def QuadraticResidueCodeOddPair(n, F):
"""
Quadratic residue codes of a given odd prime length and base ring
either don't exist at all or occur as 4-tuples - a pair of
"odd-like" codes and a pair of "even-like" codes. If n 2 is prime
then (Theorem 6.6.2 in [HP]_) a QR code exists over GF(q) iff q is a
quadratic residue mod n.
They are constructed as "odd-like" duadic codes associated the
splitting (Q,N) mod n, where Q is the set of non-zero quadratic
residues and N is the non-residues.
EXAMPLES::
sage: QuadraticResidueCodeOddPair(17,GF(13))
(Linear code of length 17, dimension 9 over Finite Field of size 13,
Linear code of length 17, dimension 9 over Finite Field of size 13)
sage: QuadraticResidueCodeOddPair(17,GF(2))
(Linear code of length 17, dimension 9 over Finite Field of size 2,
Linear code of length 17, dimension 9 over Finite Field of size 2)
sage: QuadraticResidueCodeOddPair(13,GF(9,"z"))
(Linear code of length 13, dimension 7 over Finite Field in z of size 3^2,
Linear code of length 13, dimension 7 over Finite Field in z of size 3^2)
sage: C1 = QuadraticResidueCodeOddPair(17,GF(2))[1]
sage: C1x = C1.extended_code()
sage: C2 = QuadraticResidueCodeOddPair(17,GF(2))[0]
sage: C2x = C2.extended_code()
sage: C2x.spectrum(); C1x.spectrum()
[1, 0, 0, 0, 0, 0, 102, 0, 153, 0, 153, 0, 102, 0, 0, 0, 0, 0, 1]
[1, 0, 0, 0, 0, 0, 102, 0, 153, 0, 153, 0, 102, 0, 0, 0, 0, 0, 1]
sage: C2x == C1x.dual_code()
True
sage: C3 = QuadraticResidueCodeOddPair(7,GF(2))[0]
sage: C3x = C3.extended_code()
sage: C3x.spectrum()
[1, 0, 0, 0, 14, 0, 0, 0, 1]
sage: C3x.is_self_dual()
True
This is consistent with Theorem 6.6.14 in [HP]_.
"""
from sage.coding.code_constructions import is_a_splitting
q = F.order()
Q = quadratic_residues(n)
Q.remove(0) # non-zero quad residues
N = range(1, n)
tmp = [N.remove(x) for x in Q] # non-zero quad non-residues
if n.is_prime() and n > 2 and not (q in Q):
raise ValueError, "No quadratic residue code exists for these parameters."
if not (is_a_splitting(Q, N, n)):
raise TypeError, "No quadratic residue code exists for these parameters."
return DuadicCodeOddPair(F, Q, N)
def QuadraticResidueCodeOddPair(n, F):
"""
Quadratic residue codes of a given odd prime length and base ring
either don't exist at all or occur as 4-tuples - a pair of
"odd-like" codes and a pair of "even-like" codes. If n 2 is prime
then (Theorem 6.6.2 in [HP]_) a QR code exists over GF(q) iff q is a
quadratic residue mod n.
They are constructed as "odd-like" duadic codes associated the
splitting (Q,N) mod n, where Q is the set of non-zero quadratic
residues and N is the non-residues.
EXAMPLES::
sage: QuadraticResidueCodeOddPair(17,GF(13))
(Linear code of length 17, dimension 9 over Finite Field of size 13,
Linear code of length 17, dimension 9 over Finite Field of size 13)
sage: QuadraticResidueCodeOddPair(17,GF(2))
(Linear code of length 17, dimension 9 over Finite Field of size 2,
Linear code of length 17, dimension 9 over Finite Field of size 2)
sage: QuadraticResidueCodeOddPair(13,GF(9,"z"))
(Linear code of length 13, dimension 7 over Finite Field in z of size 3^2,
Linear code of length 13, dimension 7 over Finite Field in z of size 3^2)
sage: C1 = QuadraticResidueCodeOddPair(17,GF(2))[1]
sage: C1x = C1.extended_code()
sage: C2 = QuadraticResidueCodeOddPair(17,GF(2))[0]
sage: C2x = C2.extended_code()
sage: C2x.spectrum(); C1x.spectrum()
[1, 0, 0, 0, 0, 0, 102, 0, 153, 0, 153, 0, 102, 0, 0, 0, 0, 0, 1]
[1, 0, 0, 0, 0, 0, 102, 0, 153, 0, 153, 0, 102, 0, 0, 0, 0, 0, 1]
sage: C2x == C1x.dual_code()
True
sage: C3 = QuadraticResidueCodeOddPair(7,GF(2))[0]
sage: C3x = C3.extended_code()
sage: C3x.spectrum()
[1, 0, 0, 0, 14, 0, 0, 0, 1]
sage: C3x.is_self_dual()
True
This is consistent with Theorem 6.6.14 in [HP]_.
