def RandomLinearCodeGuava(n,k,F):
r"""
The method used is to first construct a `k \times n` matrix of the block
form `(I,A)`, where `I` is a `k \times k` identity matrix and `A` is a
`k \times (n-k)` matrix constructed using random elements of `F`. Then
the columns are permuted using a randomly selected element of the symmetric
group `S_n`.
INPUT:
- ``n,k`` -- integers with `n>k>1`.
OUTPUT:
Returns a "random" linear code with length `n`, dimension `k` over field `F`.
EXAMPLES::
sage: C = codes.RandomLinearCodeGuava(30,15,GF(2)); C # optional - gap_packages (Guava package)
Linear code of length 30, dimension 15 over Finite Field of size 2
sage: C = codes.RandomLinearCodeGuava(10,5,GF(4,'a')); C # optional - gap_packages (Guava package)
Linear code of length 10, dimension 5 over Finite Field in a of size 2^2
AUTHOR: David Joyner (11-2005)
"""
current_randstate().set_seed_gap()
q = F.order()
gap.eval("C:=RandomLinearCode("+str(n)+","+str(k)+", GF("+str(q)+"))")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i,j)),F) for j in range(1,n+1)] for i in range(1,k+1)]
MS = MatrixSpace(F,k,n)
return LinearCode(MS(G))
def RandomLinearCodeGuava(n,k,F):
r"""
The method used is to first construct a `k \times n` matrix of the block
form `(I,A)`, where `I` is a `k \times k` identity matrix and `A` is a
`k \times (n-k)` matrix constructed using random elements of `F`. Then
the columns are permuted using a randomly selected element of the symmetric
group `S_n`.
INPUT:
Integers `n,k`, with `n>k>1`.
OUTPUT:
Returns a "random" linear code with length n, dimension k over field F.
EXAMPLES::
sage: C = RandomLinearCodeGuava(30,15,GF(2)); C # optional - gap_packages (Guava package)
Linear code of length 30, dimension 15 over Finite Field of size 2
sage: C = RandomLinearCodeGuava(10,5,GF(4,'a')); C # optional - gap_packages (Guava package)
Linear code of length 10, dimension 5 over Finite Field in a of size 2^2
AUTHOR: David Joyner (11-2005)
"""
current_randstate().set_seed_gap()
q = F.order()
gap.eval("C:=RandomLinearCode("+str(n)+","+str(k)+", GF("+str(q)+"))")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i,j)),F) for j in range(1,n+1)] for i in range(1,k+1)]
MS = MatrixSpace(F,k,n)
return LinearCode(MS(G))
def BinaryReedMullerCode(r, k):
r"""
The binary 'Reed-Muller code' with dimension k and
order r is a code with length `2^k` and minimum distance `2^k-r`
(see for example, section 1.10 in [HP]_). By definition, the
`r^{th}` order binary Reed-Muller code of length `n=2^m`, for
`0 \leq r \leq m`, is the set of all vectors `(f(p)\ |\ p \in GF(2)^m)`,
where `f` is a multivariate polynomial of degree at most `r`
in `m` variables.
INPUT:
- ``r, k`` -- positive integers with `2^k>r`.
OUTPUT:
Returns the binary 'Reed-Muller code' with dimension `k` and order `r`.
EXAMPLE::
sage: C = codes.BinaryReedMullerCode(2,4); C # optional - gap_packages (Guava package)
Linear code of length 16, dimension 11 over Finite Field of size 2
sage: C.minimum_distance() # optional - gap_packages (Guava package)
4
sage: C.generator_matrix() # optional - gap_packages (Guava package)
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1]
AUTHOR: David Joyner (11-2005)
"""
F = GF(2)
gap.load_package("guava")
gap.eval("C:=ReedMullerCode(" + str(r) + ", " + str(k) + ")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[
gfq_gap_to_sage(gap.eval("G[" + str(i) + "][" + str(j) + "]"), F)
for j in range(1, n + 1)
] for i in range(1, k + 1)]
MS = MatrixSpace(F, k, n)
return LinearCode(MS(G))
def BinaryReedMullerCode(r,k):
r"""
The binary 'Reed-Muller code' with dimension k and
order r is a code with length `2^k` and minimum distance `2^k-r`
(see for example, section 1.10 in [HP]_). By definition, the
`r^{th}` order binary Reed-Muller code of length `n=2^m`, for
`0 \leq r \leq m`, is the set of all vectors `(f(p)\ |\ p \\in GF(2)^m)`,
where `f` is a multivariate polynomial of degree at most `r`
in `m` variables.
