def printAllCyclicCodes(factorPolynomials):#выводим все циклические коды, которые можем сделать
s = ''
product = np.array([])
for i in range(0, len(factorPolynomials)):
if i == 0:
product = factorPolynomials[i]
else:
product = GF2.multPoly(product, factorPolynomials[i])
s += '(' + GF2.polyToString(factorPolynomials[i]) + ') '
print(s + '= ' + GF2.polyToString(product))
print()
numberCodes = 2**(len(factorPolynomials)) - 2
n = degree(product)
print('Всего возможны', numberCodes, 'различных циклических кодов длиной', n)
print('так как мы можем найти', numberCodes, 'различных неприводимых двоичных полиномов')
print(GF2.polyToString(product))
print(np.bitwise_and(1, 3))
print('Code <- Generator polynomial')
for i in range(0, numberCodes):
s = ''
gp = np.array([]) # generator polynomial
for j in range(0, len(factorPolynomials)):
if np.bitwise_and(i+1, 2**j) > 0:
if s =='':
gp = factorPolynomials[j]
else:
gp = GF2.multPoly(gp, factorPolynomials[j])
s += '(' + GF2.polyToString(factorPolynomials[j]) + ')'
print('Ccyc(' + str(n) + ', ' + str(degree(gp)) + ') <- g' + str(i+1) + ' = ' + s + ' = ' + GF2.polyToString(gp))
def __init__(self, g, n):
assert g[0] == 1, \
"g0 must equal to 1"
assert n >= degree(g), \
"n=%i must be >= degree(g)=%i" % (n, degree(g))
self._g = g[:n]; #auto remove too much dangling zeros
self._n = n;
def __init__(self, g, n):
assert g[0] == 1, \
"g0 must equal to 1"
assert n >= degree(g), \
"n=%i must be >= degree(g)=%i" % (n, degree(g))
self._g = g[:n]; #auto remove too much dangling zeros
self._n = n
def encode(m, g, systematic = True):
if systematic:
r = degree(g) # r = n - k
Xr = X(r) # X^(n-k)
XrmX = GF2.multPoly(Xr, m) # X^(n-k) * m(X)
p = GF2.modPoly(XrmX, g) # p(X) = (X^(n-k) * m(X)) mod g(X)
c = GF2.addPoly(p, XrmX) # c(X) = p(X) + (X^(n-k) * m(X))
else:
c = GF2.multPoly(m, g)
return c.astype(int)
def gToG(g, n, systematic = True, verbose = False):#получаем образующую матрицу
k = n - degree(g)
g = padEnd(g, n)
G = np.empty([k, n])
for i in range(0, k):
G[i,:] = shift(g, i)
# матрицу необходимо повернуть
# -> systematic form
if systematic:
G = makeSystematic(G, verbose)
return G.astype(int)
def printAllCyclicCodes(factorPolynomials):
"""Generates all cyclic codes that can be created from
the given factor polynomials.
(slide 28)
Args:
factorPolynomials: factor polynomials in a python array
"""
s = ''
product = np.array([])
for i in range(0, len(factorPolynomials)):
if i == 0:
product = factorPolynomials[i]
else:
product = GF2.multPoly(product, factorPolynomials[i])
s += '(' + GF2.polyToString(factorPolynomials[i]) + ') '
print(s + '= ' + GF2.polyToString(product))
print()
numberCodes = 2**(len(factorPolynomials)) - 2
n = degree(product)
print('There are', numberCodes, 'different cyclic codes of length', n,
'as')
print('we can find', numberCodes,
'different generator polynomials that are')
print('the factors of', GF2.polyToString(product))
print(np.bitwise_and(1, 3))
print('Code <- Generator polynomial')
for i in range(0, numberCodes):
s = ''
gp = np.array([]) # generator polynomial
for j in range(0, len(factorPolynomials)):
if np.bitwise_and(i + 1, 2**j) > 0:
if s == '':
gp = factorPolynomials[j]
else:
gp = GF2.multPoly(gp, factorPolynomials[j])
s += '(' + GF2.polyToString(factorPolynomials[j]) + ')'
print('Ccyc(' + str(n) + ', ' + str(degree(gp)) + ') <- g' +
str(i + 1) + ' = ' + s + ' = ' + GF2.polyToString(gp))
def printAllCyclicCodes(factorPolynomials):
"""Generates all cyclic codes that can be created from
the given factor polynomials.
(slide 28)
Args:
factorPolynomials: factor polynomials in a python array
"""
s = ''
product = np.array([])
for i in range(0, len(factorPolynomials)):
if i == 0:
product = factorPolynomials[i]
else:
product = GF2.multPoly(product, factorPolynomials[i])
s += '(' + GF2.polyToString(factorPolynomials[i]) + ') '
print(s + '= ' + GF2.polyToString(product))
print()
numberCodes = 2**(len(factorPolynomials)) - 2
n = degree(product)
print('There are', numberCodes, 'different cyclic codes of length', n, 'as')
print('we can find', numberCodes, 'different generator polynomials that are')
print('the factors of', GF2.polyToString(product))
print(np.bitwise_and(1, 3))
print('Code <- Generator polynomial')
for i in range(0, numberCodes):
s = ''
gp = np.array([]) # generator polynomial
for j in range(0, len(factorPolynomials)):
if np.bitwise_and(i+1, 2**j) > 0:
if s =='':
gp = factorPolynomials[j]
else:
gp = GF2.multPoly(gp, factorPolynomials[j])
s += '(' + GF2.polyToString(factorPolynomials[j]) + ')'
print('Ccyc(' + str(n) + ', ' + str(degree(gp)) + ') <- g' + str(i+1) + ' = ' + s + ' = ' + GF2.polyToString(gp))
def encode(m, g, systematic = True):
"""Encoding of cyclic code (in systematic form)
(slide 23)
ATTENTION: Dangling zeros in returned codeword are cut away.
