construction_3_6
sage: from sage.combinat.designs.orthogonal_arrays_recursive import construction_3_6
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: k=8;n=95
sage: is_orthogonal_array(construction_3_6(*find_construction_3_6(k,n)[1]),k,n,2)
True
"""
from orthogonal_arrays import wilson_construction
OA = OA_and_oval(n)
OA = [B[:k+i] for B in OA]
OA = [B[:k] + [x if x==0 else None for x in B[k:]] for B in OA]
OA = wilson_construction(OA,k,n,m,i,[1]*i)
assert is_orthogonal_array(OA,k,n*m+i)
return OA
def construction_3_5(k,n,m,r,s,t):
r"""
Return an `OA(k,nm+r+s+t)`.
This is exactly Wilson's construction with three truncated groups
except we make sure that all blocks have size `>k`, so we don't
need a `OA(k,m+0)` but only `OA(k,m+1),OA(k,m+2),OA(k,m+3)`.
This is construction 3.5 from [AC07]_.
INPUT:
- ``k,n,m`` (integers)
- ``r,s,t`` (integers) -- sizes of the three truncated groups,
such that `r\leq s` and `(q-r-1)(q-s) \geq (q-s-1)*(q-r)`.
The following designs must be available :
`OA(k,n),OA(k,r),OA(k,s),OA(k,t),OA(k,m+1),OA(k,m+2),OA(k,m+3)`.
.. SEEALSO::
:func:`find_construction_3_5`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays_recursive import find_construction_3_5
sage: from sage.combinat.designs.orthogonal_arrays_recursive import construction_3_5
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: k=8;n=111
sage: is_orthogonal_array(construction_3_5(*find_construction_3_5(k,n)[1]),k,n,2)
True
"""
from orthogonal_arrays import wilson_construction, OA_relabel
assert r <= s
q = n
assert (q-r-1)*(q-s) >= (q-s-1)*(q-r)
master_design = orthogonal_array(k+3,q)
# group k+1 has cardinality r
# group k+2 has cardinality s
# group k+3 has cardinality t
# Taking q-s blocks going through 0 in the last block
blocks_crossing_0 = [B[-3:] for B in master_design if B[-1] == 0][:q-s]
# defining the undeleted points of the groups k+1,k+2
group_k_1 = [x[0] for x in blocks_crossing_0]
group_k_1 = [x for x in range(q) if x not in group_k_1][:r]
group_k_2 = [x[1] for x in blocks_crossing_0]
group_k_2 = [x for x in range(q) if x not in group_k_2][:s]
# All blocks that have a deleted point in groups k+1 and k+2 MUST contain a
# point in group k+3
group_k_3 = [B[-1] for B in master_design if B[-3] not in group_k_1 and B[-2] not in group_k_2]
group_k_3 = list(set(group_k_3))
assert len(group_k_3) <= t
group_k_3.extend([x for x in range(q) if x not in group_k_3])
group_k_3 = group_k_3[:t]
# Relabelling the OA
r1 = [None]*q
r2 = [None]*q
r3 = [None]*q
for i,x in enumerate(group_k_1):
r1[x] = i
for i,x in enumerate(group_k_2):
r2[x] = i
for i,x in enumerate(group_k_3):
r3[x] = i
OA = OA_relabel(master_design, k+3,q, matrix=[range(q)]*k+[r1,r2,r3])
OA = wilson_construction(OA,k,q,m,3,[r,s,t], check=False)
return OA
def construction_3_4(k,n,m,r,s):
r"""
Return a `OA(k,nm+rs)`.
This is Wilson's construction applied to a truncated `OA(k+r+1,n)` with `r`
columns of size `1` and one column of size `s`.
The unique elements of the `r` truncated columns are picked so that a block
`B_0` contains them all.
- If there exists an `OA(k,m+r+1)` the column of size `s` is truncated in
order to intersect `B_0`.
- Otherwise, if there exists an `OA(k,m+r)`, the last column must not
intersect `B_0`
This is construction 3.4 from [AC07]_.
INPUT:
- ``k,n,m,r,s`` (integers) -- we assume that `s<n` and `1\leq r,s`
The following designs must be available:
`OA(k,n),OA(k,m),OA(k,m+1),OA(k,m+2),OA(k,s)`. Additionnally, it requires
either a `OA(k,m+r)` or a `OA(k,m+r+1)`.
.. SEEALSO::
:func:`find_construction_3_4`
EXAMPLES::
sage: from sage.combinat.designs.orthogonal_arrays_recursive import find_construction_3_4
sage: from sage.combinat.designs.orthogonal_arrays_recursive import construction_3_4
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: k=8;n=196
sage: is_orthogonal_array(construction_3_4(*find_construction_3_4(k,n)[1]),k,n,2)
True
"""
from orthogonal_arrays import wilson_construction, OA_relabel
assert s<n
master_design = orthogonal_array(k+r+1,n)
# Defines the first k+r columns of the matrix of labels
matrix = [range(n)]*k + [[None]*n]*(r) + [[None]*n]
B0 = master_design[0]
for i in range(k,k+r):
matrix[i][B0[i]] = 0
# Last column
if orthogonal_array(k,m+r,existence=True):
last_group = [x for x in range(s+1) if x != B0[-1]][:s]
elif orthogonal_array(k,m+r+1,existence=True):
last_group = [x for x in range(s+1) if x != B0[-1]][:s-1] + [B0[-1]]
else:
raise Exception
for i,x in enumerate(last_group):
matrix[-1][x] = i
OA = OA_relabel(master_design,k+r+1,n, matrix=matrix)
OA = wilson_construction(OA,k,n,m,r+1,[1]*r+[s],check=False)
return OA