"""
from sage.coding.code_constructions import is_a_splitting
q = F.order()
Q = quadratic_residues(n)
Q.remove(0) # non-zero quad residues
N = range(1, n)
tmp = [N.remove(x) for x in Q] # non-zero quad non-residues
if (n.is_prime() and n > 2 and not (q in Q)):
raise ValueError, "No quadratic residue code exists for these parameters."
if not (is_a_splitting(Q, N, n)):
raise TypeError, "No quadratic residue code exists for these parameters."
return DuadicCodeOddPair(F, Q, N)
def DuadicCodeOddPair(F,S1,S2):
"""
Constructs the "odd pair" of duadic codes associated to the
"splitting" S1, S2 of n.
.. warning::
Maybe the splitting should be associated to a sum of
q-cyclotomic cosets mod n, where q is a *prime*.
EXAMPLES::
sage: from sage.coding.code_constructions import is_a_splitting
sage: n = 11; q = 3
sage: C = cyclotomic_cosets(q,n); C
[[0], [1, 3, 4, 5, 9], [2, 6, 7, 8, 10]]
sage: S1 = C[1]
sage: S2 = C[2]
sage: is_a_splitting(S1,S2,11)
(True, 2)
sage: DuadicCodeOddPair(GF(q),S1,S2)
(Linear code of length 11, dimension 6 over Finite Field of size 3,
Linear code of length 11, dimension 6 over Finite Field of size 3)
This is consistent with Theorem 6.1.3 in [HP]_.
"""
n = max(S1+S2)+1
if not(is_a_splitting(S1,S2,n)):
raise TypeError, "%s, %s must be a splitting of %s."%(S1,S2,n)
q = F.order()
k = Mod(q,n).multiplicative_order()
FF = GF(q**k,"z")
z = FF.gen()
zeta = z**((q**k-1)/n)
P1 = PolynomialRing(FF,"x")
x = P1.gen()
g1 = prod([x-zeta**i for i in S1+[0]])
g2 = prod([x-zeta**i for i in S2+[0]])
j = sum([x**i/n for i in range(n)])
P2 = PolynomialRing(F,"x")
x = P2.gen()
coeffs1 = [lift2smallest_field(c)[0] for c in (g1+j).coeffs()]
coeffs2 = [lift2smallest_field(c)[0] for c in (g2+j).coeffs()]
gg1 = P2(coeffs1)
gg2 = P2(coeffs2)
C1 = CyclicCodeFromGeneratingPolynomial(n,gg1)
C2 = CyclicCodeFromGeneratingPolynomial(n,gg2)
return C1,C2
def DuadicCodeOddPair(F,S1,S2):
"""
Constructs the "odd pair" of duadic codes associated to the
"splitting" S1, S2 of n.
.. warning::
Maybe the splitting should be associated to a sum of
q-cyclotomic cosets mod n, where q is a *prime*.
EXAMPLES::
sage: from sage.coding.code_constructions import is_a_splitting
sage: n = 11; q = 3
sage: C = cyclotomic_cosets(q,n); C
[[0], [1, 3, 4, 5, 9], [2, 6, 7, 8, 10]]
sage: S1 = C[1]
sage: S2 = C[2]
sage: is_a_splitting(S1,S2,11)
(True, 2)
sage: codes.DuadicCodeOddPair(GF(q),S1,S2)
(Linear code of length 11, dimension 6 over Finite Field of size 3,
Linear code of length 11, dimension 6 over Finite Field of size 3)
This is consistent with Theorem 6.1.3 in [HP]_.
