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:
Check whether 

Test whether 

Compute the left stabilizer of the block 

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

Return a ( 

Return a triple 

Make a product of two Hadamard difference sets. 

Check whether 

Return a difference set. 

Given a subset 

Search for a radical difference family on 

Return a 

Return a difference set made of a cyclotomic coset in the finite field 

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

Return a difference set in either \(C_3 \times C_3 \times C_4\) or \(C_3 \times C_3 \times C_2 \times C_2\) with parameters \(v=36\), \(k=15\), \(\lambda=6\). 

Return a difference set on \(GF(p) \times GF(p+2)\). 
REFERENCES:
 BJL991
T. Beth, D. Jungnickel, H. Lenz “Design theory Vol. I.” Second edition. Encyclopedia of Mathematics and its Applications, 69. Cambridge University Press, (1999).
 BLJ992
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, 9 (1939), 353–399.
 Bu95(1,2)
M. Buratti “On simple radical difference families”, J. Combinatorial Designs, 3 (1995) 161–168.
 Tu1965
R. J. Turyn “Character sum and difference sets” Pacific J. Math. 15 (1965) 319–346.
 Tu1984
R. J. Turyn “A special class of Williamson matrices and difference sets” J. Combinatorial Theory (A) 36 (1984) 111–115.
 Wi72(1,2)
R. M. Wilson “Cyclotomy and difference families in elementary Abelian groups”, J. Number Theory, 4 (1972) 17–47.
Functions¶

sage.combinat.designs.difference_family.
are_hadamard_difference_set_parameters
(v, k, lmbda)¶ Check whether
(v,k,lmbda)
is of the form(4N^2, 2N^2  N, N^2  N)
.INPUT:
(v,k,lmbda)
– parameters of a difference set
EXAMPLES:
sage: from sage.combinat.designs.difference_family import are_hadamard_difference_set_parameters sage: are_hadamard_difference_set_parameters(36, 15, 6) True sage: are_hadamard_difference_set_parameters(60, 13, 5) False

sage.combinat.designs.difference_family.
are_mcfarland_1973_parameters
(v, k, lmbda, return_parameters=False)¶ Test whether
(v,k,lmbda)
is a triple that can be obtained from the construction from [McF1973].See
mcfarland_1973_construction()
.INPUT:
v
,k
,lmbda
 (integers) parameters of the difference familyreturn_parameters
– (boolean, defaultFalse
) ifTrue
return a pair(True, (q, s))
so that(q,s)
can be used in the functionmcfarland_1973_construction()
to actually build a(v,k,lmbda)
difference family. Or(False, None)
if the construction is not possible.
EXAMPLES:
sage: from sage.combinat.designs.difference_family import are_mcfarland_1973_parameters sage: are_mcfarland_1973_parameters(64, 28, 12) True sage: are_mcfarland_1973_parameters(64, 28, 12, return_parameters=True) (True, (2, 2)) sage: are_mcfarland_1973_parameters(60, 13, 5) False sage: are_mcfarland_1973_parameters(98125, 19500, 3875) True sage: are_mcfarland_1973_parameters(98125, 19500, 3875, True) (True, (5, 3)) sage: from sage.combinat.designs.difference_family import are_mcfarland_1973_parameters sage: for v in range(1, 100): ....: for k in range(1,30): ....: for l in range(1,15): ....: if are_mcfarland_1973_parameters(v,k,l): ....: answer, (q,s) = are_mcfarland_1973_parameters(v,k,l,return_parameters=True) ....: print("{} {} {} {} {}".format(v,k,l,q,s)) ....: assert answer is True ....: assert designs.difference_family(v,k,l,existence=True) is True ....: G,D = designs.difference_family(v,k,l) 16 6 2 2 1 45 12 3 3 1 64 28 12 2 2 96 20 4 4 1

sage.combinat.designs.difference_family.
block_stabilizer
(G, B)¶ Compute the left stabilizer of the block
B
under the action ofG
.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 ofG
.
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 = list(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 range(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) is True] sage: 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)
Todo
Do improvements due to Zhen and Wu 1999.

