Difference families

This module gathers everything related to difference families. One can build a difference family (or check that it can be built) with difference_family():

sage: G,F = designs.difference_family(13,4,1)

It defines the following functions:

is_difference_family() Check if the input is a (k, l)-difference family.
difference_family() Return a (k, l)-difference family on an Abelian group of size v.
radical_difference_family() Return a radical difference family.
radical_difference_set() Return a radical difference set.
singer_difference_set() Return a difference set associated to hyperplanes in a projective space.
df_q_6_1() Return a difference family with parameter k=6 on a finite field.
one_radical_difference_family() Return a radical difference family using an exhaustive search.
twin_prime_powers_difference_set() Return a twin prime powers difference family.

REFERENCES:

[BJL99-1]T. Beth, D. Jungnickel, H. Lenz “Design theory Vol. I.” Second edition. Encyclopedia of Mathematics and its Applications, 69. Cambridge University Press, (1999).
[BLJ99-2]T. Beth, D. Jungnickel, H. Lenz “Design theory Vol. II.” Second edition. Encyclopedia of Mathematics and its Applications, 78. Cambridge University Press, (1999).
[Bo39]R. C. Bose, “On the construction of balanced incomplete block designs”, Ann. Eugenics, vol. 9, (1939), 353–399.
[Bu95](1, 2) M. Buratti “On simple radical difference families”, J. of Combinatorial Designs, vol. 3, no. 2 (1995)
[Wi72](1, 2, 3) R. M. Wilson “Cyclotomy and difference families in elementary Abelian groups”, J. of Num. Th., 4 (1972), pp. 17-47.

Functions

sage.combinat.designs.difference_family.block_stabilizer(G, B)

Compute the left stabilizer of the block B under the action of G.

This function return the list of all x\in G such that x\cdot B=B (as a set).

INPUT:

  • G – a group (additive or multiplicative).
  • B – a subset of G.

EXAMPLES:

sage: from sage.combinat.designs.difference_family import block_stabilizer

sage: Z8 = Zmod(8)
sage: block_stabilizer(Z8, [Z8(0),Z8(2),Z8(4),Z8(6)])
[0, 2, 4, 6]
sage: block_stabilizer(Z8, [Z8(0),Z8(2)])
[0]

sage: C = cartesian_product([Zmod(4),Zmod(3)])
sage: block_stabilizer(C, [C((0,0)),C((2,0)),C((0,1)),C((2,1))])
[(0, 0), (2, 0)]

sage: b = map(Zmod(45),[1, 3, 7, 10, 22, 25, 30, 35, 37, 38, 44])
sage: block_stabilizer(Zmod(45),b)
[0]
sage.combinat.designs.difference_family.df_q_6_1(K, existence=False, check=True)

Return a (q,6,1)-difference family over the finite field K.

The construction uses Theorem 11 of [Wi72].

EXAMPLES:

sage: from sage.combinat.designs.difference_family import is_difference_family, df_q_6_1
sage: prime_powers = [v for v in xrange(31,500,30) if is_prime_power(v)]
sage: parameters = [v for v in prime_powers if df_q_6_1(GF(v,'a'), existence=True)]
sage: print parameters
[31, 151, 181, 211, 241, 271, 331, 361, 421]
sage: for v in parameters:
....:     K = GF(v, 'a')
....:     df = df_q_6_1(K, check=True)
....:     assert is_difference_family(K, df, v, 6, 1)
sage.combinat.designs.difference_family.difference_family(v, k, l=1, existence=False, explain_construction=False, check=True)

Return a (k, l)-difference family on an Abelian group of cardinality v.

Let G be a finite Abelian group. For a given subset D of G, we define \Delta D to be the multi-set of differences \Delta D = \{x - y; x \in D,
y \in D, x \not= y\}. A (G,k,\lambda)-difference family is a collection of k-subsets of G, D = \{D_1, D_2, \ldots, D_b\} such that the union of the difference sets \Delta D_i for i=1,...b, seen as a multi-set, contains each element of G \backslash \{0\} exactly \lambda-times.