"""
n = max(S1+S2)+1
if not(is_a_splitting(S1,S2,n)):
raise TypeError, "%s, %s must be a splitting of %s."%(S1,S2,n)
q = F.order()
k = Mod(q,n).multiplicative_order()
FF = GF(q**k,"z")
z = FF.gen()
zeta = z**((q**k-1)/n)
P1 = PolynomialRing(FF,"x")
x = P1.gen()
g1 = prod([x-zeta**i for i in S1+[0]])
g2 = prod([x-zeta**i for i in S2+[0]])
j = sum([x**i/n for i in range(n)])
P2 = PolynomialRing(F,"x")
x = P2.gen()
coeffs1 = [lift2smallest_field(c)[0] for c in (g1+j).coeffs()]
coeffs2 = [lift2smallest_field(c)[0] for c in (g2+j).coeffs()]
gg1 = P2(coeffs1)
gg2 = P2(coeffs2)
C1 = CyclicCodeFromGeneratingPolynomial(n,gg1)
C2 = CyclicCodeFromGeneratingPolynomial(n,gg2)
return C1,C2
def QuadraticResidueCodeEvenPair(n, F):
"""
Quadratic residue codes of a given odd prime length and base ring
either don't exist at all or occur as 4-tuples - a pair of
"odd-like" codes and a pair of "even-like" codes. If n 2 is prime
then (Theorem 6.6.2 in [HP]_) a QR code exists over GF(q) iff q is a
quadratic residue mod n.
They are constructed as "even-like" duadic codes associated the
splitting (Q,N) mod n, where Q is the set of non-zero quadratic
residues and N is the non-residues.
EXAMPLES::
sage: codes.QuadraticResidueCodeEvenPair(17,GF(13))
(Linear code of length 17, dimension 8 over Finite Field of size 13,
Linear code of length 17, dimension 8 over Finite Field of size 13)
sage: codes.QuadraticResidueCodeEvenPair(17,GF(2))
(Linear code of length 17, dimension 8 over Finite Field of size 2,
Linear code of length 17, dimension 8 over Finite Field of size 2)
sage: codes.QuadraticResidueCodeEvenPair(13,GF(9,"z"))
(Linear code of length 13, dimension 6 over Finite Field in z of size 3^2,
Linear code of length 13, dimension 6 over Finite Field in z of size 3^2)
sage: C1 = codes.QuadraticResidueCodeEvenPair(7,GF(2))[0]
sage: C1.is_self_orthogonal()
True
sage: C2 = codes.QuadraticResidueCodeEvenPair(7,GF(2))[1]
sage: C2.is_self_orthogonal()
True
sage: C3 = codes.QuadraticResidueCodeOddPair(17,GF(2))[0]
sage: C4 = codes.QuadraticResidueCodeEvenPair(17,GF(2))[1]
sage: C3 == C4.dual_code()
True
This is consistent with Theorem 6.6.9 and Exercise 365 in [HP]_.
"""
q = F.order()
Q = quadratic_residues(n)
Q.remove(0) # non-zero quad residues
N = range(1, n)
tmp = [N.remove(x) for x in Q] # non-zero quad non-residues
if n.is_prime() and n > 2 and not (q in Q):
raise ValueError, "No quadratic residue code exists for these parameters."
if not (is_a_splitting(Q, N, n)):
raise TypeError, "No quadratic residue code exists for these parameters."
return DuadicCodeEvenPair(F, Q, N)
def DuadicCodeEvenPair(F,S1,S2):
r"""
Constructs the "even pair" of duadic codes associated to the
"splitting" (see the docstring for ``is_a_splitting``
for the definition) S1, S2 of n.
.. warning::
Maybe the splitting should be associated to a sum of
q-cyclotomic cosets mod n, where q is a *prime*.
EXAMPLES::
sage: from sage.coding.code_constructions import is_a_splitting
sage: n = 11; q = 3
sage: C = cyclotomic_cosets(q,n); C
[[0], [1, 3, 4, 5, 9], [2, 6, 7, 8, 10]]
sage: S1 = C[1]
sage: S2 = C[2]
sage: is_a_splitting(S1,S2,11)
(True, 2)
sage: DuadicCodeEvenPair(GF(q),S1,S2)
(Linear code of length 11, dimension 5 over Finite Field of size 3,
Linear code of length 11, dimension 5 over Finite Field of size 3)
"""
n = max(S1+S2)+1
if not(is_a_splitting(S1,S2,n)):
raise TypeError, "%s, %s must be a splitting of %s."%(S1,S2,n)
q = F.order()
k = Mod(q,n).multiplicative_order()
FF = GF(q**k,"z")
z = FF.gen()
zeta = z**((q**k-1)/n)
P1 = PolynomialRing(FF,"x")
x = P1.gen()
g1 = prod([x-zeta**i for i in S1+[0]])
g2 = prod([x-zeta**i for i in S2+[0]])
P2 = PolynomialRing(F,"x")
x = P2.gen()
gg1 = P2([lift2smallest_field(c)[0] for c in g1.coeffs()])
gg2 = P2([lift2smallest_field(c)[0] for c in g2.coeffs()])
C1 = CyclicCodeFromGeneratingPolynomial(n,gg1)
C2 = CyclicCodeFromGeneratingPolynomial(n,gg2)
return C1,C2
def DuadicCodeEvenPair(F,S1,S2):
r"""
Constructs the "even pair" of duadic codes associated to the
"splitting" (see the docstring for ``is_a_splitting``
for the definition) S1, S2 of n.
