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 = Zmod(n).cyclotomic_cosets(q); 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 sage: codes.DuadicCodeOddPair(GF(q),S1,S2) ([11, 6] Cyclic Code over GF(3), [11, 6] Cyclic Code over GF(3)) This is consistent with Theorem 6.1.3 in [HP2003]_. """ from sage.misc.stopgap import stopgap stopgap( "The function DuadicCodeOddPair has several issues which may cause wrong results", 25896) from .cyclic_code import CyclicCode n = len(S1) + len(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).coefficients(sparse=False) ] coeffs2 = [ _lift2smallest_field(c)[0] for c in (g2 + j).coefficients(sparse=False) ] gg1 = P2(coeffs1) gg2 = P2(coeffs2) gg1 = gcd(gg1, x**n - 1) gg2 = gcd(gg2, x**n - 1) C1 = CyclicCode(length=n, generator_pol=gg1) C2 = CyclicCode(length=n, generator_pol=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 = Zmod(n).cyclotomic_cosets(q); 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 sage: codes.DuadicCodeEvenPair(GF(q),S1,S2) ([11, 5] Cyclic Code over GF(3), [11, 5] Cyclic Code over GF(3)) """ from sage.misc.stopgap import stopgap stopgap( "The function DuadicCodeEvenPair has several issues which may cause wrong results", 25896) from .cyclic_code import CyclicCode n = len(S1) + len(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.coefficients(sparse=False)]) gg2 = P2( [_lift2smallest_field(c)[0] for c in g2.coefficients(sparse=False)]) C1 = CyclicCode(length=n, generator_pol=gg1) C2 = CyclicCode(length=n, generator_pol=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 = Zmod(n).cyclotomic_cosets(q); 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 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 = len(S1) + len(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).coefficients(sparse=False) ] coeffs2 = [ lift2smallest_field(c)[0] for c in (g2 + j).coefficients(sparse=False) ] 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 = Zmod(n).cyclotomic_cosets(q); 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 sage: codes.DuadicCodeOddPair(GF(q),S1,S2) ([11, 6] Cyclic Code over GF(3), [11, 6] Cyclic Code over GF(3)) This is consistent with Theorem 6.1.3 in [HP2003]_. """ from .cyclic_code import CyclicCode n = len(S1) + len(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).coefficients(sparse=False)] coeffs2 = [_lift2smallest_field(c)[0] for c in (g2+j).coefficients(sparse=False)] gg1 = P2(coeffs1) gg2 = P2(coeffs2) gg1 = gcd(gg1, x**n - 1) gg2 = gcd(gg2, x**n - 1) C1 = CyclicCode(length = n, generator_pol = gg1) C2 = CyclicCode(length = n, generator_pol = 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 = Zmod(n).cyclotomic_cosets(q); 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 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 = len(S1) + len(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.coefficients(sparse=False)]) gg2 = P2( [lift2smallest_field(c)[0] for c in g2.coefficients(sparse=False)]) 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 = Zmod(n).cyclotomic_cosets(q); 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 sage: codes.DuadicCodeEvenPair(GF(q),S1,S2) ([11, 5] Cyclic Code over GF(3), [11, 5] Cyclic Code over GF(3)) """ from .cyclic_code import CyclicCode n = len(S1) + len(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.coefficients(sparse=False)]) gg2 = P2([_lift2smallest_field(c)[0] for c in g2.coefficients(sparse=False)]) C1 = CyclicCode(length = n, generator_pol = gg1) C2 = CyclicCode(length = n, generator_pol = gg2) return C1,C2
def check_consistency(self, n): """ Check that the pseudo-Conway polynomials of degree dividing `n` in this lattice satisfy the required compatibility conditions. EXAMPLES:: sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice sage: PCL = PseudoConwayLattice(2, use_database=False) sage: PCL.check_consistency(6) sage: PCL.check_consistency(60) # long time """ p = self.p K = FiniteField(p**n, modulus = self.polynomial(n), names='a') a = K.gen() for m in n.divisors(): assert (a**((p**n-1)//(p**m-1))).minimal_polynomial() == self.polynomial(m)
