def HammingCode(r,F):
r"""
Implements the Hamming codes.
The `r^{th}` Hamming code over `F=GF(q)` is an
`[n,k,d]` code with length `n=(q^r-1)/(q-1)`,
dimension `k=(q^r-1)/(q-1) - r` and minimum distance
`d=3`. The parity check matrix of a Hamming code has rows
consisting of all nonzero vectors of length r in its columns,
modulo a scalar factor so no parallel columns arise. A Hamming code
is a single error-correcting code.
INPUT:
- ``r`` - an integer 2
- ``F`` - a finite field.
OUTPUT: Returns the r-th q-ary Hamming code.
EXAMPLES::
sage: HammingCode(3,GF(2))
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C = HammingCode(3,GF(3)); C
Linear code of length 13, dimension 10 over Finite Field of size 3
sage: C.minimum_distance()
3
sage: C.minimum_distance(algorithm='gap') # long time, check d=3
3
sage: C = HammingCode(3,GF(4,'a')); C
Linear code of length 21, dimension 18 over Finite Field in a of size 2^2
"""
q = F.order()
n = (q**r-1)/(q-1)
k = n-r
MS = MatrixSpace(F,n,r)
X = ProjectiveSpace(r-1,F)
PFn = [list(p) for p in X.point_set(F).points(F)]
H = MS(PFn).transpose()
Cd = LinearCode(H)
# Hamming code always has distance 3, so we provide the distance.
return LinearCode(Cd.dual_code().gen_mat(), d=3)
def HammingCode(r, F):
r"""
Implements the Hamming codes.
The `r^{th}` Hamming code over `F=GF(q)` is an
`[n,k,d]` code with length `n=(q^r-1)/(q-1)`,
dimension `k=(q^r-1)/(q-1) - r` and minimum distance
`d=3`. The parity check matrix of a Hamming code has rows
consisting of all nonzero vectors of length r in its columns,
modulo a scalar factor so no parallel columns arise. A Hamming code
is a single error-correcting code.
INPUT:
- ``r`` - an integer 2
- ``F`` - a finite field.
OUTPUT: Returns the r-th q-ary Hamming code.
EXAMPLES::
sage: HammingCode(3,GF(2))
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C = HammingCode(3,GF(3)); C
Linear code of length 13, dimension 10 over Finite Field of size 3
sage: C.minimum_distance()
3
sage: C.minimum_distance(algorithm='gap') # long time, check d=3
3
sage: C = HammingCode(3,GF(4,'a')); C
Linear code of length 21, dimension 18 over Finite Field in a of size 2^2
"""
q = F.order()
n = (q**r - 1) / (q - 1)
k = n - r
MS = MatrixSpace(F, n, r)
X = ProjectiveSpace(r - 1, F)
PFn = [list(p) for p in X.point_set(F).points(F)]
H = MS(PFn).transpose()
Cd = LinearCode(H)
# Hamming code always has distance 3, so we provide the distance.
return LinearCode(Cd.dual_code().gen_mat(), d=3)
def projective_embedding(self, i=None, PP=None):
"""
Returns a morphism from this space into an ambient projective space
of the same dimension.
INPUT:
- ``i`` - integer (default: dimension of self = last
coordinate) determines which projective embedding to compute. The
embedding is that which has a 1 in the i-th coordinate, numbered
from 0.
- ``PP`` - (default: None) ambient projective space, i.e.,
codomain of morphism; this is constructed if it is not
given.
EXAMPLES::
sage: AA = AffineSpace(2, QQ, 'x')
sage: pi = AA.projective_embedding(0); pi
Scheme morphism:
From: Affine Space of dimension 2 over Rational Field
To: Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x0, x1) to
(1 : x0 : x1)
sage: z = AA(3,4)
sage: pi(z)
(1/4 : 3/4 : 1)
sage: pi(AA(0,2))
(1/2 : 0 : 1)
sage: pi = AA.projective_embedding(1); pi
Scheme morphism:
From: Affine Space of dimension 2 over Rational Field
To: Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x0, x1) to
(x0 : 1 : x1)
sage: pi(z)
(3/4 : 1/4 : 1)
sage: pi = AA.projective_embedding(2)
sage: pi(z)
(3 : 4 : 1)
"""
n = self.dimension_relative()
if i is None:
try:
i = self._default_embedding_index
except AttributeError:
i = int(n)
else:
i = int(i)
try:
return self.__projective_embedding[i]
except AttributeError:
self.__projective_embedding = {}
except KeyError:
pass
if PP is None:
from sage.schemes.generic.projective_space import ProjectiveSpace
PP = ProjectiveSpace(n, self.base_ring())
R = self.coordinate_ring()
v = list(R.gens())
if n < 0 or n >self.dimension_relative():
raise ValueError, \
"Argument i (=%s) must be between 0 and %s, inclusive"%(i,n)
v.insert(i, R(1))
phi = self.hom(v, PP)
self.__projective_embedding[i] = phi
return phi
def parametrization(self, point=None, morphism=True):
r"""
Return a parametrization `f` of ``self`` together with the
inverse of `f`.
