This module implements combinatorial designs that cannot be obtained by more general constructions. Most of them come from the Handbook of Combinatorial Designs [DesignHandbook].
All this would only be a dream without the mathematical knowledge and help of Julian R. Abel.
These functions can all be obtained through the designs.<tab> functions.
This module implements:
OA(11,640), OA(15,224), OA(7,66), OA(7,68), OA(8,69), OA(9,135), OA(10,520), OA(14,524), OA(7,74), OA(8,76), OA(10,205), OA(11,80), OA(20,352), OA(7,18), OA(20,416), OA(10,1620), OA(10,469), OA(20,544), OA(10,796), OA(11,160), OA(16,176), OA(15,896), OA(9,40), OA(9,1612), OA(16,208), OA(25,1262), OA(15,112), OA(9,1078), OA(17,560), OA(9,120), OA(11,185), OA(12,522), OA(11,254)
2 MOLS of order 10, 5 MOLS of order 12, 3 MOLS of order 18, 4 MOLS of order 14, 4 MOLS of order 15
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REFERENCES:
[DesignHandbook] | (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27) Handbook of Combinatorial Designs (2ed) Charles Colbourn, Jeffrey Dinitz Chapman & Hall/CRC 2012 |
Return a -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] | (1, 2) Haim Hanani, Balanced incomplete block designs and related designs, http://dx.doi.org/10.1016/0012-365X(75)90040-0, Discrete Mathematics, Volume 11, Issue 3, 1975, Pages 255-369. |
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -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
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -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)
Return a -difference matrix.
As explained in [JulianAbel13].
REFERENCES:
[JulianAbel13] | (1, 2) 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)
Return a -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)
Return a -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)
Return a pair of MOLS of order 10
Data obtained from http://www.cecm.sfu.ca/organics/papers/lam/paper/html/POLS10/POLS10.html
EXAMPLES:
sage: from sage.combinat.designs.latin_squares import are_mutually_orthogonal_latin_squares
sage: from sage.combinat.designs.database import MOLS_10_2
sage: MOLS = MOLS_10_2()
sage: print are_mutually_orthogonal_latin_squares(MOLS)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(2,10)
True
Return 5 MOLS of order 12
These MOLS have been found by Brendan McKay.
EXAMPLES:
sage: from sage.combinat.designs.latin_squares import are_mutually_orthogonal_latin_squares
sage: from sage.combinat.designs.database import MOLS_12_5
sage: MOLS = MOLS_12_5()
sage: print are_mutually_orthogonal_latin_squares(MOLS)
True
Return four MOLS of order 14
These MOLS were shared by Ian Wanless. The first proof of existence was given in [Todorov12].
EXAMPLES:
sage: from sage.combinat.designs.latin_squares import are_mutually_orthogonal_latin_squares
sage: from sage.combinat.designs.database import MOLS_14_4
sage: MOLS = MOLS_14_4()
sage: print are_mutually_orthogonal_latin_squares(MOLS)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(4,14)
True
REFERENCE:
[Todorov12] | D.T. Todorov, Four mutually orthogonal Latin squares of order 14, Journal of Combinatorial Designs 2012, vol.20 n.8 pp.363-367 |
Return 4 MOLS of order 15.
These MOLS were shared by Ian Wanless.
EXAMPLES:
sage: from sage.combinat.designs.latin_squares import are_mutually_orthogonal_latin_squares
sage: from sage.combinat.designs.database import MOLS_15_4
sage: MOLS = MOLS_15_4()
sage: print are_mutually_orthogonal_latin_squares(MOLS)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(4,15)
True
Return 3 MOLS of order 18.
These MOLS were shared by Ian Wanless.
EXAMPLES:
sage: from sage.combinat.designs.latin_squares import are_mutually_orthogonal_latin_squares
sage: from sage.combinat.designs.database import MOLS_18_3
sage: MOLS = MOLS_18_3()
sage: print are_mutually_orthogonal_latin_squares(MOLS)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(3,18)
True
Returns an OA(10,1620)
This is obtained through the generalized Brouwer-van Rees
construction. Indeed, and there exists an
.
Note
This function should be removed once find_brouwer_van_rees_with_one_truncated_column() can handle all incomplete orthogonal arrays obtained through 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: print 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 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
containing a Baer subplane (i.e. a
)
and a point
. Among the
lines of
containing
:
lines intersect
on
points
lines intersect
on
point
As those lines are disjoint outside of
we can use them as groups to build a GDD on
points. By keeping only 9 lines of the second kind, however, we obtain a
-GDD of type 12^5.16^9.
We complete it into a PBD by adding a block
for each group
. We then build an OA from this PBD using the fact that all blocks of size 9 are disjoint.
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: print 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
Return an OA(10,469)
This construction appears in [Brouwer80]. It is based on the same technique used in brouwer_separable_design().
