def __init__(self, q, name="a", modulus=None, repr="poly", cache=False):
"""
Initialize ``self``.
EXAMPLES::
sage: k.<a> = GF(2^3)
sage: j.<b> = GF(3^4)
sage: k == j
False
sage: GF(2^3,'a') == copy(GF(2^3,'a'))
True
sage: TestSuite(GF(2^3, 'a')).run()
"""
self._kwargs = {}
if repr not in ['int', 'log', 'poly']:
raise ValueError, "Unknown representation %s" % repr
q = Integer(q)
if q < 2:
raise ValueError, "q must be a prime power"
F = q.factor()
if len(F) > 1:
raise ValueError, "q must be a prime power"
p = F[0][0]
k = F[0][1]
if q >= 1 << 16:
raise ValueError, "q must be < 2^16"
import constructor
FiniteField.__init__(self,
constructor.FiniteField(p),
name,
normalize=False)
self._kwargs['repr'] = repr
self._kwargs['cache'] = cache
if modulus is None or modulus == 'conway':
if k == 1:
modulus = 'random' # this will use the gfq_factory_pk function.
elif ConwayPolynomials().has_polynomial(p, k):
from sage.rings.finite_rings.conway_polynomials import conway_polynomial
modulus = conway_polynomial(p, k)
elif modulus is None:
modulus = 'random'
else:
raise ValueError, "Conway polynomial not found"
from sage.rings.polynomial.all import is_Polynomial
if is_Polynomial(modulus):
modulus = modulus.list()
self._cache = Cache_givaro(self, p, k, modulus, repr, cache)
def __init__(self, q, name="a", modulus=None, repr="poly", cache=False):
"""
Initialize ``self``.
EXAMPLES::
sage: k.<a> = GF(2^3)
sage: j.<b> = GF(3^4)
sage: k == j
False
sage: GF(2^3,'a') == copy(GF(2^3,'a'))
True
sage: TestSuite(GF(2^3, 'a')).run()
"""
self._kwargs = {}
if repr not in ['int', 'log', 'poly']:
raise ValueError("Unknown representation %s"%repr)
q = Integer(q)
if q < 2:
raise ValueError("q must be a prime power")
F = q.factor()
if len(F) > 1:
raise ValueError("q must be a prime power")
p = F[0][0]
k = F[0][1]
if q >= 1<<16:
raise ValueError("q must be < 2^16")
import constructor
FiniteField.__init__(self, constructor.FiniteField(p), name, normalize=False)
self._kwargs['repr'] = repr
self._kwargs['cache'] = cache
if modulus is None or modulus == 'conway':
if k == 1:
modulus = 'random' # this will use the gfq_factory_pk function.
elif ConwayPolynomials().has_polynomial(p, k):
from sage.rings.finite_rings.conway_polynomials import conway_polynomial
modulus = conway_polynomial(p, k)
elif modulus is None:
modulus = 'random'
else:
raise ValueError("Conway polynomial not found")
from sage.rings.polynomial.all import is_Polynomial
if is_Polynomial(modulus):
modulus = modulus.list()
self._cache = Cache_givaro(self, p, k, modulus, repr, cache)
def q3_minus_one_matrix(K):
r"""
Return a companion matrix in `GL(3, K)` whose multiplicative order is `q^3 - 1`.
This function is used in :func:`HughesPlane`
EXAMPLES::
sage: from sage.combinat.designs.block_design import q3_minus_one_matrix
sage: m = q3_minus_one_matrix(GF(3))
sage: m.multiplicative_order() == 3**3 - 1
True
sage: m = q3_minus_one_matrix(GF(4,'a'))
sage: m.multiplicative_order() == 4**3 - 1
True
sage: m = q3_minus_one_matrix(GF(5))
sage: m.multiplicative_order() == 5**3 - 1
True
sage: m = q3_minus_one_matrix(GF(9,'a'))
sage: m.multiplicative_order() == 9**3 - 1
True
"""
q = K.cardinality()
M = MatrixSpace(K, 3)
if q.is_prime():
from sage.rings.finite_rings.conway_polynomials import conway_polynomial
try:
a, b, c, _ = conway_polynomial(q, 3)
except RuntimeError: # the polynomial is not in the database
pass
else:
return M([0, 0, -a, 1, 0, -b, 0, 1, -c])
m = M()
m[1, 0] = m[2, 1] = K.one()
while True:
m[0, 2] = K._random_nonzero_element()
m[1, 2] = K.random_element()
m[2, 2] = K.random_element()
if m.multiplicative_order() == q**3 - 1:
return m
def q3_minus_one_matrix(K):
r"""
Return a companion matrix in `GL(3, K)` whose multiplicative order is `q^3 - 1`.
