This module gathers code related to difference matrices. One can build those objects (or know if they can be built) with difference_matrix():
sage: G,DM = designs.difference_matrix(9,5,1)
Return a -difference matrix
A matrix is a
-difference matrix if it has size
, its entries belong to the group
of cardinality
, and
for any two rows
of
and
there are exactly
values
such that
.
INPUT:
k – (integer) number of columns. If k=None it is set to the largest value available.
g – (integer) cardinality of the group
lmbda – (integer; default: 1) – number of times each element of
appears as a difference.
check – (boolean) Whether to check that output is correct before returning it. As this is expected to be useless (but we are cautious guys), you may want to disable it whenever you want speed. Set to True by default.
existence (boolean) – instead of building the design, return:
- True – meaning that Sage knows how to build the design
- Unknown – meaning that Sage does not know how to build the design, but that the design may exist (see sage.misc.unknown).
- False – meaning that the design does not exist.
Note
When k=None and existence=True the function returns an
integer, i.e. the largest such that we can build a
-DM.
EXAMPLES:
sage: G,M = designs.difference_matrix(25,10); G
Finite Field in x of size 5^2
sage: designs.difference_matrix(993,None,existence=1)
32
Here we print for each the maximum possible
for which Sage knows
how to build a
-difference matrix:
sage: for g in range(2,30):
....: k_max = designs.difference_matrix(g=g,k=None,existence=True)
....: print "{:2} {}".format(g, k_max)
....: _ = designs.difference_matrix(g,k_max)
2 2
3 3
4 4
5 5
6 2
7 7
8 8
9 9
10 2
11 11
12 6
13 13
14 2
15 3
16 16
17 17
18 2
19 19
20 4
21 6
22 2
23 23
24 8
25 25
26 2
27 27
28 6
29 29
TESTS:
sage: designs.difference_matrix(10,12,1,existence=True)
False
sage: designs.difference_matrix(10,12,1)
Traceback (most recent call last):
...
EmptySetError: No (10,12,1)-Difference Matrix exists as k(=12)>g(=10)
sage: designs.difference_matrix(10,9,1,existence=True)
Unknown
sage: designs.difference_matrix(10,9,1)
Traceback (most recent call last):
...
NotImplementedError: I don't know how to build a (10,9,1)-Difference Matrix!
Return the product of the (G1,k,lmbda1) and (G2,k,lmbda2) difference matrices M1 and M2.
The result is a -difference matrix.
INPUT:
EXAMPLES:
sage: from sage.combinat.designs.difference_matrices import (
....: difference_matrix_product,
....: is_difference_matrix)
sage: G1,M1 = designs.difference_matrix(11,6)
sage: G2,M2 = designs.difference_matrix(7,6)
sage: G,M = difference_matrix_product(6,M1,G1,1,M2,G2,1)
sage: G1
Finite Field of size 11
sage: G2
Finite Field of size 7
sage: G
The cartesian product of (Finite Field of size 11, Finite Field of size 7)
sage: is_difference_matrix(M,G,6,1)
True
Try to find a product decomposition construction for difference matrices.
INPUT:
OUTPUT:
A pair of pairs (g1,lmbda),(g2,lmbda2) if Sage knows how to build
and
difference matrices and False
otherwise.
EXAMPLES:
sage: from sage.combinat.designs.difference_matrices import find_product_decomposition
sage: find_product_decomposition(77,6)
((7, 1), (11, 1))
sage: find_product_decomposition(616,7)
((7, 1), (88, 1))
sage: find_product_decomposition(24,10)
False