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 = [a for a in G if a - F.one() in sG]
B = [b for b in G if b + F.one() in sG]
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 find_traces(k, r, l, rl = None):
'''
Compute traces such that the charpoly of the Frobenius has a root of order n
modulo l.
INPUT:
- ``k`` -- the base field.
- ``r`` -- the degree of the extension.
- ``l`` -- a prime number.
EXAMPLES::
sage: from ellrains import find_traces
sage: r = 281
sage: l = 3373
sage: k = GF(1747)
sage: find_traces(k, r, l)
[4, 14, 18, 43, 57, 3325, 3337, 3348, 3354, 3357, 3364]
'''
L = GF(l)
q = L(k.order())
bound = 4*k.characteristic()
if rl is None:
rl = ZZ((l-1)/r)
# Primitive root of unity of order r
zeta = L.multiplicative_generator()**rl
# Candidate eigenvalue
lmbd = L(1)
# Traces
T = []
for i in xrange(1, r):
lmbd *= zeta
# Ensure that the order is r
if r.gcd(i) != 1:
continue
if not lmbd.multiplicative_order() == r:
print l, r, i
trc = lmbd + q/lmbd
# Check Hasse bound
if trc.lift_centered()**2 < bound:
T.append(trc)
return T
def H_value(self, p, f, t, ring=None):
"""
Return the trace of the Frobenius, computed in terms of Gauss sums
using the hypergeometric trace formula.
INPUT:
- `p` -- a prime number
- `f` -- an integer such that `q = p^f`
- `t` -- a rational parameter
- ``ring`` -- optional (default ``UniversalCyclotomicfield``)
The ring could be also ``ComplexField(n)`` or ``QQbar``.
OUTPUT:
an integer
.. WARNING::
This is apparently working correctly as can be tested
using ComplexField(70) as value ring.
Using instead UniversalCyclotomicfield, this is much
slower than the `p`-adic version :meth:`padic_H_value`.
EXAMPLES:
With values in the UniversalCyclotomicField (slow)::
sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/2]*4,[0]*4))
sage: [H.H_value(3,i,-1) for i in range(1,3)]
[0, -12]
sage: [H.H_value(5,i,-1) for i in range(1,3)]
[-4, 276]
sage: [H.H_value(7,i,-1) for i in range(1,3)] # not tested
[0, -476]
sage: [H.H_value(11,i,-1) for i in range(1,3)] # not tested
[0, -4972]
sage: [H.H_value(13,i,-1) for i in range(1,3)] # not tested
[-84, -1420]
With values in ComplexField::
sage: [H.H_value(5,i,-1, ComplexField(60)) for i in range(1,3)]
[-4, 276]
Check issue from :trac:`28404`::
sage: H1 = Hyp(cyclotomic=([1,1,1],[6,2]))
sage: H2 = Hyp(cyclotomic=([6,2],[1,1,1]))
sage: [H1.H_value(5,1,i) for i in range(2,5)]
[1, -4, -4]
sage: [H2.H_value(5,1,QQ(i)) for i in range(2,5)]
[-4, 1, -4]
REFERENCES:
- [BeCoMe]_ (Theorem 1.3)
- [Benasque2009]_
"""
alpha = self._alpha
beta = self._beta
t = QQ(t)
if 0 in alpha:
return self._swap.H_value(p, f, ~t, ring)
if ring is None:
ring = UniversalCyclotomicField()
gamma = self.gamma_array()
q = p ** f
m = {r: beta.count(QQ((r, q - 1))) for r in range(q - 1)}
D = -min(self.zigzag(x, flip_beta=True) for x in alpha + beta)
# also: D = (self.weight() + 1 - m[0]) // 2
M = self.M_value()
Fq = GF(q)
gen = Fq.multiplicative_generator()
zeta_q = ring.zeta(q - 1)
tM = Fq(M / t)
for k in range(q - 1):
if gen ** k == tM:
teich = zeta_q ** k
break
gauss_table = [gauss_sum(zeta_q ** r, Fq) for r in range(q - 1)]