INPUT:
- ``r, k`` -- positive integers with `2^k>r`.
OUTPUT:
Returns the binary 'Reed-Muller code' with dimension `k` and order `r`.
EXAMPLE::
sage: C = codes.BinaryReedMullerCode(2,4); C # optional - gap_packages (Guava package)
Linear code of length 16, dimension 11 over Finite Field of size 2
sage: C.minimum_distance() # optional - gap_packages (Guava package)
4
sage: C.gen_mat() # optional - gap_packages (Guava package)
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1]
AUTHOR: David Joyner (11-2005)
"""
F = GF(2)
gap.load_package("guava")
gap.eval("C:=ReedMullerCode("+str(r)+", "+str(k)+")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G["+str(i)+"]["+str(j)+"]"),F) for j in range(1,n+1)] for i in range(1,k+1)]
MS = MatrixSpace(F,k,n)
return LinearCode(MS(G))
def QuasiQuadraticResidueCode(p):
r"""
A (binary) quasi-quadratic residue code (or QQR code), as defined by
Proposition 2.2 in [BM]_, has a generator matrix in the block form `G=(Q,N)`.
Here `Q` is a `p \times p` circulant matrix whose top row
is `(0,x_1,...,x_{p-1})`, where `x_i=1` if and only if `i`
is a quadratic residue `\mod p`, and `N` is a `p \times p` circulant
matrix whose top row is `(0,y_1,...,y_{p-1})`, where `x_i+y_i=1` for all `i`.
INPUT:
- ``p`` -- a prime `>2`.
OUTPUT:
Returns a QQR code of length `2p`.
EXAMPLES::
sage: C = codes.QuasiQuadraticResidueCode(11); C # optional - gap_packages (Guava package)
Linear code of length 22, dimension 11 over Finite Field of size 2
REFERENCES:
.. [BM] Bazzi and Mitter, {\it Some constructions of codes from group actions}, (preprint
March 2003, available on Mitter's MIT website).
.. [Jresidue] D. Joyner, {\it On quadratic residue codes and hyperelliptic curves},
(preprint 2006)
These are self-orthogonal in general and self-dual when $p \\equiv 3 \\pmod 4$.
AUTHOR: David Joyner (11-2005)
"""
F = GF(2)
gap.load_package("guava")
gap.eval("C:=QQRCode(" + str(p) + ")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[
gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i, j)), F)
for j in range(1, n + 1)
] for i in range(1, k + 1)]
MS = MatrixSpace(F, k, n)
return LinearCode(MS(G))
def QuasiQuadraticResidueCode(p):
r"""
A (binary) quasi-quadratic residue code (or QQR code), as defined by
Proposition 2.2 in [BM]_, has a generator matrix in the block form `G=(Q,N)`.
Here `Q` is a `p \times p` circulant matrix whose top row
is `(0,x_1,...,x_{p-1})`, where `x_i=1` if and only if `i`
is a quadratic residue `\mod p`, and `N` is a `p \times p` circulant
matrix whose top row is `(0,y_1,...,y_{p-1})`, where `x_i+y_i=1` for all `i`.
INPUT:
- ``p`` -- a prime `>2`.
OUTPUT:
Returns a QQR code of length `2p`.
EXAMPLES::
sage: C = codes.QuasiQuadraticResidueCode(11); C # optional - gap_packages (Guava package)
Linear code of length 22, dimension 11 over Finite Field of size 2
REFERENCES:
.. [BM] Bazzi and Mitter, {\it Some constructions of codes from group actions}, (preprint
March 2003, available on Mitter's MIT website).
.. [Jresidue] \D. Joyner, {\it On quadratic residue codes and hyperelliptic curves},
(preprint 2006)
These are self-orthogonal in general and self-dual when $p \\equiv 3 \\pmod 4$.
AUTHOR: David Joyner (11-2005)
"""
F = GF(2)
gap.load_package("guava")
gap.eval("C:=QQRCode(" + str(p) + ")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i, j)), F)
for j in range(1, n + 1)] for i in range(1, k + 1)]
MS = MatrixSpace(F, k, n)
return LinearCode(MS(G))
def QuasiQuadraticResidueCode(p):
r"""
A (binary) quasi-quadratic residue code (or QQR code), as defined by
Proposition 2.2 in [BM2003]_, has a generator matrix in the block form `G=(Q,N)`.