Julian R. Abel’s instructions:
Brouwer notes that a cyclic
(or
-BIBD) can be obtained with a base block containing
and
points in each residue class mod 3. Thus, by reducing the
to its points congruent to
one obtains a
-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
, one can build a matrix with size
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
with the rows of the matrix one obtains a parallel class of a resolvable
.
Add to this the parallel class of all blocks
to obtain a resolvable
equivalent to an
.
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: print 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
Return an OA(10,520).
This design is built by the slightly more general construction 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: print 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
Returns an OA(10,796)
Construction shared by Julian R. Abel, from [AC07]:
Truncate one block of a
to size
, then add an extra point. Form a block on each group plus the extra point: we obtain a
-PBD in which only the extra point lies in more than one block of size
(and each other point lies in exactly 1 such block).
For each block
(of size
say) not containing the extra point, construct a
on
. For each block
(of size
or
) containing the extra point, construct a
on
, the size
hole being on
where
is the extra point. Finally form
extra block of size
on
.
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: print 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
Returns an OA(11,160)
Published by Julian R. Abel in [AbelThesis]. Uses the fact that is a product of a power of
and a prime number.
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_11_160
sage: OA = OA_11_160()
sage: print is_orthogonal_array(OA,11,160,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(11,160)
True
Returns an OA(11,185)
The construction is given in [Greig99]. In Julian R. Abel’s words:
Start with awith a
points Fano subplane; outside this plane there are
points on a line of the subplane and
other points. Greig notes that the subdesign consisting of these
points is a
. Now add the
points of a line disjoint from this subdesign (e.g. a line of the Fano subplane). This line will intersect every line of the
point subdesign in
point. Thus the new line sizes are
and
, plus a unique line of size
, giving a
-PBD and an
.
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_11_185
sage: OA = OA_11_185()
sage: print is_orthogonal_array(OA,11,185,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(11,185)
True
Return an OA(11,254)
This constructions appears in [Greig99].
From a cyclic whose base blocks contains 7,9, and 4 points in the
congruence classes mod 3, build a
by ignoring the
points of a congruence class. There exist
,
which gives the
.
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: print 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
Returns an OA(11,640)
Published by Julian R. Abel in [AbelThesis] (uses the fact that is the product of a power of
and a prime number).
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: print 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
Return an OA(11,80)
As explained in the Handbook III.3.76 [DesignHandbook]. Uses the fact that
and that
is prime.
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_11_80
sage: OA = OA_11_80()
sage: print is_orthogonal_array(OA,11,80,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(11,80)
True
Return an OA(12,522)
This design is built by the slightly more general construction 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: print 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 an OA(14,524)
This design is built by the slightly more general construction 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: print 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
Returns an OA(15,112)
Published by Julian R. Abel in [AbelThesis]. Uses the fact that 112 = and that
is prime.
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_15_112
sage: OA = OA_15_112()
sage: print is_orthogonal_array(OA,15,112,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(15,112)
True
Returns an OA(15,224)
Published by Julian R. Abel in [AbelThesis] (uses the fact that is a product of a power of
and a prime number).
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: print 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
Returns an OA(15,896)
Uses the fact that is the product of a power of
and
a prime number.
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: print 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
Returns an OA(16,176)
Published by Julian R. Abel in [AbelThesis]. Uses the fact that is a product of a power of
and a prime number.
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_16_176
sage: OA = OA_16_176()
sage: print is_orthogonal_array(OA,16,176,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(16,176)
True
Returns an OA(16,208)
Published by Julian R. Abel in [AbelThesis]. Uses the fact that is a product of
and a prime number.
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: print 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
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: print 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
Returns an OA(20,352)
Published by Julian R. Abel in [AbelThesis] (uses the fact that is the product of a power of
and a prime number).
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: print 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
Returns an OA(20,416)
Published by Julian R. Abel in [AbelThesis] (uses the fact that is the product of a power of
and a prime number).
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: print 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
Returns an OA(20,544)
Published by Julian R. Abel in [AbelThesis] (uses the fact that
is the product of a power of
and a prime number).
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: print 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
Returns an OA(25,1262)
The construction is given in [Greig99]. In Julian R. Abel’s words:
Start with a cyclicor
-BIBD whose base block contains respectively
and
point in the residue classes mod 3. In the resulting BIBD, remove one of the three classes: the result is a
-PBD, from which the
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: print 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
Return an .
The consruction shared by Julian R. Abel works for OA(10,520), OA(12,522), and OA(14,524).
Let
and
. Build a
. Remove
points contained in a common block, add a new point
and create a block
for every (possibly truncated) group
. The result is a
. Note that all blocks of size
only intersect on
, and that the unique block
of size
intersects all blocks of size
on one point. Now:
- Build an
for each block of size 16
- Build an
for each block of size 17
- Build an
for each block of size 31 (with the hole on
).