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 cardinalityv
.Let \(G\) be a finite Abelian group. For a given subset \(D\) of \(G\), we define \(\Delta D\) to be the multiset 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 multiset, contains each element of \(G \backslash \{0\}\) exactly \(\lambda\)times.
When there is only one block, i.e. \(\lambda(v  1) = k(k1)\), 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 momentNotImplementedError
is raised.INPUT:
v,k,l
– parameters of the difference family. Ifl
is not provided it is assumed to be1
.existence
– ifTrue
, then return eitherTrue
if Sage knows how to build such design,Unknown
if it does not andFalse
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
). IfTrue
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, ifexistence
isTrue
a troolean or ifexplain_construction
isTrue
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 Ring of integers modulo 15 sage: D [[0, 1, 2, 4, 5, 8, 10]] sage: print(designs.difference_family(15,7,3,explain_construction=True)) Singer difference set sage: print(designs.difference_family(91,10,1,explain_construction=True)) Singer difference set sage: print(designs.difference_family(64,28,12, explain_construction=True)) McFarland 1973 construction sage: print(designs.difference_family(576, 276, 132, explain_construction=True)) Hadamard difference set product from N1=2 and N2=3
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), int(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), int(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 range(2,100): ....: constructions = [] ....: for k in range(2,10): ....: for l in range(1,10): ....: ret = designs.difference_family(v,k,l,existence=True) ....: if ret is 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), (7,1) 97: (2,1), (3,1), (3,2), (4,1), (4,3), (6,5), (8,7), (9,3)
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 groupG
.EXAMPLES:
sage: from sage.combinat.designs.difference_family import group_law sage: group_law(Zmod(3)) (0, <builtin function add>, <builtin function neg>) sage: group_law(SymmetricGroup(5)) ((), <builtin function mul>, <builtin function inv>) sage: group_law(VectorSpace(QQ,3)) ((0, 0, 0), <builtin function add>, <builtin function neg>)

sage.combinat.designs.difference_family.
hadamard_difference_set_product
(G1, D1, G2, D2)¶ Make a product of two Hadamard difference sets.
This product construction appears in [Tu1984].
INPUT:
G1,D1
,G2,D2
– two Hadamard difference sets
EXAMPLES:
sage: from sage.combinat.designs.difference_family import hadamard_difference_set_product sage: from sage.combinat.designs.difference_family import is_difference_family sage: G1,D1 = designs.difference_family(16,6,2) sage: G2,D2 = designs.difference_family(36,15,6) sage: G11,D11 = hadamard_difference_set_product(G1,D1,G1,D1) sage: assert is_difference_family(G11, D11, 256, 120, 56) sage: assert designs.difference_family(256, 120, 56, existence=True) is True sage: G12,D12 = hadamard_difference_set_product(G1,D1,G2,D2) sage: assert is_difference_family(G12, D12, 576, 276, 132) sage: assert designs.difference_family(576, 276, 132, existence=True) is True

sage.combinat.designs.difference_family.
hadamard_difference_set_product_parameters
(N)¶ Check whether a product construction is available for Hadamard difference set with parameter
N
.This function looks for two integers \(N_1\) and \(N_2\) greater than \(1\) and so that \(N = 2 N_1 N_2\) and there exists Hadamard difference set with parameters \((4 N_i^2, 2N_i^2  N_i, N_i^2  N_i)\). If such pair exists, the output is the pair
(N_1, N_2)
otherwise it isNone
.INPUT:
N
– positive integer
EXAMPLES:
sage: from sage.combinat.designs.difference_family import hadamard_difference_set_product_parameters sage: hadamard_difference_set_product_parameters(8) (2, 2)

sage.combinat.designs.difference_family.
is_difference_family
(G, D, v=None, k=None, l=None, verbose=False)¶ Check whether
D
forms a difference family in the groupG
.INPUT:
G
– group of cardinalityv
D
– a set ofk
subsets ofG
v
,k
andl
– optional parameters of the difference familyverbose
 whether to print additional information
See also
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, abd, b+c+d], ....: [c, a+bd, b+c, ab+d, a+b+c], ....: [ab+c+d, abcd, a+cd, bc+d, a+b], ....: [bd, a+b+d, ab+cd, ab+c, b+c+d]] sage: is_difference_family(G, D) True
The following example has a third block with a nontrivial 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)