When there is only one block, i.e. \lambda(v - 1) = k(k-1), then a (G,k,\lambda)-difference family is also called a difference set.

See also Wikipedia article Difference_set.

If there is no such difference family, an EmptySetError is raised and if there is no construction at the moment NotImplementedError is raised.

INPUT:

  • v,k,l – parameters of the difference family. If l is not provided it is assumed to be 1.
  • existence – if True, then return either True if Sage knows how to build such design, Unknown if it does not and False if it knows that the design does not exist.
  • explain_construction – instead of returning a difference family, returns a string that explains the construction used.
  • check – boolean (default: True). If True then the result of the computation is checked before being returned. This should not be needed but ensures that the output is correct.

OUTPUT:

A pair (G,D) made of a group G and a difference family D on that group. Or, if existence is True a troolean or if explain_construction is True a string.

EXAMPLES:

sage: G,D = designs.difference_family(73,4)
sage: G
Finite Field of size 73
sage: D
[[0, 1, 5, 18],
 [0, 3, 15, 54],
 [0, 9, 45, 16],
 [0, 27, 62, 48],
 [0, 8, 40, 71],
 [0, 24, 47, 67]]

sage: print designs.difference_family(73, 4, explain_construction=True)
The database contains a (73,4)-evenly distributed set

sage: G,D = designs.difference_family(15,7,3)
sage: G
The cartesian product of (Finite Field of size 3, Finite Field of size 5)
sage: D
[[(1, 1), (1, 4), (2, 2), (2, 3), (0, 0), (1, 0), (2, 0)]]
sage: print designs.difference_family(15,7,3,explain_construction=True)
Twin prime powers difference family

sage: print designs.difference_family(91,10,1,explain_construction=True)
Singer difference set

For k=6,7 we look at the set of small prime powers for which a construction is available:

sage: def prime_power_mod(r,m):
....:     k = m+r
....:     while True:
....:         if is_prime_power(k):
....:             yield k
....:         k += m

sage: from itertools import islice
sage: l6 = {True:[], False: [], Unknown: []}
sage: for q in islice(prime_power_mod(1,30), 60):
....:     l6[designs.difference_family(q,6,existence=True)].append(q)
sage: l6[True]
[31, 121, 151, 181, 211, ...,  3061, 3121, 3181]
sage: l6[Unknown]
[61]
sage: l6[False]
[]

sage: l7 = {True: [], False: [], Unknown: []}
sage: for q in islice(prime_power_mod(1,42), 60):
....:     l7[designs.difference_family(q,7,existence=True)].append(q)
sage: l7[True]
[169, 337, 379, 421, 463, 547, 631, 673, 757, 841, 883, 967, ...,  4621, 4957, 5167]
sage: l7[Unknown]
[43, 127, 211, 2017, 2143, 2269, 2311, 2437, 2521, 2647, ..., 4999, 5041, 5209]
sage: l7[False]
[]

List available constructions:

