def szekeres_difference_set_pair(m, check=True):
r"""
Construct Szekeres `(2m+1,m,1)`-cyclic difference family
Let `4m+3` be a prime power. Theorem 3 in [Sz69]_ contains a construction of a pair
of *complementary difference sets* `A`, `B` in the subgroup `G` of the quadratic
residues in `F_{4m+3}^*`. Namely `|A|=|B|=m`, `a\in A` whenever `a-1\in G`, `b\in B`
whenever `b+1 \in G`. See also Theorem 2.6 in [SWW72]_ (there the formula for `B` is
correct, as opposed to (4.2) in [Sz69]_, where the sign before `1` is wrong.
In modern terminology, for `m>1` the sets `A` and `B` form a
:func:`difference family<sage.combinat.designs.difference_family>` with parameters `(2m+1,m,1)`.
I.e. each non-identity `g \in G` can be expressed uniquely as `xy^{-1}` for `x,y \in A` or `x,y \in B`.
Other, specific to this construction, properties of `A` and `B` are: for `a` in `A` one has
`a^{-1}` not in `A`, whereas for `b` in `B` one has `b^{-1}` in `B`.
INPUT:
- ``m`` (integer) -- dimension of the matrix
- ``check`` (default: ``True``) -- whether to check `A` and `B` for correctness
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import szekeres_difference_set_pair
sage: G,A,B=szekeres_difference_set_pair(6)
sage: G,A,B=szekeres_difference_set_pair(7)
REFERENCE:
.. [Sz69] \G. Szekeres,
Tournaments and Hadamard matrices,
Enseignement Math. (2) 15(1969), 269-278
"""
from sage.rings.finite_rings.finite_field_constructor import GF
F = GF(4 * m + 3)
t = F.multiplicative_generator()**2
G = F.cyclotomic_cosets(t, cosets=[F.one()])[0]
sG = set(G)
A = filter(lambda a: a - F.one() in sG, G)
B = filter(lambda b: b + F.one() in sG, G)
if check:
from itertools import product, chain
assert (len(A) == len(B) == m)
if m > 1:
assert (sG == set(
[xy[0] / xy[1] for xy in chain(product(A, A), product(B, B))]))
assert (all(F.one() / b + F.one() in sG for b in B))
assert (not any(F.one() / a - F.one() in sG for a in A))
return G, A, B
def szekeres_difference_set_pair(m, check=True):
r"""
Construct Szekeres `(2m+1,m,1)`-cyclic difference family
Let `4m+3` be a prime power. Theorem 3 in [Sz69]_ contains a construction of a pair
of *complementary difference sets* `A`, `B` in the subgroup `G` of the quadratic
residues in `F_{4m+3}^*`. Namely `|A|=|B|=m`, `a\in A` whenever `a-1\in G`, `b\in B`
whenever `b+1 \in G`. See also Theorem 2.6 in [SWW72]_ (there the formula for `B` is
correct, as opposed to (4.2) in [Sz69]_, where the sign before `1` is wrong.
In modern terminology, for `m>1` the sets `A` and `B` form a
:func:`difference family<sage.combinat.designs.difference_family>` with parameters `(2m+1,m,1)`.
I.e. each non-identity `g \in G` can be expressed uniquely as `xy^{-1}` for `x,y \in A` or `x,y \in B`.
Other, specific to this construction, properties of `A` and `B` are: for `a` in `A` one has
`a^{-1}` not in `A`, whereas for `b` in `B` one has `b^{-1}` in `B`.
INPUT:
- ``m`` (integer) -- dimension of the matrix
- ``check`` (default: ``True``) -- whether to check `A` and `B` for correctness
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import szekeres_difference_set_pair
sage: G,A,B=szekeres_difference_set_pair(6)
sage: G,A,B=szekeres_difference_set_pair(7)
REFERENCE:
.. [Sz69] \G. Szekeres,
Tournaments and Hadamard matrices,
Enseignement Math. (2) 15(1969), 269-278
"""
from sage.rings.finite_rings.finite_field_constructor import GF
F = GF(4*m+3)
t = F.multiplicative_generator()**2
G = F.cyclotomic_cosets(t, cosets=[F.one()])[0]
sG = set(G)
A = filter(lambda a: a-F.one() in sG, G)
B = filter(lambda b: b+F.one() in sG, G)
if check:
from itertools import product, chain
assert(len(A)==len(B)==m)
if m>1:
assert(sG==set([xy[0]/xy[1] for xy in chain(product(A,A), product(B,B))]))
assert(all(F.one()/b+F.one() in sG for b in B))
assert(not any(F.one()/a-F.one() in sG for a in A))
return G,A,B
