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Matthias Koeppe
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sage.combinat: More # optional
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‎src/sage/combinat/designs/bibd.py

Lines changed: 27 additions & 28 deletions
Original file line numberDiff line numberDiff line change
@@ -4,7 +4,7 @@
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This module gathers everything related to Balanced Incomplete Block Designs. One can build a
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BIBD (or check that it can be built) with :func:`balanced_incomplete_block_design`::
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7-
sage: BIBD = designs.balanced_incomplete_block_design(7,3,1)
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sage: BIBD = designs.balanced_incomplete_block_design(7,3,1) # optional - sage.schemes
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In particular, Sage can build a `(v,k,1)`-BIBD when one exists for all `k\leq
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5`. The following functions are available:
@@ -90,7 +90,7 @@ def biplane(n, existence=False):
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sage: designs.biplane(4) # optional - sage.rings.finite_rings
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(16,6,2)-Balanced Incomplete Block Design
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sage: designs.biplane(7, existence=True)
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sage: designs.biplane(7, existence=True) # optional - sage.schemes
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True
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sage: designs.biplane(11)
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(79,13,2)-Balanced Incomplete Block Design
@@ -153,9 +153,10 @@ def balanced_incomplete_block_design(v, k, lambd=1, existence=False, use_LJCR=Fa
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EXAMPLES::
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156-
sage: designs.balanced_incomplete_block_design(7, 3, 1).blocks()
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sage: designs.balanced_incomplete_block_design(7, 3, 1).blocks() # optional - sage.schemes
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[[0, 1, 3], [0, 2, 4], [0, 5, 6], [1, 2, 6], [1, 4, 5], [2, 3, 5], [3, 4, 6]]
158-
sage: B = designs.balanced_incomplete_block_design(66, 6, 1, use_LJCR=True) # optional - internet
158+
sage: B = designs.balanced_incomplete_block_design(66, 6, 1, # optional - internet
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....: use_LJCR=True)
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sage: B # optional - internet
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(66,6,1)-Balanced Incomplete Block Design
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sage: B.blocks() # optional - internet
@@ -387,33 +388,33 @@ def BruckRyserChowla_check(v, k, lambd):
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Nonexistence of projective planes of order 6 and 14
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sage: from sage.combinat.designs.bibd import BruckRyserChowla_check
390-
sage: BruckRyserChowla_check(43,7,1)
391+
sage: BruckRyserChowla_check(43,7,1) # optional - sage.schemes
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False
392-
sage: BruckRyserChowla_check(211,15,1)
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sage: BruckRyserChowla_check(211,15,1) # optional - sage.schemes
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False
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Existence of symmetric BIBDs with parameters `(79,13,2)` and `(56,11,2)`
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sage: from sage.combinat.designs.bibd import BruckRyserChowla_check
398-
sage: BruckRyserChowla_check(79,13,2)
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sage: BruckRyserChowla_check(79,13,2) # optional - sage.schemes
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True
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sage: BruckRyserChowla_check(56,11,2)
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sage: BruckRyserChowla_check(56,11,2) # optional - sage.schemes
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True
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TESTS:
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Test some non-symmetric parameters::
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sage: from sage.combinat.designs.bibd import BruckRyserChowla_check
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sage: BruckRyserChowla_check(89,11,3)
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sage: BruckRyserChowla_check(89,11,3) # optional - sage.schemes
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Unknown
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sage: BruckRyserChowla_check(25,23,2)
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sage: BruckRyserChowla_check(25,23,2) # optional - sage.schemes
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Unknown
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Clearly wrong parameters satisfying the theorem::
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sage: from sage.combinat.designs.bibd import BruckRyserChowla_check
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sage: BruckRyserChowla_check(13,25,50)
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sage: BruckRyserChowla_check(13,25,50) # optional - sage.schemes
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True
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"""
@@ -569,9 +570,9 @@ def BIBD_from_TD(v,k,existence=False):
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First construction::
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sage: from sage.combinat.designs.bibd import BIBD_from_TD
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sage: BIBD_from_TD(25,5,existence=True)
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sage: BIBD_from_TD(25,5,existence=True) # optional - sage.schemes
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True
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sage: _ = BlockDesign(25,BIBD_from_TD(25,5))
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sage: _ = BlockDesign(25,BIBD_from_TD(25,5)) # optional - sage.schemes
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Second construction::
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@@ -866,8 +867,8 @@ def BIBD_from_PBD(PBD, v, k, check=True, base_cases=None):
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sage: from sage.combinat.designs.bibd import PBD_4_5_8_9_12
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sage: from sage.combinat.designs.bibd import BIBD_from_PBD
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sage: from sage.combinat.designs.bibd import is_pairwise_balanced_design
869-
sage: PBD = PBD_4_5_8_9_12(17)
870-
sage: bibd = is_pairwise_balanced_design(BIBD_from_PBD(PBD,52,4),52,[4])
870+
sage: PBD = PBD_4_5_8_9_12(17) # optional - sage.schemes
871+
sage: bibd = is_pairwise_balanced_design(BIBD_from_PBD(PBD,52,4),52,[4]) # optional - sage.schemes
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"""
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if base_cases is None:
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base_cases = {}
@@ -904,11 +905,11 @@ def _relabel_bibd(B,n,p=None):
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- ``n`` (integer) -- number of points.