sage: for v in xrange(2,100):
....:     constructions = []
....:     for k in xrange(2,10):
....:         for l in xrange(1,10):
....:             if designs.difference_family(v,k,l,existence=True):
....:                 constructions.append((k,l))
....:                 _ = designs.difference_family(v,k,l)
....:     if constructions:
....:         print "%2d: %s"%(v, ', '.join('(%d,%d)'%(k,l) for k,l in constructions))
 3: (2,1)
 4: (3,2)
 5: (2,1), (4,3)
 6: (5,4)
 7: (2,1), (3,1), (3,2), (4,2), (6,5)
 8: (7,6)
 9: (2,1), (4,3), (8,7)
10: (9,8)
11: (2,1), (4,6), (5,2), (5,4), (6,3)
13: (2,1), (3,1), (3,2), (4,1), (4,3), (5,5), (6,5)
15: (3,1), (4,6), (5,6), (7,3)
16: (3,2), (5,4), (6,2)
17: (2,1), (4,3), (5,5), (8,7)
19: (2,1), (3,1), (3,2), (4,2), (6,5), (9,4), (9,8)
21: (3,1), (4,3), (5,1), (6,3), (6,5)
22: (4,2), (6,5), (7,4), (8,8)
23: (2,1)
25: (2,1), (3,1), (3,2), (4,1), (4,3), (6,5), (7,7), (8,7)
27: (2,1), (3,1)
28: (3,2), (6,5)
29: (2,1), (4,3), (7,3), (7,6), (8,4), (8,6)
31: (2,1), (3,1), (3,2), (4,2), (5,2), (5,4), (6,1), (6,5)
33: (3,1), (5,5), (6,5)
34: (4,2)
35: (5,2)
37: (2,1), (3,1), (3,2), (4,1), (4,3), (6,5), (9,2), (9,8)
39: (3,1), (6,5)
40: (3,2), (4,1)
41: (2,1), (4,3), (5,1), (5,4), (6,3), (8,7)
43: (2,1), (3,1), (3,2), (4,2), (6,5), (7,2), (7,3), (7,6), (8,4)
45: (3,1), (5,1)
46: (4,2), (6,2)
47: (2,1)
49: (2,1), (3,1), (3,2), (4,1), (4,3), (6,5), (8,7), (9,3)
51: (3,1), (5,2), (6,3)
52: (4,1)
53: (2,1), (4,3)
55: (3,1), (9,4)
57: (3,1), (7,3), (8,1)
59: (2,1)
61: (2,1), (3,1), (3,2), (4,1), (4,3), (5,1), (5,4), (6,2), (6,3), (6,5)
63: (3,1)
64: (3,2), (4,1), (7,2), (7,6), (9,8)
65: (5,1)
67: (2,1), (3,1), (3,2), (6,5)
69: (3,1)
71: (2,1), (5,2), (5,4), (7,3), (7,6), (8,4)
73: (2,1), (3,1), (3,2), (4,1), (4,3), (6,5), (8,7), (9,1), (9,8)
75: (3,1), (5,2)
76: (4,1)
79: (2,1), (3,1), (3,2), (6,5)
81: (2,1), (3,1), (4,3), (5,1), (5,4), (8,7)
83: (2,1)
85: (4,1), (7,2), (7,3), (8,2)
89: (2,1), (4,3), (8,7)
91: (6,1)
97: (2,1), (3,1), (3,2), (4,1), (4,3), (6,5), (8,7), (9,3)

TESTS:

Check more of the Wilson constructions from [Wi72]:

sage: Q5 = [241, 281,421,601,641, 661, 701, 821,881]
sage: Q9 = [73, 1153, 1873, 2017]
sage: Q15 = [76231]
sage: Q4 = [13, 73, 97, 109, 181, 229, 241, 277, 337, 409, 421, 457]
sage: Q8 = [1009, 3137, 3697]
sage: for Q,k in [(Q4,4),(Q5,5),(Q8,8),(Q9,9),(Q15,15)]:
....:     for q in Q:
....:         assert designs.difference_family(q,k,1,existence=True) is True
....:         _ = designs.difference_family(q,k,1)

Check Singer difference sets:

sage: sgp = lambda q,d: ((q**(d+1)-1)//(q-1), (q**d-1)//(q-1), (q**(d-1)-1)//(q-1))

sage: for q in range(2,10):
....:     if is_prime_power(q):
....:         for d in [2,3,4]:
....:           v,k,l = sgp(q,d)
....:           assert designs.difference_family(v,k,l,existence=True) is True
....:           _ = designs.difference_family(v,k,l)

Check twin primes difference sets:

sage: for p in [3,5,7,9,11]:
....:     v = p*(p+2); k = (v-1)/2;  lmbda = (k-1)/2
....:     G,D = designs.difference_family(v,k,lmbda)