Here `Q` is a `p \times p` circulant matrix whose top row
is `(0,x_1,...,x_{p-1})`, where `x_i=1` if and only if `i`
is a quadratic residue `\mod p`, and `N` is a `p \times p` circulant
matrix whose top row is `(0,y_1,...,y_{p-1})`, where `x_i+y_i=1` for all `i`.
INPUT:
- ``p`` -- a prime `>2`.
OUTPUT:
Returns a QQR code of length `2p`.
EXAMPLES::
sage: C = codes.QuasiQuadraticResidueCode(11); C # optional - gap_packages (Guava package)
[22, 11] linear code over GF(2)
REFERENCES:
- [BM2003]_
- [Joy2006]_
These are self-orthogonal in general and self-dual when $p \\equiv 3 \\pmod 4$.
AUTHOR: David Joyner (11-2005)
"""
F = GF(2)
gap.load_package("guava")
gap.eval("C:=QQRCode(" + str(p) + ")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[
gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i, j)), F)
for j in range(1, n + 1)
] for i in range(1, k + 1)]
MS = MatrixSpace(F, k, n)
return LinearCode(MS(G))
def QuasiQuadraticResidueCode(p):
r"""
A (binary) quasi-quadratic residue code (or QQR code).
Follows the definition of Proposition 2.2 in [BM]. The code has a generator
matrix in the block form `G=(Q,N)`. Here `Q` is a `p \times p` circulant
matrix whose top row is `(0,x_1,...,x_{p-1})`, where `x_i=1` if and only if
`i` is a quadratic residue `\mod p`, and `N` is a `p \times p` circulant
matrix whose top row is `(0,y_1,...,y_{p-1})`, where `x_i+y_i=1` for all
`i`.
INPUT:
- ``p`` -- a prime `>2`.
OUTPUT:
Returns a QQR code of length `2p`.
EXAMPLES::
sage: C = codes.QuasiQuadraticResidueCode(11); C # optional - gap_packages (Guava package)
[22, 11] linear code over GF(2)
These are self-orthogonal in general and self-dual when $p \\equiv 3 \\pmod 4$.
AUTHOR: David Joyner (11-2005)
"""
if not is_package_installed('gap_packages'):
raise PackageNotFoundError('gap_packages')
F = GF(2)
gap.load_package("guava")
gap.eval("C:=QQRCode(" + str(p) + ")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i, j)), F)
for j in range(1, n + 1)] for i in range(1, k + 1)]
MS = MatrixSpace(F, k, n)
return LinearCode(MS(G))
def RandomLinearCodeGuava(n, k, F):
r"""
The method used is to first construct a `k \times n` matrix of the block
form `(I,A)`, where `I` is a `k \times k` identity matrix and `A` is a
`k \times (n-k)` matrix constructed using random elements of `F`. Then
the columns are permuted using a randomly selected element of the symmetric
group `S_n`.
INPUT:
- ``n,k`` -- integers with `n>k>1`.
OUTPUT:
Returns a "random" linear code with length `n`, dimension `k` over field `F`.
EXAMPLES::
sage: C = codes.RandomLinearCodeGuava(30,15,GF(2)); C # optional - gap_packages (Guava package)
[30, 15] linear code over GF(2)
sage: C = codes.RandomLinearCodeGuava(10,5,GF(4,'a')); C # optional - gap_packages (Guava package)
[10, 5] linear code over GF(4)
AUTHOR: David Joyner (11-2005)
"""
current_randstate().set_seed_gap()
q = F.order()
if not is_package_installed('gap_packages'):
raise PackageNotFoundError('gap_packages')
gap.load_package("guava")
gap.eval("C:=RandomLinearCode("+str(n)+","+str(k)+", GF("+str(q)+"))")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i, j)), F)
for j in range(1, n + 1)] for i in range(1, k + 1)]
MS = MatrixSpace(F, k, n)
return LinearCode(MS(G))
def _element_constructor_(self, x):
r"""
Coerce ``x`` into the finite field.
INPUT:
- ``x`` -- object
OUTPUT:
If possible, makes a finite field element from ``x``.