sage.combinat.designs.difference_family.
mcfarland_1973_construction
(q, s)¶ Return a difference set.
The difference set returned has the following parameters
\[v = \frac{q^{s+1}(q^{s+1}+q2)}{q1}, k = \frac{q^s (q^{s+1}1)}{q1}, \lambda = \frac{q^s(q^s1)}{q1}\]This construction is due to [McF1973].
INPUT:
q
,s
 (integers) parameters for the difference set (see the above formulas for the expression ofv
,k
,l
in terms ofq
ands
)
See also
The function
are_mcfarland_1973_parameters()
makes the translation between the parameters \((q,s)\) corresponding to a given triple \((v,k,\lambda)\).REFERENCES:
 McF1973(1,2,3)
Robert L. McFarland “A family of difference sets in noncyclic groups” J. Combinatorial Theory (A) 15 (1973) 1–10. doi:10.1016/00973165(73)900319
EXAMPLES:
sage: from sage.combinat.designs.difference_family import ( ....: mcfarland_1973_construction, is_difference_family) sage: G,D = mcfarland_1973_construction(3, 1) sage: assert is_difference_family(G, D, 45, 12, 3) sage: G,D = mcfarland_1973_construction(2, 2) sage: assert is_difference_family(G, D, 64, 28, 12)

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 (ix)%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 \(k(k1)\) divides \(q1\).
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 \(B=\{x:x^{k1}=1\}\cup \{0\}\) (for \(k\) even). Equivalently, \(K^*\) decomposes as:
\[K^* = \Delta (x_1 B) \cup \cdots \cup \Delta (x_k B) = x_1 \Delta B \cup \cdots \cup x_k \Delta B.\]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 \(C\)).
\[\begin{split}\begin{array}{llll} (k\text{ odd} ) & \Delta B &= \{r^ir^j:r^i\neq r^j\} &= \pm C\cdot \{r^i1: 0 < i \leq m\}\\ (k\text{ even}) & \Delta B &= \{r^ir^j:r^i\neq r^j\}\cup C &= \pm C\cdot \{r^i1: 0 < i < m\}\cup \pm C \end{array}\end{split}\]where
\[(k\text{ odd})\ m = (k1)/2 \quad \text{and} \quad (k\text{ even})\ m = k/2.\]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 \(q = t k(k1) + 1\). 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 \(k1\)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 fieldk
– positive integer, the size of the blocksl
– the \(\lambda\) parameter (default to \(1\))existence
– ifTrue
, then return eitherTrue
if Sage knows how to build such design,Unknown
if it does not andFalse
if it knows that the design does not exist.check
– boolean (default:True
). IfTrue
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)) ....: list_q = [] ....: for q in range(k*(k1)+1, 2000, k*(k1)): ....: if is_prime_power(q): ....: K = GF(q,'a') ....: if radical_difference_family(K, k, existence=True) is True: ....: list_q.append(q) ....: _ = radical_difference_family(K,k) ....: print(" ".join(str(p) for p in list_q)) 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 parametersk
andl
.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(xy 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(xy 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*(k1))): ....: v = k*(k1)//l + 1 ....: if is_prime_power(v) and radical_difference_set(GF(v,'a'),k,l,existence=True) is 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:
\[v = \frac{q^{d+1}1}{q1}, \quad k = \frac{q^d1}{q1}, \quad \lambda = \frac{q^{d1}1}{q1}.\]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.
turyn_1965_3x3xK
(k=4)¶ Return a difference set in either \(C_3 \times C_3 \times C_4\) or \(C_3 \times C_3 \times C_2 \times C_2\) with parameters \(v=36\), \(k=15\), \(\lambda=6\).
This example appears in [Tu1965].
INPUT:
k
– either2
(to get a difference set in \(C_3 \times C_3 \times C_2 \times C_2\)) or4
(to get a difference set in \(C_3 \times C_3 \times C_3 \times C_4\)).
EXAMPLES:
sage: from sage.combinat.designs.difference_family import turyn_1965_3x3xK sage: from sage.combinat.designs.difference_family import is_difference_family sage: G,D = turyn_1965_3x3xK(4) sage: assert is_difference_family(G, D, 36, 15, 6) sage: G,D = turyn_1965_3x3xK(2) sage: assert is_difference_family(G, D, 36, 15, 6)

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 \(y\) nonsquares
For more information see Wikipedia article Difference_set.
INPUT:
check
– boolean (default:True
). IfTrue
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)]]