This function is used in :func:`HughesPlane`
EXAMPLES::
sage: from sage.combinat.designs.block_design import q3_minus_one_matrix
sage: m = q3_minus_one_matrix(GF(3))
sage: m.multiplicative_order() == 3**3 - 1
True
sage: m = q3_minus_one_matrix(GF(4,'a'))
sage: m.multiplicative_order() == 4**3 - 1
True
sage: m = q3_minus_one_matrix(GF(5))
sage: m.multiplicative_order() == 5**3 - 1
True
sage: m = q3_minus_one_matrix(GF(9,'a'))
sage: m.multiplicative_order() == 9**3 - 1
True
"""
q = K.cardinality()
M = MatrixSpace(K, 3)
if q.is_prime():
from sage.rings.finite_rings.conway_polynomials import conway_polynomial
try:
a,b,c,_ = conway_polynomial(q, 3)
except RuntimeError: # the polynomial is not in the database
pass
else:
return M([0,0,-a,1,0,-b,0,1,-c])
m = M()
m[1,0] = m[2,1] = K.one()
while True:
m[0,2] = K._random_nonzero_element()
m[1,2] = K.random_element()
m[2,2] = K.random_element()
if m.multiplicative_order() == q**3 - 1:
return m
def random_q3_minus_one_matrix(K):
r"""
Return a companion matrix in `GL(3, K)` whose multiplicative order is `q^3 - 1`.
EXAMPLES::
sage: m = random_q3_minus_one_matrix(GF(3))
sage: m.multiplicative_order() == 3**3 - 1
True
sage: m = random_q3_minus_one_matrix(GF(4,'a'))
sage: m.multiplicative_order() == 4**3 - 1
True
sage: m = random_q3_minus_one_matrix(GF(5))
sage: m.multiplicative_order() == 5**3 - 1
True
sage: m = random_q3_minus_one_matrix(GF(9,'a'))
sage: m.multiplicative_order() == 9**3 - 1
True
"""
q = K.cardinality()
M = MatrixSpace(K, 3)
if q.is_prime():
from sage.rings.finite_rings.conway_polynomials import conway_polynomial
try:
a,b,c,_ = conway_polynomial(q, 3)
except RuntimeError: # the polynomial is not in the database
pass
else:
return M([0,0,-a,1,0,-b,0,1,-c])
while True:
a = K._random_nonzero_element()
b = K.random_element()
c = K.random_element()
m = M([0,0,-a,1,0,-b,0,1,-c])
if m.multiplicative_order() == q**3 - 1:
return m
def singer_difference_set(q,d):
r"""
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:
.. MATH::
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
"""
q = Integer(q)
assert q.is_prime_power()
assert d >= 2
from sage.rings.finite_rings.constructor import GF
from sage.rings.finite_rings.conway_polynomials import conway_polynomial
from sage.rings.finite_rings.integer_mod_ring import Zmod
# build a polynomial c over GF(q) such that GF(q)[x] / (c(x)) is a
# GF(q**(d+1)) and such that x is a multiplicative generator.
p,e = q.factor()[0]
c = conway_polynomial(p,e*(d+1))
if e != 1: # i.e. q is not a prime, so we factorize c over GF(q) and pick
# one of its factor
K = GF(q,'z')
c = c.change_ring(K).factor()[0][0]
else:
K = GF(q)
z = c.parent().gen()
# Now we consider the GF(q)-subspace V spanned by (1,z,z^2,...,z^(d-1)) inside
# GF(q^(d+1)). The multiplication by z is an automorphism of the
# GF(q)-projective space built from GF(q^(d+1)). The difference family is
# obtained by taking the integers i such that z^i belong to V.
powers = [0]
i = 1
x = z
k = (q**d-1)//(q-1)
while len(powers) < k:
if x.degree() <= (d-1):
powers.append(i)
x = (x*z).mod(c)
i += 1
return Zmod((q**(d+1)-1)//(q-1)), [powers]
def singer_difference_set(q, d):
r"""
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:
.. MATH::
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
"""
q = Integer(q)
assert q.is_prime_power()
assert d >= 2
from sage.rings.finite_rings.constructor import GF
from sage.rings.finite_rings.conway_polynomials import conway_polynomial
from sage.rings.finite_rings.integer_mod_ring import Zmod
# build a polynomial c over GF(q) such that GF(q)[x] / (c(x)) is a
# GF(q**(d+1)) and such that x is a multiplicative generator.
p, e = q.factor()[0]
c = conway_polynomial(p, e * (d + 1))
if e != 1: # i.e. q is not a prime, so we factorize c over GF(q) and pick
# one of its factor
K = GF(q, 'z')
c = c.change_ring(K).factor()[0][0]
else:
K = GF(q)
z = c.parent().gen()
# Now we consider the GF(q)-subspace V spanned by (1,z,z^2,...,z^(d-1)) inside
# GF(q^(d+1)). The multiplication by z is an automorphism of the
# GF(q)-projective space built from GF(q^(d+1)). The difference family is
# obtained by taking the integers i such that z^i belong to V.
powers = [0]
i = 1
x = z
k = (q**d - 1) // (q - 1)
while len(powers) < k:
if x.degree() <= (d - 1):
powers.append(i)
x = (x * z).mod(c)
i += 1
return Zmod((q**(d + 1) - 1) // (q - 1)), [powers]