EXAMPLES::
sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k = FiniteField_ext_pari(3^4, 'a')
sage: b = k(5) # indirect doctest
sage: b.parent()
Finite Field in a of size 3^4
sage: a = k.gen()
sage: k(a + 2)
a + 2
Univariate polynomials coerce into finite fields by evaluating
the polynomial at the field's generator::
sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: R.<x> = QQ[]
sage: k, a = FiniteField_ext_pari(5^2, 'a').objgen()
sage: k(R(2/3))
4
sage: k(x^2)
a + 3
sage: R.<x> = GF(5)[]
sage: k(x^3-2*x+1)
2*a + 4
sage: x = polygen(QQ)
sage: k(x^25)
a
sage: Q, q = FiniteField_ext_pari(5^7, 'q').objgen()
sage: L = GF(5)
sage: LL.<xx> = L[]
sage: Q(xx^2 + 2*xx + 4)
q^2 + 2*q + 4
Multivariate polynomials only coerce if constant::
sage: R = k['x,y,z']; R
Multivariate Polynomial Ring in x, y, z over Finite Field in a of size 5^2
sage: k(R(2))
2
sage: R = QQ['x,y,z']
sage: k(R(1/5))
Traceback (most recent call last):
...
TypeError: unable to coerce
Gap elements can also be coerced into finite fields::
sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: F = FiniteField_ext_pari(8, 'a')
sage: a = F.multiplicative_generator(); a
a
sage: b = gap(a^3); b
Z(2^3)^3
sage: F(b)
a + 1
sage: a^3
a + 1
sage: a = GF(13)(gap('0*Z(13)')); a
0
sage: a.parent()
Finite Field of size 13
sage: F = GF(16, 'a')
sage: F(gap('Z(16)^3'))
a^3
sage: F(gap('Z(16)^2'))
a^2
You can also call a finite extension field with a string
to produce an element of that field, like this::
sage: k = GF(2^8, 'a')
sage: k('a^200')
a^4 + a^3 + a^2
This is especially useful for fast conversions from Singular etc.
to ``FiniteField_ext_pariElements``.
AUTHORS:
- David Joyner (2005-11)
- Martin Albrecht (2006-01-23)
- Martin Albrecht (2006-03-06): added coercion from string
"""
if isinstance(x, element_ext_pari.FiniteField_ext_pariElement):
if x.parent() is self:
return x
elif x.parent() == self:
# canonically isomorphic finite fields
return element_ext_pari.FiniteField_ext_pariElement(self, x)
else:
# This is where we *would* do coercion from one finite field to another...
raise TypeError, "no coercion defined"
elif sage.interfaces.gap.is_GapElement(x):
from sage.interfaces.gap import gfq_gap_to_sage
try:
return gfq_gap_to_sage(x, self)
except (ValueError, IndexError, TypeError):
raise TypeError, "no coercion defined"
if isinstance(x, (int, long, integer.Integer, rational.Rational,
pari.pari_gen, list)):
return element_ext_pari.FiniteField_ext_pariElement(self, x)
elif isinstance(x, multi_polynomial_element.MPolynomial):
if x.is_constant():
return self(x.constant_coefficient())
else:
raise TypeError, "no coercion defined"
elif isinstance(x, polynomial_element.Polynomial):
if x.is_constant():
return self(x.constant_coefficient())
else:
return x.change_ring(self)(self.gen())
elif isinstance(x, str):
x = x.replace(self.variable_name(),'a')
x = pari.pari(x)
t = x.type()
if t == 't_POL':
if (x.variable() == 'a' \
and x.polcoeff(0).type()[2] == 'I'): #t_INT and t_INTMOD
return self(x)
if t[2] == 'I': #t_INT and t_INTMOD
return self(x)
raise TypeError, "string element does not match this finite field"
try:
if x.parent() == self.vector_space():
x = pari.pari('+'.join(['%s*a^%s'%(x[i], i) for i in range(self.degree())]))
return element_ext_pari.FiniteField_ext_pariElement(self, x)
except AttributeError:
pass
try:
return element_ext_pari.FiniteField_ext_pariElement(self, integer.Integer(x))
except TypeError, msg:
raise TypeError, "%s\nno coercion defined"%msg
def _element_constructor_(self, x):
r"""
Coerce ``x`` into the finite field.