.. warning::
Maybe the splitting should be associated to a sum of
q-cyclotomic cosets mod n, where q is a *prime*.
EXAMPLES::
sage: from sage.coding.code_constructions import is_a_splitting
sage: n = 11; q = 3
sage: C = cyclotomic_cosets(q,n); C
[[0], [1, 3, 4, 5, 9], [2, 6, 7, 8, 10]]
sage: S1 = C[1]
sage: S2 = C[2]
sage: is_a_splitting(S1,S2,11)
(True, 2)
sage: codes.DuadicCodeEvenPair(GF(q),S1,S2)
(Linear code of length 11, dimension 5 over Finite Field of size 3,
Linear code of length 11, dimension 5 over Finite Field of size 3)
"""
n = max(S1+S2)+1
if not(is_a_splitting(S1,S2,n)):
raise TypeError, "%s, %s must be a splitting of %s."%(S1,S2,n)
q = F.order()
k = Mod(q,n).multiplicative_order()
FF = GF(q**k,"z")
z = FF.gen()
zeta = z**((q**k-1)/n)
P1 = PolynomialRing(FF,"x")
x = P1.gen()
g1 = prod([x-zeta**i for i in S1+[0]])
g2 = prod([x-zeta**i for i in S2+[0]])
P2 = PolynomialRing(F,"x")
x = P2.gen()
gg1 = P2([lift2smallest_field(c)[0] for c in g1.coeffs()])
gg2 = P2([lift2smallest_field(c)[0] for c in g2.coeffs()])
C1 = CyclicCodeFromGeneratingPolynomial(n,gg1)
C2 = CyclicCodeFromGeneratingPolynomial(n,gg2)
return C1,C2
def QuadraticResidueCodeEvenPair(n,F):
"""
Quadratic residue codes of a given odd prime length and base ring
either don't exist at all or occur as 4-tuples - a pair of
"odd-like" codes and a pair of "even-like" codes. If n 2 is prime
then (Theorem 6.6.2 in [HP]_) a QR code exists over GF(q) iff q is a
quadratic residue mod n.
They are constructed as "even-like" duadic codes associated the
splitting (Q,N) mod n, where Q is the set of non-zero quadratic
residues and N is the non-residues.
EXAMPLES::
sage: codes.QuadraticResidueCodeEvenPair(17,GF(13))
(Linear code of length 17, dimension 8 over Finite Field of size 13,
Linear code of length 17, dimension 8 over Finite Field of size 13)
sage: codes.QuadraticResidueCodeEvenPair(17,GF(2))
(Linear code of length 17, dimension 8 over Finite Field of size 2,
Linear code of length 17, dimension 8 over Finite Field of size 2)
sage: codes.QuadraticResidueCodeEvenPair(13,GF(9,"z"))
(Linear code of length 13, dimension 6 over Finite Field in z of size 3^2,
Linear code of length 13, dimension 6 over Finite Field in z of size 3^2)
sage: C1 = codes.QuadraticResidueCodeEvenPair(7,GF(2))[0]
sage: C1.is_self_orthogonal()
True
sage: C2 = codes.QuadraticResidueCodeEvenPair(7,GF(2))[1]
sage: C2.is_self_orthogonal()
True
sage: C3 = codes.QuadraticResidueCodeOddPair(17,GF(2))[0]
sage: C4 = codes.QuadraticResidueCodeEvenPair(17,GF(2))[1]
sage: C3 == C4.dual_code()
True
This is consistent with Theorem 6.6.9 and Exercise 365 in [HP]_.
"""
q = F.order()
Q = quadratic_residues(n); Q.remove(0) # non-zero quad residues
N = range(1,n); tmp = [N.remove(x) for x in Q] # non-zero quad non-residues
if (n.is_prime() and n>2 and not(q in Q)):
raise ValueError, "No quadratic residue code exists for these parameters."
if not(is_a_splitting(Q,N,n)):
raise TypeError, "No quadratic residue code exists for these parameters."
return DuadicCodeEvenPair(F,Q,N)
