Database of small combinatorial designs

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:

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

Functions

sage.combinat.designs.database.DM_12_6_1()

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](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.
sage.combinat.designs.database.DM_21_6_1()

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)
sage.combinat.designs.database.DM_24_8_1()

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)
sage.combinat.designs.database.DM_273_17_1()

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)
sage.combinat.designs.database.DM_28_6_1()

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)
sage.combinat.designs.database.DM_33_6_1()

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)
sage.combinat.designs.database.DM_35_6_1()

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)
sage.combinat.designs.database.DM_36_9_1()

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)
sage.combinat.designs.database.DM_39_6_1()

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
sage.combinat.designs.database.DM_44_6_1()

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)
sage.combinat.designs.database.DM_45_7_1()

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)
sage.combinat.designs.database.DM_48_9_1()

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)
sage.combinat.designs.database.DM_51_6_1()

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)
sage.combinat.designs.database.DM_52_6_1()

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)
sage.combinat.designs.database.DM_55_7_1()

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)
sage.combinat.designs.database.DM_56_8_1()

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)
sage.combinat.designs.database.DM_57_8_1()

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)
sage.combinat.designs.database.DM_60_6_1()

Return a (60,6,1)-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)
sage.combinat.designs.database.DM_75_8_1()

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)
sage.combinat.designs.database.DM_993_32_1()

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)
sage.combinat.designs.database.MOLS_10_2()

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
sage.combinat.designs.database.MOLS_12_5()

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
sage.combinat.designs.database.MOLS_14_4()

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
sage.combinat.designs.database.MOLS_15_4()

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
sage.combinat.designs.database.MOLS_18_3()

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
sage.combinat.designs.database.OA_10_1620()

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 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
sage.combinat.designs.database.OA_10_205()

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.

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
sage.combinat.designs.database.OA_10_469()

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 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: 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
sage.combinat.designs.database.OA_10_520()

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
sage.combinat.designs.database.OA_10_796()

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: 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
sage.combinat.designs.database.OA_11_160()

Returns an OA(11,160)

Published by Julian R. Abel in [AbelThesis]. Uses the fact that 160 = 2^5
\times 5 is a product of a power of 2 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
sage.combinat.designs.database.OA_11_185()

Returns an OA(11,185)

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

Start with a PG(2,16) with a 7 points Fano subplane; outside this plane there are 7(17-3) = 98 points on a line of the subplane and 273-98-7 = 168 other points. Greig notes that the subdesign consisting of these 168 points is a (168, \{10,12\})-PBD. Now add the 17 points of a line disjoint from this subdesign (e.g. a line of the Fano subplane). This line will intersect every line of the 168 point subdesign in 1 point. Thus the new line sizes are 11 and 13, plus a unique line of size 17, giving a (185,\{11,13,17\}-PBD and an OA(11,185).

EXAMPLES:

sage: from sage.combinat.designs.designs_pyx import is_orthogonal_array
sage: from sage.combinat.designs.database import OA_11_185
sage: OA = OA_11_185()
sage: print is_orthogonal_array(OA,11,185,2)
True

The design is available from the general constructor:

sage: designs.orthogonal_arrays.is_available(11,185)
True
sage.combinat.designs.database.OA_11_254()

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).

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
sage.combinat.designs.database.OA_11_640()

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).

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
sage.combinat.designs.database.OA_11_80()

Return an OA(11,80)

As explained in the Handbook III.3.76 [DesignHandbook]. Uses the fact that 80 = 2^4 \times 5 and that 5 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
sage.combinat.designs.database.OA_12_522()

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
sage.combinat.designs.database.OA_14_524()

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
sage.combinat.designs.database.OA_15_112()

Returns an OA(15,112)

Published by Julian R. Abel in [AbelThesis]. Uses the fact that 112 = 2^4
\times 7 and that 7 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
sage.combinat.designs.database.OA_15_224()

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).

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
sage.combinat.designs.database.OA_15_896()

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.

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
sage.combinat.designs.database.OA_16_176()

Returns an OA(16,176)

Published by Julian R. Abel in [AbelThesis]. Uses the fact that 176 = 2^4
\times 11 is a product of a power of 2 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
sage.combinat.designs.database.OA_16_208()

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.

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
sage.combinat.designs.database.OA_17_560()

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
sage.combinat.designs.database.OA_20_352()

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).

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
sage.combinat.designs.database.OA_20_416()

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).

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
sage.combinat.designs.database.OA_20_544()

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).

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
sage.combinat.designs.database.OA_25_1262()

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: 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
sage.combinat.designs.database.OA_520_plus_x(x)

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

The consruction shared by Julian R. Abel works for OA(10,520), OA(12,522), and 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 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
sage.combinat.designs.database.OA_7_18()

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
sage.combinat.designs.database.OA_7_66()

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
sage.combinat.designs.database.OA_7_68()

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
sage.combinat.designs.database.OA_7_74()

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
sage.combinat.designs.database.OA_8_69()

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
sage.combinat.designs.database.OA_8_76()

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
sage.combinat.designs.database.OA_9_1078()

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 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
sage.combinat.designs.database.OA_9_120()

Return an OA(9,120)

Construction shared by Julian R. Abel:

From a resolvable (120,8,1)-BIBD, one can obtain 7 MOLS(120) or a resolvable TD(8,120) by forming a resolvable TD(8,8) - 8.TD(8,1) on I_8 \times B for each block B in the BIBD. This gives a TD(8,120)
- 120 TD(8,1) (which is resolvable as the BIBD is resolvable).

See also

RBIBD_120_8_1()

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
sage.combinat.designs.database.OA_9_135()

Return an OA(9,135)

Construction shared by Julian R. Abel:

This design can be built by Wilson’s method (135 = 8.16 + 7) applied to an Orthogonal Array OA(9+7,16) 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 PG(2,2) (the projective plane over GF(2)) is a subdesign in PG(2,16) (the projective plane over GF(16)); in a cyclic PG(2,16) or BIBD(273,17,1) the points \equiv 0
\pmod{39} form such a subdesign (note that 273=16^2 + 16 +1 and 273 = 39 \times 7 and 7 = 2^2 + 2 + 1).

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 (273,17,1) 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
sage.combinat.designs.database.OA_9_1612()

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 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
sage.combinat.designs.database.OA_9_40()

Return an OA(9,40)

As explained in the Handbook III.3.62 [DesignHandbook]. Uses the fact that 40 = 2^3 \times 5 and that 5 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
sage.combinat.designs.database.QDM_19_6_1_1_1()

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].

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
sage.combinat.designs.database.QDM_21_5_1_1_1()

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].

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
sage.combinat.designs.database.QDM_21_6_1_1_5()

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].

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
sage.combinat.designs.database.QDM_25_6_1_1_5()

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].

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
sage.combinat.designs.database.QDM_33_6_1_1_1()

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].

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
sage.combinat.designs.database.QDM_35_7_1_1_7()

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].

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
sage.combinat.designs.database.QDM_37_6_1_1_1()

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].

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
sage.combinat.designs.database.QDM_45_7_1_1_9()

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].

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
sage.combinat.designs.database.QDM_54_7_1_1_8()

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].

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
sage.combinat.designs.database.QDM_57_9_1_1_8()

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

Used to build an OA(9,65)

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
sage.combinat.designs.database.RBIBD_120_8_1()

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.

See also

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)
sage.combinat.designs.database.cyclic_shift(l, i)

x.__init__(...) initializes x; see help(type(x)) for signature

sage.combinat.designs.database.f()

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

Used to build an OA(9,65)

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

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