If ``point`` is specified, then that point is used
for the parametrization. Otherwise, use ``self.rational_point()``
to find a point.
If ``morphism`` is True, then `f` is returned in the form
of a Scheme morphism. Otherwise, it is a tuple of polynomials
that gives the parametrization.
EXAMPLES:
An example over a finite field ::
sage: c = Conic(GF(2), [1,1,1,1,1,0])
sage: c.parametrization()
(Scheme morphism:
From: Projective Space of dimension 1 over Finite Field of size 2
To: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y
+ y^2 + x*z + y*z
Defn: Defined on coordinates by sending (x : y) to
(x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),
Scheme morphism:
From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y
+ y^2 + x*z + y*z
To: Projective Space of dimension 1 over Finite Field of size 2
Defn: Defined on coordinates by sending (x : y : z) to
(y : x))
An example with ``morphism = False`` ::
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism = False); (p, i)
([-2*x*y, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y
A ``ValueError`` is raised if ``self`` has no rational point ::
sage: C = Conic(x^2 + y^2 + 7*z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + 7*z^2 has no rational points over Rational Field!
A ``ValueError`` is raised if ``self`` is not smooth ::
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
"""
if (not self._parametrization is None) and not point:
par = self._parametrization
else:
if not self.is_smooth():
raise ValueError, "The conic self (=%s) is not smooth, hence does not have a parametrization." % self
if point == None:
point = self.rational_point()
point = Sequence(point)
B = self.base_ring()
Q = PolynomialRing(B, 'x,y')
[x, y] = Q.gens()
gens = self.ambient_space().gens()
P = PolynomialRing(B, 4, ['X', 'Y', 'T0', 'T1'])
[X, Y, T0, T1] = P.gens()
c3 = [j for j in range(2,-1,-1) if point[j] != 0][0]
c1 = [j for j in range(3) if j != c3][0]
c2 = [j for j in range(3) if j != c3 and j != c1][0]
L = [0,0,0]
L[c1] = Y*T1*point[c1] + Y*T0
L[c2] = Y*T1*point[c2] + X*T0
L[c3] = Y*T1*point[c3]
bezout = P(self.defining_polynomial()(L) / T0)
t = [bezout([x,y,0,-1]),bezout([x,y,1,0])]
par = (tuple([Q(p([x,y,t[0],t[1]])/y) for p in L]),
tuple([gens[m]*point[c3]-gens[c3]*point[m]
for m in [c2,c1]]))
if self._parametrization is None:
self._parametrization = par
if not morphism:
return par
P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
return P1.hom(par[0],self), self.Hom(P1)(par[1], check = False)
def parametrization(self, point=None, morphism=True):
r"""
Return a parametrization `f` of ``self`` together with the
inverse of `f`.
If ``point`` is specified, then that point is used
for the parametrization. Otherwise, use ``self.rational_point()``
to find a point.
If ``morphism`` is True, then `f` is returned in the form
of a Scheme morphism. Otherwise, it is a tuple of polynomials
that gives the parametrization.
EXAMPLES:
An example over a finite field ::
sage: c = Conic(GF(2), [1,1,1,1,1,0])
sage: c.parametrization()
(Scheme morphism:
From: Projective Space of dimension 1 over Finite Field of size 2
To: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y
+ y^2 + x*z + y*z
Defn: Defined on coordinates by sending (x : y) to
(x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),
Scheme morphism:
From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y
+ y^2 + x*z + y*z
To: Projective Space of dimension 1 over Finite Field of size 2
Defn: Defined on coordinates by sending (x : y : z) to
(y : x))
An example with ``morphism = False`` ::
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism = False); (p, i)
([-2*x*y, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y
A ``ValueError`` is raised if ``self`` has no rational point ::
sage: C = Conic(x^2 + y^2 + 7*z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + 7*z^2 has no rational points over Rational Field!
A ``ValueError`` is raised if ``self`` is not smooth ::
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
"""
if (not self._parametrization is None) and not point:
par = self._parametrization
else:
if not self.is_smooth():
raise ValueError, "The conic self (=%s) is not smooth, hence does not have a parametrization." % self
if point == None:
point = self.rational_point()
point = Sequence(point)
B = self.base_ring()
Q = PolynomialRing(B, 'x,y')
[x, y] = Q.gens()
gens = self.ambient_space().gens()
P = PolynomialRing(B, 4, ['X', 'Y', 'T0', 'T1'])
[X, Y, T0, T1] = P.gens()
c3 = [j for j in range(2, -1, -1) if point[j] != 0][0]
c1 = [j for j in range(3) if j != c3][0]
c2 = [j for j in range(3) if j != c3 and j != c1][0]
L = [0, 0, 0]
L[c1] = Y * T1 * point[c1] + Y * T0
L[c2] = Y * T1 * point[c2] + X * T0
L[c3] = Y * T1 * point[c3]
bezout = P(self.defining_polynomial()(L) / T0)
t = [bezout([x, y, 0, -1]), bezout([x, y, 1, 0])]
par = (tuple([Q(p([x, y, t[0], t[1]]) / y) for p in L]),
tuple([
gens[m] * point[c3] - gens[c3] * point[m]
for m in [c2, c1]
]))
if self._parametrization is None:
self._parametrization = par
if not morphism:
return par
P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
return P1.hom(par[0], self), self.Hom(P1)(par[1], check=False)
def parametrization(self, point=None, morphism=True):
r"""
Return a parametrization `f` of ``self`` together with the
inverse of `f`.