Check the database:

sage: from sage.combinat.designs.database import DF,EDS
sage: for v,k,l in DF:
....:     assert designs.difference_family(v,k,l,existence=True) is True
....:     df = designs.difference_family(v,k,l,check=True)

sage: for k in EDS:
....:     for v in EDS[k]:
....:         assert designs.difference_family(v,k,1,existence=True) is True
....:         df = designs.difference_family(v,k,1,check=True)

Check a failing construction (trac ticket #17528):

sage: designs.difference_family(9,3)
Traceback (most recent call last):
...
NotImplementedError: No construction available for (9,3,1)-difference family

Todo

Implement recursive constructions from Buratti “Recursive for difference matrices and relative difference families” (1998) and Jungnickel “Composition theorems for difference families and regular planes” (1978)

sage.combinat.designs.difference_family.group_law(G)

Return a triple (identity, operation, inverse) that define the operations on the group G.

EXAMPLES:

sage: from sage.combinat.designs.difference_family import group_law
sage: group_law(Zmod(3))
(0, <built-in function add>, <built-in function neg>)
sage: group_law(SymmetricGroup(5))
((), <built-in function mul>, <built-in function inv>)
sage: group_law(VectorSpace(QQ,3))
((0, 0, 0), <built-in function add>, <built-in function neg>)
sage.combinat.designs.difference_family.is_difference_family(G, D, v=None, k=None, l=None, verbose=False)

Check wether D forms a difference family in the group G.

INPUT:

  • G - group of cardinality v
  • D - a set of k-subsets of G
  • v, k and l - optional parameters of the difference family
  • verbose - whether to print additional information

EXAMPLES:

sage: from sage.combinat.designs.difference_family import is_difference_family
sage: G = Zmod(21)
sage: D = [[0,1,4,14,16]]
sage: is_difference_family(G, D, 21, 5)
True

sage: G = Zmod(41)
sage: D = [[0,1,4,11,29],[0,2,8,17,21]]
sage: is_difference_family(G, D, verbose=True)
Too few:
  5 is obtained 0 times in blocks []
  14 is obtained 0 times in blocks []
  27 is obtained 0 times in blocks []
  36 is obtained 0 times in blocks []
Too much:
  4 is obtained 2 times in blocks [0, 1]
  13 is obtained 2 times in blocks [0, 1]
  28 is obtained 2 times in blocks [0, 1]
  37 is obtained 2 times in blocks [0, 1]
False
sage: D = [[0,1,4,11,29],[0,2,8,17,22]]
sage: is_difference_family(G, D)
True

sage: G = Zmod(61)
sage: D = [[0,1,3,13,34],[0,4,9,23,45],[0,6,17,24,32]]
sage: is_difference_family(G, D)
True

sage: G = AdditiveAbelianGroup([3]*4)
sage: a,b,c,d = G.gens()
sage: D = [[d, -a+d, -c+d, a-b-d, b+c+d],
....:      [c, a+b-d, -b+c, a-b+d, a+b+c],
....:      [-a-b+c+d, a-b-c-d, -a+c-d, b-c+d, a+b],
....:      [-b-d, a+b+d, a-b+c-d, a-b+c, -b+c+d]]
sage: is_difference_family(G, D)
True

The following example has a third block with a non-trivial stabilizer:

sage: G = Zmod(15)
sage: D = [[0,1,4],[0,2,9],[0,5,10]]
sage: is_difference_family(G,D,verbose=True)
It is a (15,3,1)-difference family
True

The function also supports multiplicative groups (non necessarily Abelian):

sage: G = DihedralGroup(8)
sage: x,y = G.gens()
sage: i = G.one()
sage: D1 = [[i,x,x^4], [i,x^2, y*x], [i,x^5,y], [i,x^6,y*x^2], [i,x^7,y*x^5]]
sage: is_difference_family(G, D1, 16, 3, 2)
True
sage: from sage.combinat.designs.bibd import BIBD_from_difference_family
sage: bibd = BIBD_from_difference_family(G,D1,lambd=2)

