A Steiner Quadruple System on points is a family
of
-sets, such that any set
of size three is a subset of
exactly one member of
.
This module implements Haim Hanani’s constructive proof that a Steiner Quadruple
System exists if and only if . Hanani’s proof consists in 6
different constructions that build a large Steiner Quadruple System from a smaller
one, and though it does not give a very clear understanding of why it works (to say the
least)... it does !
The constructions have been implemented while reading two papers simultaneously, for one of them sometimes provides the informations that the other one does not. The first one is Haim Hanani’s original paper [Hanani60], and the other one is a paper from Horan and Hurlbert which goes through all constructions [HH12].
It can be used through the designs object:
sage: designs.steiner_quadruple_system(8)
((0, 1, 2, 3), (0, 1, 6, 7), (0, 5, 2, 7), (0, 5, 6, 3), (4, 1, 2, 7),
(4, 1, 6, 3), (4, 5, 2, 3), (4, 5, 6, 7), (0, 1, 4, 5), (0, 2, 4, 6),
(0, 3, 4, 7), (1, 2, 5, 6), (1, 3, 5, 7), (2, 3, 6, 7))
REFERENCES:
[Hanani60] | (1, 2, 3, 4, 5) Haim Hanani, On quadruple systems, pages 145–157, vol. 12, Canadadian Journal of Mathematics, 1960 http://cms.math.ca/cjm/v12/cjm1960v12.0145-0157.pdf |
[HH12] | Victoria Horan and Glenn Hurlbert, Overlap Cycles for Steiner Quadruple Systems, 2012, http://arxiv.org/abs/1204.3215 |
AUTHORS:
This module’s main function is the following :
steiner_quadruple_system() | Returns a Steiner Quadruple System on ![]() |
This function redistributes its work among 6 constructions :
Construction ![]() |
two_n() | Returns a Steiner Quadruple System on ![]() |
Construction ![]() |
three_n_minus_two() | Returns a Steiner Quadruple System on ![]() |
Construction ![]() |
three_n_minus_eight() | Returns a Steiner Quadruple System on ![]() |
Construction ![]() |
three_n_minus_four() | Returns a Steiner Quadruple System on ![]() |
Construction ![]() |
four_n_minus_six() | Returns a Steiner Quadruple System on ![]() |
Construction ![]() |
twelve_n_minus_ten() | Returns a Steiner Quadruple System on ![]() |
It also defines two specific Steiner Quadruple Systems that the constructions
require, i.e.`SQS_{14}` and as well as the systems of pairs
and
(see [Hanani60]).
Returns the collection of pairs
For more information on this system, see [Hanani60].
EXAMPLE:
sage: from sage.combinat.designs.steiner_quadruple_systems import P
sage: P(3,4)
[(0, 5), (2, 7), (4, 1), (6, 3)]
Returns the collection of pairs
For more information on this system, see [Hanani60].
EXAMPLE:
sage: from sage.combinat.designs.steiner_quadruple_systems import barP
sage: barP(3,4)
[(0, 4), (3, 5), (1, 2)]
Returns the 1-factorization of
For more information on this system, see [Hanani60].
EXAMPLE:
sage: from sage.combinat.designs.steiner_quadruple_systems import barP_system
sage: barP_system(3)
[[(4, 3), (2, 5)],
[(0, 5), (4, 1)],
[(0, 2), (1, 3)],
[(1, 5), (4, 2), (0, 3)],
[(0, 4), (3, 5), (1, 2)],
[(0, 1), (2, 3), (4, 5)]]
Returns a Steiner Quadruple System on points.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.steiner_quadruple_systems import four_n_minus_six, is_steiner_quadruple_system
sage: for n in xrange(4, 20):
....: if (n%6) in [2,4]:
....: sqs = designs.steiner_quadruple_system(n)
....: if not is_steiner_quadruple_system(4*n-6, four_n_minus_six(n, sqs)):
....: print "Something is wrong !"
Tests if is a Steiner Quadruple System on
.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.steiner_quadruple_systems import is_steiner_quadruple_system
sage: is_steiner_quadruple_system(8,designs.steiner_quadruple_system(8))
True
Relabels the set so that is in
.