def BCHCode(n, delta, F, b=0): r""" A 'Bose-Chaudhuri-Hockenghem code' (or BCH code for short) is the largest possible cyclic code of length n over field F=GF(q), whose generator polynomial has zeros (which contain the set) `Z = \{a^{b},a^{b+1}, ..., a^{b+delta-2}\}`, where a is a primitive `n^{th}` root of unity in the splitting field `GF(q^m)`, b is an integer `0\leq b\leq n-delta+1` and m is the multiplicative order of q modulo n. (The integers `b,...,b+delta-2` typically lie in the range `1,...,n-1`.) The integer `delta \geq 1` is called the "designed distance". The length n of the code and the size q of the base field must be relatively prime. The generator polynomial is equal to the least common multiple of the minimal polynomials of the elements of the set `Z` above. Special cases are b=1 (resulting codes are called 'narrow-sense' BCH codes), and `n=q^m-1` (known as 'primitive' BCH codes). It may happen that several values of delta give rise to the same BCH code. The largest one is called the Bose distance of the code. The true minimum distance, d, of the code is greater than or equal to the Bose distance, so `d\geq delta`. EXAMPLES:: sage: FF.<a> = GF(3^2,"a") sage: x = PolynomialRing(FF,"x").gen() sage: L = [b.minpoly() for b in [a,a^2,a^3]]; g = LCM(L) sage: f = x^(8)-1 sage: g.divides(f) True sage: C = codes.CyclicCode(8,g); C Linear code of length 8, dimension 4 over Finite Field of size 3 sage: C.minimum_distance() 4 sage: C = codes.BCHCode(8,3,GF(3),1); C Linear code of length 8, dimension 4 over Finite Field of size 3 sage: C.minimum_distance() 4 sage: C = codes.BCHCode(8,3,GF(3)); C Linear code of length 8, dimension 5 over Finite Field of size 3 sage: C.minimum_distance() 3 sage: C = codes.BCHCode(26, 5, GF(5), b=1); C Linear code of length 26, dimension 10 over Finite Field of size 5 """ q = F.order() R = IntegerModRing(n) m = R(q).multiplicative_order() FF = GF(q ** m, "z") z = FF.gen() e = z.multiplicative_order() / n a = z ** e # order n P = PolynomialRing(F, "x") x = P.gen() L1 = [] for coset in R.cyclotomic_cosets(q, range(b, b + delta - 1)): L1.extend(P((a ** j).minpoly()) for j in coset) g = P(LCM(L1)) if not (g.divides(x ** n - 1)): raise ValueError("BCH codes does not exist with the given input.") return CyclicCodeFromGeneratingPolynomial(n, g)
def BCHCode(n, delta, F, b=0): r""" A 'Bose-Chaudhuri-Hockenghem code' (or BCH code for short) is the largest possible cyclic code of length n over field F=GF(q), whose generator polynomial has zeros (which contain the set) `Z = \{a^{b},a^{b+1}, ..., a^{b+delta-2}\}`, where a is a primitive `n^{th}` root of unity in the splitting field `GF(q^m)`, b is an integer `0\leq b\leq n-delta+1` and m is the multiplicative order of q modulo n. (The integers `b,...,b+delta-2` typically lie in the range `1,...,n-1`.) The integer `delta \geq 1` is called the "designed distance". The length n of the code and the size q of the base field must be relatively prime. The generator polynomial is equal to the least common multiple of the minimal polynomials of the elements of the set `Z` above. Special cases are b=1 (resulting codes are called 'narrow-sense' BCH codes), and `n=q^m-1` (known as 'primitive' BCH codes). It may happen that several values of delta give rise to the same BCH code. The largest one is called the Bose distance of the code. The true minimum distance, d, of the code is greater than or equal to the Bose distance, so `d\geq delta`. EXAMPLES:: sage: FF.<a> = GF(3^2,"a") sage: x = PolynomialRing(FF,"x").gen() sage: L = [b.minpoly() for b in [a,a^2,a^3]]; g = LCM(L) sage: f = x^(8)-1 sage: g.divides(f) True sage: C = codes.CyclicCode(8,g); C Linear code of length 8, dimension 4 over Finite Field of size 3 sage: C.minimum_distance() 4 sage: C = codes.BCHCode(8,3,GF(3),1); C Linear code of length 8, dimension 4 over Finite Field of size 3 sage: C.minimum_distance() 4 sage: C = codes.BCHCode(8,3,GF(3)); C Linear code of length 8, dimension 5 over Finite Field of size 3 sage: C.minimum_distance() 3 sage: C = codes.BCHCode(26, 5, GF(5), b=1); C Linear code of length 26, dimension 10 over Finite Field of size 5 """ q = F.order() R = IntegerModRing(n) m = R(q).multiplicative_order() FF = GF(q**m, "z") z = FF.gen() e = z.multiplicative_order() / n a = z**e # order n P = PolynomialRing(F, "x") x = P.gen() L1 = [] for coset in R.cyclotomic_cosets(q, range(b, b + delta - 1)): L1.extend(P((a**j).minpoly()) for j in coset) g = P(LCM(L1)) if not (g.divides(x**n - 1)): raise ValueError("BCH codes does not exist with the given input.") return CyclicCodeFromGeneratingPolynomial(n, g)