INPUT:
- ``x`` -- object
OUTPUT:
If possible, makes a finite field element from ``x``.
EXAMPLES::
sage: k = FiniteField(3^4, 'a', impl='pari_mod')
sage: b = k(5) # indirect doctest
sage: b.parent()
Finite Field in a of size 3^4
sage: a = k.gen()
sage: k(a + 2)
a + 2
Univariate polynomials coerce into finite fields by evaluating
the polynomial at the field's generator::
sage: R.<x> = QQ[]
sage: k.<a> = FiniteField(5^2, impl='pari_mod')
sage: k(R(2/3))
4
sage: k(x^2)
a + 3
sage: R.<x> = GF(5)[]
sage: k(x^3-2*x+1)
2*a + 4
sage: x = polygen(QQ)
sage: k(x^25)
a
sage: Q.<q> = FiniteField(5^7, impl='pari_mod')
sage: L = GF(5)
sage: LL.<xx> = L[]
sage: Q(xx^2 + 2*xx + 4)
q^2 + 2*q + 4
Multivariate polynomials only coerce if constant::
sage: R = k['x,y,z']; R
Multivariate Polynomial Ring in x, y, z over Finite Field in a of size 5^2
sage: k(R(2))
2
sage: R = QQ['x,y,z']
sage: k(R(1/5))
Traceback (most recent call last):
...
TypeError: unable to coerce
Gap elements can also be coerced into finite fields::
sage: F = FiniteField(8, 'a', impl='pari_mod')
sage: a = F.multiplicative_generator(); a
a
sage: b = gap(a^3); b
Z(2^3)^3
sage: F(b)
a + 1
sage: a^3
a + 1
sage: a = GF(13)(gap('0*Z(13)')); a
0
sage: a.parent()
Finite Field of size 13
sage: F = GF(16, 'a', impl='pari_mod')
sage: F(gap('Z(16)^3'))
a^3
sage: F(gap('Z(16)^2'))
a^2
You can also call a finite extension field with a string
to produce an element of that field, like this::
sage: k = GF(2^8, 'a', impl='pari_mod')
sage: k('a^200')
a^4 + a^3 + a^2
This is especially useful for fast conversions from Singular etc.
to ``FiniteField_ext_pariElements``.
AUTHORS:
- David Joyner (2005-11)
- Martin Albrecht (2006-01-23)
- Martin Albrecht (2006-03-06): added coercion from string
"""
if isinstance(x, element_ext_pari.FiniteField_ext_pariElement):
if x.parent() is self:
return x
elif x.parent() == self:
# canonically isomorphic finite fields
return element_ext_pari.FiniteField_ext_pariElement(self, x)
else:
raise TypeError("no coercion defined")
elif sage.interfaces.gap.is_GapElement(x):
from sage.interfaces.gap import gfq_gap_to_sage
try:
return gfq_gap_to_sage(x, self)
except (ValueError, IndexError, TypeError):
raise TypeError("no coercion defined")
if isinstance(x, (int, long, integer.Integer, rational.Rational,
pari.pari_gen, list)):
return element_ext_pari.FiniteField_ext_pariElement(self, x)
elif isinstance(x, multi_polynomial_element.MPolynomial):
if x.is_constant():
return self(x.constant_coefficient())
else:
raise TypeError("no coercion defined")
elif isinstance(x, polynomial_element.Polynomial):
if x.is_constant():
return self(x.constant_coefficient())
else:
return x.change_ring(self)(self.gen())
elif isinstance(x, str):
x = x.replace(self.variable_name(),'a')
x = pari.pari(x)
t = x.type()
if t == 't_POL':
if (x.variable() == 'a' \
and x.polcoeff(0).type()[2] == 'I'): #t_INT and t_INTMOD
return self(x)
if t[2] == 'I': #t_INT and t_INTMOD
return self(x)
raise TypeError("string element does not match this finite field")
try:
if x.parent() == self.vector_space():
x = pari.pari('+'.join(['%s*a^%s'%(x[i], i) for i in range(self.degree())]))
return element_ext_pari.FiniteField_ext_pariElement(self, x)
except AttributeError:
pass
try:
return element_ext_pari.FiniteField_ext_pariElement(self, integer.Integer(x))
except TypeError as msg:
raise TypeError("%s\nno coercion defined"%msg)