INPUT:
EXAMPLE:
sage: from sage.combinat.designs.steiner_quadruple_systems import relabel_system
sage: designs.steiner_quadruple_system(8)
((0, 1, 2, 3), (0, 1, 6, 7), (0, 5, 2, 7), (0, 5, 6, 3), (4, 1, 2, 7),
(4, 1, 6, 3), (4, 5, 2, 3), (4, 5, 6, 7), (0, 1, 4, 5), (0, 2, 4, 6),
(0, 3, 4, 7), (1, 2, 5, 6), (1, 3, 5, 7), (2, 3, 6, 7))
sage: relabel_system(8,designs.steiner_quadruple_system(8))
((4, 5, 6, 7), (0, 1, 4, 5), (1, 2, 4, 6), (0, 2, 4, 7), (1, 3, 5, 6),
(0, 3, 5, 7), (2, 3, 6, 7), (0, 1, 2, 3), (2, 3, 4, 5), (0, 3, 4, 6),
(1, 3, 4, 7), (0, 2, 5, 6), (1, 2, 5, 7), (0, 1, 6, 7))
Returns a Steiner Quadruple System on points.
INPUT:
EXAMPLES:
sage: designs.steiner_quadruple_system(4)
((0, 1, 2, 3),)
sage: designs.steiner_quadruple_system(8)
((0, 1, 2, 3), (0, 1, 6, 7), (0, 5, 2, 7), (0, 5, 6, 3), (4, 1, 2, 7),
(4, 1, 6, 3), (4, 5, 2, 3), (4, 5, 6, 7), (0, 1, 4, 5), (0, 2, 4, 6),
(0, 3, 4, 7), (1, 2, 5, 6), (1, 3, 5, 7), (2, 3, 6, 7))
TESTS:
sage: for n in xrange(4, 100): # long time
....: if (n%6) in [2,4]: # long time
....: sqs = designs.steiner_quadruple_system(n, check=True) # long time
Returns a Steiner Quadruple System on points.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.steiner_quadruple_systems import three_n_minus_eight, is_steiner_quadruple_system
sage: for n in xrange(4, 30):
....: if (n%12) == 2:
....: sqs = designs.steiner_quadruple_system(n)
....: if not is_steiner_quadruple_system(3*n-8, three_n_minus_eight(n, sqs)):
....: print "Something is wrong !"
Returns a Steiner Quadruple System on points.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.steiner_quadruple_systems import three_n_minus_four, is_steiner_quadruple_system
sage: for n in xrange(4, 30):
....: if n%12 == 10:
....: sqs = designs.steiner_quadruple_system(n)
....: if not is_steiner_quadruple_system(3*n-4, three_n_minus_four(n, sqs)):
....: print "Something is wrong !"
Returns a Steiner Quadruple System on points.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.steiner_quadruple_systems import three_n_minus_two, is_steiner_quadruple_system
sage: for n in xrange(4, 30):
....: if (n%6) in [2,4]:
....: sqs = designs.steiner_quadruple_system(n)
....: if not is_steiner_quadruple_system(3*n-2, three_n_minus_two(n, sqs)):
....: print "Something is wrong !"
Returns a Steiner Quadruple System on points.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.steiner_quadruple_systems import twelve_n_minus_ten, is_steiner_quadruple_system
sage: for n in xrange(4, 15):
....: if (n%6) in [2,4]:
....: sqs = designs.steiner_quadruple_system(n)
....: if not is_steiner_quadruple_system(12*n-10, twelve_n_minus_ten(n, sqs)):
....: print "Something is wrong !"
Returns a Steiner Quadruple System on points.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.steiner_quadruple_systems import two_n, is_steiner_quadruple_system
sage: for n in xrange(4, 30):
....: if (n%6) in [2,4]:
....: sqs = designs.steiner_quadruple_system(n)
....: if not is_steiner_quadruple_system(2*n, two_n(n, sqs)):
....: print "Something is wrong !"