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 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 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 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: codes.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: codes.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: codes.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 = codes.QuadraticResidueCodeOddPair(17,GF(2))[1]
sage: C1x = C1.extended_code()
sage: C2 = codes.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 = codes.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]_.
TESTS::
sage: codes.QuadraticResidueCodeOddPair(9,GF(2))
Traceback (most recent call last):
...
ValueError: the argument n must be an odd prime
"""
from sage.misc.misc import srange
from sage.categories.finite_fields import FiniteFields
if F not in FiniteFields():
raise ValueError("the argument F must be a finite field")
q = F.order()
n = Integer(n)
if n <= 2 or not n.is_prime():
raise ValueError("the argument n must be an odd prime")
Q = quadratic_residues(n); Q.remove(0) # non-zero quad residues
N = [x for x in srange(1,n) if x not in Q] # non-zero quad non-residues
if q not in Q:
raise ValueError("the order of the finite field must be a quadratic residue modulo n")
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: codes.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: codes.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: codes.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 = codes.QuadraticResidueCodeOddPair(17,GF(2))[1]
sage: C1x = C1.extended_code()
sage: C2 = codes.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 = codes.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]_.
TESTS::
sage: codes.QuadraticResidueCodeOddPair(9,GF(2))
Traceback (most recent call last):
...
ValueError: the argument n must be an odd prime
"""
from sage.misc.misc import srange
from sage.categories.finite_fields import FiniteFields
if F not in FiniteFields():
raise ValueError("the argument F must be a finite field")
q = F.order()
n = Integer(n)
if n <= 2 or not n.is_prime():
raise ValueError("the argument n must be an odd prime")
Q = quadratic_residues(n)
Q.remove(0) # non-zero quad residues
N = [x for x in srange(1, n) if x not in Q] # non-zero quad non-residues
if q not in Q:
raise ValueError(
"the order of the finite field must be a quadratic residue modulo n"
)
return DuadicCodeOddPair(F, Q, N)
