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 [HP2003]_) 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)) # known bug (#25896) ([17, 9] Cyclic Code over GF(13), [17, 9] Cyclic Code over GF(13)) sage: codes.QuadraticResidueCodeOddPair(17, GF(2)) ([17, 9] Cyclic Code over GF(2), [17, 9] Cyclic Code over GF(2)) sage: codes.QuadraticResidueCodeOddPair(13, GF(9,"z")) # known bug (#25896) ([13, 7] Cyclic Code over GF(9), [13, 7] Cyclic Code over GF(9)) 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: C3 = codes.QuadraticResidueCodeOddPair(7, GF(2))[0] sage: C3x = C3.extended_code() sage: C3x.spectrum() [1, 0, 0, 0, 14, 0, 0, 0, 1] This is consistent with Theorem 6.6.14 in [HP2003]_. TESTS:: sage: codes.QuadraticResidueCodeOddPair(9,GF(2)) Traceback (most recent call last): ... ValueError: the argument n must be an odd prime """ from sage.arith.srange 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: C3 = codes.QuadraticResidueCodeOddPair(7,GF(2))[0] sage: C3x = C3.extended_code() sage: C3x.spectrum() [1, 0, 0, 0, 14, 0, 0, 0, 1] 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.arith.srange 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 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 [HP2003]_) 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)) ([17, 8] Cyclic Code over GF(13), [17, 8] Cyclic Code over GF(13)) sage: codes.QuadraticResidueCodeEvenPair(17, GF(2)) ([17, 8] Cyclic Code over GF(2), [17, 8] Cyclic Code over GF(2)) sage: codes.QuadraticResidueCodeEvenPair(13,GF(9,"z")) ([13, 6] Cyclic Code over GF(9), [13, 6] Cyclic Code over GF(9)) sage: C1,C2 = codes.QuadraticResidueCodeEvenPair(7,GF(2)) sage: C1.is_self_orthogonal() True sage: C2.is_self_orthogonal() True sage: C3 = codes.QuadraticResidueCodeOddPair(17,GF(2))[0] sage: C4 = codes.QuadraticResidueCodeEvenPair(17,GF(2))[1] sage: C3.systematic_generator_matrix() == C4.dual_code().systematic_generator_matrix() True This is consistent with Theorem 6.6.9 and Exercise 365 in [HP2003]_. TESTS:: sage: codes.QuadraticResidueCodeEvenPair(14,Zmod(4)) Traceback (most recent call last): ... ValueError: the argument F must be a finite field sage: codes.QuadraticResidueCodeEvenPair(14,GF(2)) Traceback (most recent call last): ... ValueError: the argument n must be an odd prime sage: codes.QuadraticResidueCodeEvenPair(5,GF(2)) Traceback (most recent call last): ... ValueError: the order of the finite field must be a quadratic residue modulo n """ from sage.arith.srange 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 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 [HP2003]_) 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)) # known bug (#25896) ([17, 8] Cyclic Code over GF(13), [17, 8] Cyclic Code over GF(13)) sage: codes.QuadraticResidueCodeEvenPair(17, GF(2)) ([17, 8] Cyclic Code over GF(2), [17, 8] Cyclic Code over GF(2)) sage: codes.QuadraticResidueCodeEvenPair(13,GF(9,"z")) # known bug (#25896) ([13, 6] Cyclic Code over GF(9), [13, 6] Cyclic Code over GF(9)) sage: C1,C2 = codes.QuadraticResidueCodeEvenPair(7,GF(2)) sage: C1.is_self_orthogonal() True sage: C2.is_self_orthogonal() True sage: C3 = codes.QuadraticResidueCodeOddPair(17,GF(2))[0] sage: C4 = codes.QuadraticResidueCodeEvenPair(17,GF(2))[1] sage: C3.systematic_generator_matrix() == C4.dual_code().systematic_generator_matrix() True This is consistent with Theorem 6.6.9 and Exercise 365 in [HP2003]_. TESTS:: sage: codes.QuadraticResidueCodeEvenPair(14,Zmod(4)) Traceback (most recent call last): ... ValueError: the argument F must be a finite field sage: codes.QuadraticResidueCodeEvenPair(14,GF(2)) Traceback (most recent call last): ... ValueError: the argument n must be an odd prime sage: codes.QuadraticResidueCodeEvenPair(5,GF(2)) Traceback (most recent call last): ... ValueError: the order of the finite field must be a quadratic residue modulo n """ from sage.arith.srange 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 DuadicCodeEvenPair(F, Q, N)