If ``point`` is specified, then that point is used
for the parametrization. Otherwise, use ``self.rational_point()``
to find a point.
If ``morphism`` is True, then `f` is returned in the form
of a Scheme morphism. Otherwise, it is a tuple of polynomials
that gives the parametrization.
ALGORITHM:
Uses Denis Simon's pari script Qfparam.
See ``sage.quadratic_forms.qfsolve.qfparam``.
EXAMPLES ::
sage: c = Conic([1,1,-1])
sage: c.parametrization()
(Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y) to
(2*x*y : x^2 - y^2 : x^2 + y^2),
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
To: Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*x : -1/2*y + 1/2*z))
An example with ``morphism = False`` ::
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism = False); (p, i)
([-2*x*y, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y
A ``ValueError`` is raised if ``self`` has no rational point ::
sage: C = Conic(x^2 + 2*y^2 + z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!
A ``ValueError`` is raised if ``self`` is not smooth ::
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
"""
if (not self._parametrization is None) and not point:
par = self._parametrization
else:
if not self.is_smooth():
raise ValueError, "The conic self (=%s) is not smooth, hence does not have a parametrization." % self
if point == None:
point = self.rational_point()
point = Sequence(point)
Q = PolynomialRing(QQ, 'x,y')
[x, y] = Q.gens()
gens = self.ambient_space().gens()
M = self.symmetric_matrix()
M *= lcm([t.denominator() for t in M.list()])
par1 = qfparam(M, point)
B = Matrix([[par1[i][j] for j in range(3)] for i in range(3)])
# self is in the image of B and does not lie on a line,
# hence B is invertible
A = B.inverse()
par2 = [sum([A[i, j] * gens[j] for j in range(3)]) for i in [1, 0]]
par = ([Q(pol(x / y) * y**2) for pol in par1], par2)
if self._parametrization is None:
self._parametrization = par
if not morphism:
return par
P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
return P1.hom(par[0], self), self.Hom(P1)(par[1], check=False)
def parametrization(self, point=None, morphism=True):
r"""
Return a parametrization `f` of ``self`` together with the
inverse of `f`.
If ``point`` is specified, then that point is used
for the parametrization. Otherwise, use ``self.rational_point()``
to find a point.
If ``morphism`` is True, then `f` is returned in the form
of a Scheme morphism. Otherwise, it is a tuple of polynomials
that gives the parametrization.
ALGORITHM:
Uses Denis Simon's pari script Qfparam.
See ``sage.quadratic_forms.qfsolve.qfparam``.
EXAMPLES ::
sage: c = Conic([1,1,-1])
sage: c.parametrization()
(Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y) to
(2*x*y : x^2 - y^2 : x^2 + y^2),
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
To: Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*x : -1/2*y + 1/2*z))
An example with ``morphism = False`` ::
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism = False); (p, i)
([-2*x*y, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y
A ``ValueError`` is raised if ``self`` has no rational point ::
sage: C = Conic(x^2 + 2*y^2 + z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!
A ``ValueError`` is raised if ``self`` is not smooth ::
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
"""
if (not self._parametrization is None) and not point:
par = self._parametrization
else:
if not self.is_smooth():
raise ValueError, "The conic self (=%s) is not smooth, hence does not have a parametrization." % self
if point == None:
point = self.rational_point()
point = Sequence(point)
Q = PolynomialRing(QQ, 'x,y')
[x, y] = Q.gens()
gens = self.ambient_space().gens()
M = self.symmetric_matrix()
M *= lcm([ t.denominator() for t in M.list() ])
par1 = qfparam(M, point)
B = Matrix([[par1[i][j] for j in range(3)] for i in range(3)])
# self is in the image of B and does not lie on a line,
# hence B is invertible
A = B.inverse()
par2 = [sum([A[i,j]*gens[j] for j in range(3)]) for i in [1,0]]
par = ([Q(pol(x/y)*y**2) for pol in par1], par2)
if self._parametrization is None:
self._parametrization = par
if not morphism:
return par
P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
return P1.hom(par[0],self), self.Hom(P1)(par[1], check = False)