sigma = sum(q**(D + m[0] - m[r]) *
prod(gauss_table[(-v * r) % (q - 1)]**gv
for v, gv in gamma.items()) *
teich ** r
for r in range(q - 1))
resu = ZZ(-1) ** m[0] / (1 - q) * sigma
if not ring.is_exact():
resu = resu.real_part().round()
return resu
K1MASK = 2**m1 - 1
K1HIGHBIT = 2**(m1-1)
K0MASK = 2**m0 - 1
K0HIGHBIT = 2**(m0-1)
NECKLACES = 1/m1*sum(euler_phi(d)*2**(ZZ(m1/d)) for d in m1.divisors())
# 9*2*2*2 > 64
if m2 > 8:
USE_DWORD = 1
else:
USE_DWORD = 0
GF2 = GF(2)
x = polygen(GF2)
K2 = GF(2**m2, name='g2', modulus=modulus)
g2 = K2.gen()
z2 = K2.multiplicative_generator()
K1 = GF(2**m1, name='g1', modulus=modulus)
g1 = K1.gen()
z1 = K1.multiplicative_generator()
Z1 = z1.polynomial().change_ring(ZZ)(2)
K0 = GF(2**m0, name='g0', modulus=modulus)
g0 = K0.gen()
z0 = K0.multiplicative_generator()
L = [K0, K1, K2]
moduli = [k.modulus() for k in L]
F = [f.change_ring(ZZ)(2) for f in moduli]
weights = [f.hamming_weight()-2 for f in moduli]
degrees = [(f-x**f.degree()-1).exponents() for f in moduli]
FWEIGHTS = weights
F0D = degrees[0]
if F0D == [1]:
def H_value(self, p, f, t, ring=None):
"""
Return the trace of the Frobenius, computed in terms of Gauss sums
using the hypergeometric trace formula.
INPUT:
- `p` -- a prime number
- `f` -- an integer such that `q = p^f`
- `t` -- a rational parameter
- ``ring`` -- optional (default ``UniversalCyclotomicfield``)
The ring could be also ``ComplexField(n)`` or ``QQbar``.
OUTPUT:
an integer
.. WARNING::
This is apparently working correctly as can be tested
using ComplexField(70) as value ring.
Using instead UniversalCyclotomicfield, this is much
slower than the `p`-adic version :meth:`padic_H_value`.
EXAMPLES:
With values in the UniversalCyclotomicField (slow)::
sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/2]*4,[0]*4))
sage: [H.H_value(3,i,-1) for i in range(1,3)]
[0, -12]
sage: [H.H_value(5,i,-1) for i in range(1,3)]
[-4, 276]
sage: [H.H_value(7,i,-1) for i in range(1,3)] # not tested
[0, -476]
sage: [H.H_value(11,i,-1) for i in range(1,3)] # not tested
[0, -4972]
sage: [H.H_value(13,i,-1) for i in range(1,3)] # not tested
[-84, -1420]
With values in ComplexField::
sage: [H.H_value(5,i,-1, ComplexField(60)) for i in range(1,3)]
[-4, 276]
REFERENCES:
- [BeCoMe]_ (Theorem 1.3)
- [Benasque2009]_
"""
alpha = self._alpha
beta = self._beta
if 0 in alpha:
H = self.swap_alpha_beta()
return(H.H_value(p, f, ~t, ring))
if ring is None:
ring = UniversalCyclotomicField()
t = QQ(t)
gamma = self.gamma_array()
q = p ** f
m = {r: beta.count(QQ((r, q - 1))) for r in range(q - 1)}
D = -min(self.zigzag(x, flip_beta=True) for x in alpha + beta)
# also: D = (self.weight() + 1 - m[0]) // 2
M = self.M_value()
Fq = GF(q)
gen = Fq.multiplicative_generator()
zeta_q = ring.zeta(q - 1)
tM = Fq(M / t)
for k in range(q - 1):
if gen ** k == tM:
teich = zeta_q ** k
break
gauss_table = [gauss_sum(zeta_q ** r, Fq) for r in range(q - 1)]
sigma = sum(q**(D + m[0] - m[r]) *
prod(gauss_table[(-v * r) % (q - 1)] ** gv
for v, gv in gamma.items()) *
teich ** r
for r in range(q - 1))
resu = ZZ(-1) ** m[0] / (1 - q) * sigma
if not ring.is_exact():
resu = resu.real_part().round()
return resu