- Build an
for each block
of size 32 (with the holes on
and
).
- Build an
on
.
Only a row
is missing from the
This construction is used in OA(10,520), OA(12,522), and OA(14,524).
EXAMPLE:
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: print is_orthogonal_array(OA,10,520,2) # not tested (already tested in OA_10_520)
True
Return an OA(7,18)
Proved in [JulianAbel13].
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_7_18
sage: OA = OA_7_18()
sage: print is_orthogonal_array(OA,7,18,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(7,18)
True
Return an OA(7,66)
Construction shared by Julian R. Abel.
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_7_66
sage: OA = OA_7_66()
sage: print is_orthogonal_array(OA,7,66,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(7,66)
True
Return an OA(7,68)
Construction shared by Julian R. Abel.
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_7_68
sage: OA = OA_7_68()
sage: print is_orthogonal_array(OA,7,68,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(7,68)
True
Return an OA(7,74)
Construction shared by Julian R. Abel.
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_7_74
sage: OA = OA_7_74()
sage: print is_orthogonal_array(OA,7,74,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(7,74)
True
Return an OA(8,69)
Construction shared by Julian R. Abel.
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_8_69
sage: OA = OA_8_69()
sage: print is_orthogonal_array(OA,8,69,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(8,69)
True
Return an OA(8,76)
Construction shared by Julian R. Abel.
EXAMPLES:
sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_8_76
sage: OA = OA_8_76()
sage: print is_orthogonal_array(OA,8,76,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(8,76)
True
Returns an OA(9,1078)
This is obtained through the generalized Brouwer-van Rees
construction. Indeed, and there exists an
.
Note
This function should be removed once find_brouwer_van_rees_with_one_truncated_column() can handle all incomplete orthogonal arrays obtained through 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: print 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 an OA(9,120)
Construction shared by Julian R. Abel:
From a resolvable, one can obtain 7
or a resolvable
by forming a resolvable
on
for each block
in the BIBD. This gives a
(which is resolvable as the BIBD is resolvable).
See also
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_9_120
sage: OA = OA_9_120()
sage: print is_orthogonal_array(OA,9,120,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(9,120)
True
Return an OA(9,135)
Construction shared by Julian R. Abel:
This design can be built by Wilson’s method (
) applied to an Orthogonal Array
with 7 groups truncated to size 1 in such a way that a block contain 0, 1 or 3 points of the truncated groups.
This is possible, because
(the projective plane over
) is a subdesign in
(the projective plane over
); in a cyclic
or
the points
form such a subdesign (note that
and
and
).
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_9_135
sage: OA = OA_9_135()
sage: print is_orthogonal_array(OA,9,135,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(9,135)
True
As this orthogonal array requires a cyclic difference set, we check that
it is available:
sage: G,D = designs.difference_family(273,17,1)
sage: G
Ring of integers modulo 273
Returns an OA(9,1612)
This is obtained through the generalized Brouwer-van Rees
construction. Indeed, and there exists an
.
Note
This function should be removed once find_brouwer_van_rees_with_one_truncated_column() can handle all incomplete orthogonal arrays obtained through 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: print 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 an OA(9,40)
As explained in the Handbook III.3.62 [DesignHandbook]. Uses the fact that
and that
is prime.
EXAMPLES:
sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_9_40
sage: OA = OA_9_40()
sage: print is_orthogonal_array(OA,9,40,2)
True
The design is available from the general constructor:
sage: designs.orthogonal_arrays.is_available(9,40)
True
Return a -quasi-difference matrix.
Used to build an
Given in the Handbook III.3.49 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
Given in the Handbook III.3.51 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
Given in the Handbook III.3.53 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
Given in the Handbook III.3.55 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
Given in the Handbook III.3.57 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
As explained in the Handbook III.3.63 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
Given in the Handbook III.3.60 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
As explained in the Handbook III.3.71 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
As explained in the Handbook III.3.74 [DesignHandbook].
EXAMPLE:
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
Return a -quasi-difference matrix.
Used to build an
Construction shared by Julian R. Abel
EXAMPLE:
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
Return a resolvable
This function output a list L of blocks such that
L[i*15:(i+1)*15] is a partition of
.
Construction shared by Julian R. Abel:
Seiden’s method: Start with a cyclic
and let
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
non-hyperoval points.
The design is also resolvable. In the original
take any point
in the hyperoval and consider a block
containing
. The
points in
that do not belong to the hyperoval correspond to
blocks forming a parallel class in the dualised design. The other
parallel classes come in a similar way, by using point
and the other
blocks containing
.
See also
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)
x.__init__(...) initializes x; see help(type(x)) for signature
Return a -quasi-difference matrix.
Used to build an
Construction shared by Julian R. Abel
EXAMPLE:
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