TESTS:

sage: K = GF(3^2,'z')
sage: z = K.gen()
sage: D = [[1,z+1,2]]
sage: _ = is_difference_family(K, D, verbose=True)
the number of differences (=6) must be a multiple of v-1=8
sage: _
False
sage.combinat.designs.difference_family.one_cyclic_tiling(A, n)

Given a subset A of the cyclic additive group G = Z / nZ return another subset B so that A + B = G and |A| |B| = n (i.e. any element of G is uniquely expressed as a sum a+b with a in A and b in B).

EXAMPLES:

sage: from sage.combinat.designs.difference_family import one_cyclic_tiling
sage: tile = [0,2,4]
sage: m = one_cyclic_tiling(tile,6); m
[0, 3]
sage: sorted((i+j)%6 for i in tile for j in m)
[0, 1, 2, 3, 4, 5]

sage: def print_tiling(tile, translat, n):
....:     for x in translat:
....:         print ''.join('X' if (i-x)%n in tile else '.' for i in range(n))

sage: tile = [0, 1, 2, 7]
sage: m = one_cyclic_tiling(tile, 12)
sage: print_tiling(tile, m, 12)
XXX....X....
....XXX....X
...X....XXX.

sage: tile = [0, 1, 5]
sage: m = one_cyclic_tiling(tile, 12)
sage: print_tiling(tile, m, 12)
XX...X......
...XX...X...
......XX...X
..X......XX.

sage: tile = [0, 2]
sage: m = one_cyclic_tiling(tile, 8)
sage: print_tiling(tile, m, 8)
X.X.....
....X.X.
.X.X....
.....X.X

ALGORITHM:

Uses dancing links sage.combinat.dlx

sage.combinat.designs.difference_family.one_radical_difference_family(K, k)

Search for a radical difference family on K using dancing links algorithm.

For the definition of radical difference family, see radical_difference_family(). Here, we consider only radical difference family with \lambda = 1.

INPUT:

  • K – a finite field of cardinality q.
  • k – a positive integer so that k(k-1) divides q-1.

OUTPUT:

Either a difference family or None if it does not exist.

ALGORITHM:

The existence of a radical difference family is equivalent to a one dimensional tiling (or packing) problem in a cyclic group. This subsequent problem is solved by a call to the function one_cyclic_tiling().

Let K^* be the multiplicative group of the finite field K. A radical family has the form \mathcal B = \{x_1 B, \ldots, x_k B\}, where B=\{x:x^{k}=1\} (for k odd) or B=\{x:x^{k-1}=1\}\cup \{0\} (for k even). Equivalently, K^* decomposes as:

K^* = \Delta (x_1 B) \cup ... \cup \Delta (x_k B) = x_1 \Delta B \cup ... \cup x_k \Delta B

We observe that C=B\backslash 0 is a subgroup of the (cyclic) group K^*, that can thus be generated by some element r. Furthermore, we observe that \Delta B is always a union of cosets of \pm C (which is twice larger than C).

\begin{array}{llll}
(k\text{ odd} ) & \Delta B &= \{r^i-r^j:r^i\neq r^j\} &= \pm C\cdot \{r^i-1: 0 < i \leq m\}\\
(k\text{ even}) & \Delta B &= \{r^i-r^j:r^i\neq r^j\}\cup C &= \pm C\cdot \{r^i-1: 0 < i < m\}\cup \pm C
\end{array}

where

(k\text{ odd})\ m = (k-1)/2 \quad \text{and} \quad (k\text{ even})\ m = k/2.

Consequently, \mathcal B = \{x_1 B, \ldots, x_k B\} is a radical difference family if and only if \{x_1 (\Delta B/(\pm C)), \ldots, x_k
(\Delta B/(\pm C))\} is a partition of the cyclic group K^*/(\pm C).