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- ``p`` (optional) -- the point that will be labeled with n-1.
908+
- ``p`` (optional) -- the point that will be labeled with `n-1`.
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EXAMPLES::
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sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest
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sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest, optional - sage.schemes
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[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 10],
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[0, 5, 7, 11], [0, 13, 26, 39], [0, 14, 25, 28],
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[0, 15, 27, 38], [0, 16, 22, 32], [0, 17, 23, 34],
@@ -950,7 +951,7 @@ def PBD_4_5_8_9_12(v, check=True):
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EXAMPLES::
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sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest
954+
sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest, optional - sage.schemes
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[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 10],
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[0, 5, 7, 11], [0, 13, 26, 39], [0, 14, 25, 28],
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[0, 15, 27, 38], [0, 16, 22, 32], [0, 17, 23, 34],
@@ -1029,7 +1030,7 @@ def _PBD_4_5_8_9_12_closure(B):
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10301031
EXAMPLES::
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1032-
sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest
1033+
sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest, optional - sage.schemes
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[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 10],
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[0, 5, 7, 11], [0, 13, 26, 39], [0, 14, 25, 28],
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[0, 15, 27, 38], [0, 16, 22, 32], [0, 17, 23, 34],
@@ -1569,16 +1570,14 @@ def arc(self, s=2, solver=None, verbose=0, *, integrality_tolerance=1e-3):
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15701571
EXAMPLES::
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1572-
sage: B = designs.balanced_incomplete_block_design(21, 5)
1573-
sage: a2 = B.arc()
1574-
sage: a2 # random
1573+
sage: B = designs.balanced_incomplete_block_design(21, 5) # optional - sage.schemes
1574+
sage: a2 = B.arc(); a2 # random # optional - sage.schemes
15751575
[5, 9, 10, 12, 15, 20]
1576-
sage: len(a2)
1576+
sage: len(a2) # optional - sage.schemes
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6
1578-
sage: a4 = B.arc(4)
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sage: a4 # random
1578+
sage: a4 = B.arc(4); a4 # random # optional - sage.schemes
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[0, 1, 2, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20]
1581-
sage: len(a4)
1580+
sage: len(a4) # optional - sage.schemes
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16
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The `2`-arc and `4`-arc above are maximal. One can check that they
@@ -1591,9 +1590,9 @@ def arc(self, s=2, solver=None, verbose=0, *, integrality_tolerance=1e-3):
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sage: 1 + r*3
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16
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1594-
sage: B.trace(a2).is_t_design(2, return_parameters=True)
1593+
sage: B.trace(a2).is_t_design(2, return_parameters=True) # optional - sage.schemes
15951594
(True, (2, 6, 2, 1))
1596-
sage: B.trace(a4).is_t_design(2, return_parameters=True)
1595+
sage: B.trace(a4).is_t_design(2, return_parameters=True) # optional - sage.schemes
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(True, (2, 16, 4, 1))
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Some other examples which are not maximal::

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