EXAMPLES:

sage: from sage.combinat.designs.difference_family import (
....:    one_radical_difference_family,
....:    is_difference_family)

sage: one_radical_difference_family(GF(13),4)
[[0, 1, 3, 9]]

The parameters that appear in [Bu95]:

sage: df = one_radical_difference_family(GF(449), 8); df
[[0, 1, 18, 25, 176, 324, 359, 444],
 [0, 9, 88, 162, 222, 225, 237, 404],
 [0, 11, 140, 198, 275, 357, 394, 421],
 [0, 40, 102, 249, 271, 305, 388, 441],
 [0, 49, 80, 93, 161, 204, 327, 433],
 [0, 70, 99, 197, 230, 362, 403, 435],
 [0, 121, 141, 193, 293, 331, 335, 382],
 [0, 191, 285, 295, 321, 371, 390, 392]]
sage: is_difference_family(GF(449), df, 449, 8, 1)
True
sage.combinat.designs.difference_family.radical_difference_family(K, k, l=1, existence=False, check=True)

Return a (v,k,l)-radical difference family.

Let fix an integer k and a prime power q = t k(k-1) + 1. Let K be a field of cardinality q. A (q,k,1)-difference family is radical if its base blocks are either: a coset of the k-th root of unity for k odd or a coset of k-1-th root of unity and 0 if k is even (the number t is the number of blocks of that difference family).

The terminology comes from M. Buratti article [Bu95] but the first constructions go back to R. Wilson [Wi72].

INPUT:

  • K - a finite field
  • k – positive integer, the size of the blocks
  • l – the \lambda parameter (default to 1)
  • existence – if True, then return either True if Sage knows how to build such design, Unknown if it does not and False if it knows that the design does not exist.
  • check – boolean (default: True). If True then the result of the computation is checked before being returned. This should not be needed but ensures that the output is correct.

EXAMPLES:

sage: from sage.combinat.designs.difference_family import radical_difference_family

sage: radical_difference_family(GF(73),9)
[[1, 2, 4, 8, 16, 32, 37, 55, 64]]

sage: radical_difference_family(GF(281),5)
[[1, 86, 90, 153, 232],
 [4, 50, 63, 79, 85],
 [5, 36, 149, 169, 203],
 [7, 40, 68, 219, 228],
 [9, 121, 212, 248, 253],
 [29, 81, 222, 246, 265],
 [31, 137, 167, 247, 261],
 [32, 70, 118, 119, 223],
 [39, 56, 66, 138, 263],
 [43, 45, 116, 141, 217],
 [98, 101, 109, 256, 279],
 [106, 124, 145, 201, 267],
 [111, 123, 155, 181, 273],
 [156, 209, 224, 264, 271]]

sage: for k in range(5,10):
....:     print "k = {}".format(k)
....:     for q in range(k*(k-1)+1, 2000, k*(k-1)):
....:          if is_prime_power(q):
....:              K = GF(q,'a')
....:              if radical_difference_family(K, k, existence=True):
....:                  print q,
....:                  _ = radical_difference_family(K,k)
....:     print
k = 5
41 61 81 241 281 401 421 601 641 661 701 761 821 881 1181 1201 1301 1321
1361 1381 1481 1601 1681 1801 1901
k = 6
181 211 241 631 691 1531 1831 1861
k = 7
337 421 463 883 1723
k = 8
449 1009
k = 9
73 1153 1873
sage.combinat.designs.difference_family.radical_difference_set(K, k, l=1, existence=False, check=True)

Return a difference set made of a cyclotomic coset in the finite field K and with paramters k and l.

Most of these difference sets appear in chapter VI.18.48 of the Handbook of combinatorial designs.

EXAMPLES:

sage: from sage.combinat.designs.difference_family import radical_difference_set

sage: D = radical_difference_set(GF(7), 3, 1); D
[[1, 2, 4]]
sage: sorted(x-y for x in D[0] for y in D[0] if x != y)
[1, 2, 3, 4, 5, 6]

sage: D = radical_difference_set(GF(16,'a'), 6, 2)
sage: sorted(x-y for x in D[0] for y in D[0] if x != y)
[1,
 1,
 a,
 a,
 a + 1,
 a + 1,
 a^2,
 a^2,
 ...
 a^3 + a^2 + a + 1,
 a^3 + a^2 + a + 1]

sage: for k in range(2,50):
....:     for l in reversed(divisors(k*(k-1))):
....:         v = k*(k-1)//l + 1
....:         if is_prime_power(v) and radical_difference_set(GF(v,'a'),k,l,existence=True):
....:             _ = radical_difference_set(GF(v,'a'),k,l)
....:             print "{:3} {:3} {:3}".format(v,k,l)
  3   2   1
  4   3   2
  7   3   1
  5   4   3
  7   4   2
 13   4   1
 11   5   2
  7   6   5
 11   6   3
 16   6   2
  8   7   6
  9   8   7
 19   9   4
 37   9   2
 73   9   1
 11  10   9
 19  10   5
 23  11   5
 13  12  11
 23  12   6
 27  13   6
 27  14   7
 16  15  14
 31  15   7
...
 41  40  39
 79  40  20
 83  41  20
 43  42  41
 83  42  21
 47  46  45
 49  48  47
197  49  12
sage.combinat.designs.difference_family.singer_difference_set(q, d)

Return a difference set associated to the set of hyperplanes in a projective space of dimension d over GF(q).

A Singer difference set has parameters:

v = \frac{q^{d+1}-1}{q-1}, \quad
k = \frac{q^d-1}{q-1}, \quad
\lambda = \frac{q^{d-1}-1}{q-1}.

The idea of the construction is as follows. One consider the finite field GF(q^{d+1}) as a vector space of dimension d+1 over GF(q). The set of GF(q)-lines in GF(q^{d+1}) is a projective plane and its set of hyperplanes form a balanced incomplete block design.

Now, considering a multiplicative generator z of GF(q^{d+1}), we get a transitive action of a cyclic group on our projective plane from which it is possible to build a difference set.

The construction is given in details in [Stinson2004], section 3.3.

EXAMPLES:

sage: from sage.combinat.designs.difference_family import singer_difference_set, is_difference_family
sage: G,D = singer_difference_set(3,2)
sage: is_difference_family(G,D,verbose=True)
It is a (13,4,1)-difference family
True

sage: G,D = singer_difference_set(4,2)
sage: is_difference_family(G,D,verbose=True)
It is a (21,5,1)-difference family
True

sage: G,D = singer_difference_set(3,3)
sage: is_difference_family(G,D,verbose=True)
It is a (40,13,4)-difference family
True

sage: G,D = singer_difference_set(9,3)
sage: is_difference_family(G,D,verbose=True)
It is a (820,91,10)-difference family
True
sage.combinat.designs.difference_family.twin_prime_powers_difference_set(p, check=True)

Return a difference set on GF(p) \times GF(p+2).

The difference set is built from the following element of the cartesian product of finite fields GF(p) \times GF(p+2):

  • (x,0) with any x
  • (x,y) with x and y squares
  • (x,y) with x and y non-squares

For more information see Wikipedia article Difference_set.

INPUT:

  • check – boolean (default: True). If True then the result of the computation is checked before being returned. This should not be needed but ensures that the output is correct.

EXAMPLES:

sage: from sage.combinat.designs.difference_family import twin_prime_powers_difference_set
sage: G,D = twin_prime_powers_difference_set(3)
sage: G
The cartesian product of (Finite Field of size 3, Finite Field of size 5)
sage: D
[[(1, 1), (1, 4), (2, 2), (2, 3), (0, 0), (1, 0), (2, 0)]]