"""
if systematic:
r = degree(g) # r = n - k
Xr = X(r) # X^(n-k)
XrmX = GF2.multPoly(Xr, m) # X^(n-k) * m(X)
p = GF2.modPoly(XrmX, g) # p(X) = (X^(n-k) * m(X)) mod g(X)
c = GF2.addPoly(p, XrmX) # c(X) = p(X) + (X^(n-k) * m(X))
else:
c = GF2.multPoly(m, g)
return c.astype(int)
def encode(m, g, systematic=True):
"""Encoding of cyclic code (in systematic form)
(slide 23)
ATTENTION: Dangling zeros in returned codeword are cut away.
"""
if systematic:
r = degree(g) # r = n - k
Xr = X(r) # X^(n-k)
XrmX = GF2.multPoly(Xr, m) # X^(n-k) * m(X)
p = GF2.modPoly(XrmX, g) # p(X) = (X^(n-k) * m(X)) mod g(X)
c = GF2.addPoly(p, XrmX) # c(X) = p(X) + (X^(n-k) * m(X))
else:
c = GF2.multPoly(m, g)
return c.astype(int)
def gToG(g, n, systematic = True, verbose = False):
"""Builds Generator Matrix G from given Generator Polynomial.
(slide 25)
Args:
g: generator polynomial
n: code length
systematic: generator matrix in systematic form (default: True)
verbose: verbose output on how to gain systematic form (default: False)
Returns:
Generator Matrix
"""
k = n - degree(g)
g = padEnd(g, n)
G = np.empty([k, n])
for i in range(0, k):
G[i,:] = shift(g, i)
# row additions to gain embedded identity matrix on right side
# -> systematic form
if systematic:
G = makeSystematic(G, verbose)
return G.astype(int)
def gToG(g, n, systematic=True, verbose=False):
"""Builds Generator Matrix G from given Generator Polynomial.
(slide 25)
Args:
g: generator polynomial
n: code length
systematic: generator matrix in systematic form (default: True)
verbose: verbose output on how to gain systematic form (default: False)
Returns:
Generator Matrix
"""
k = n - degree(g)
g = padEnd(g, n)
G = np.empty([k, n])
for i in range(0, k):
G[i, :] = shift(g, i)
# row additions to gain embedded identity matrix on right side
# -> systematic form
if systematic:
G = makeSystematic(G, verbose)
return G.astype(int)
def k(self): # override
return self.n() - degree(self.g())
def k(self): # override
return self.n() - degree(self.g())
def decode(self, r, verbose=True):
"""Decode received polynomial r(X) using the Euclidean Algorithm.
(slide 26)
Args:
r: received polynomial
verbose: print the algorithm steps.
"""
GF = self.GF()
if verbose:
print()
print('Decode the received polynomial:')
print('r(X) = ' + self.GF().polyToString(r))
S = self.S(r, verbose)
if verbose:
print('The Euclidean algorithm is applied by constructing the ' + \
'following table:')
ri, ti = GF.HCF(X(2 * self.t()), S, verbose)
lamb = GF.monicMultiplier(ti)
errorLocationPoly = GF.multPoly(lamb, ti)
errorLocationPolyDerivative = GF.derivePoly(errorLocationPoly)
errorEvalutationPoly = GF.multPoly(lamb, ri)
if verbose:
print(u'An element \u03BB \u2208 GF(' + str(GF.q()) +
') is conveniently selected to multiply')
print(
u't_i(X) by, in order to convert it into a monic polynomial.')
print(u'This value of \u03BB is \u03BB = ' +
GF.elementToString(lamb))
print('Therefore:')
print()
print('Error location polynomial:')
print(u'\u03C3(X) = \u03BB * t_i(X) = ' +
GF.elementToString(lamb) + '(' + GF.polyToString(ti) + ')')
print(' = ' + GF.polyToString(errorLocationPoly))
print()
print(u'\u03C3\'(X) = ' +
GF.polyToString(errorLocationPolyDerivative))
print()
print('Error evaluation polynomial:')
print(u'W(X) = -\u03BB * r_i(X) = ' + GF.elementToString(lamb) +
' * ' + GF.polyToString(ri))
print(' = ' +
GF.polyToString(errorEvalutationPoly))
print()
print(
u'Performing Chien search in the error location polynomial \u03C3(X):'
)
print()
errorLocations = []
for i, root in enumerate(GF.roots(errorLocationPoly)):
j = GF.elementToExp(GF.elementFromExp(-GF.elementToExp(root)))
errorLocations.append(j)
if verbose:
print(u'\u03B1^(-j_' + str(i+1) + ') = ' + GF.elementToString(root) + \
'\t-> j_' + str(i+1) + ' = ' + str(j))
if verbose:
print('\nError values:')
errorValues = []
for i, errorLocation in enumerate(errorLocations):
alpha = GF.elementFromExp(-errorLocation)
res_W = GF.substituteElementIntoPoly(errorEvalutationPoly, alpha)
res_o = GF.substituteElementIntoPoly(errorLocationPolyDerivative,
alpha)
errorValue = GF.divElements(res_W, res_o)
errorValues.append(errorValue)
if verbose:
print('e_j' + str(i+1) + ' = W(' + GF.elementToString(alpha) + \
u') / \u03C3\'(' + GF.elementToString(alpha) + ') = ' + \
GF.elementToString(res_W) + \
' / ' + GF.elementToString(res_o) + ' = ' + \
GF.elementToString(errorValue))
e = np.zeros(degree(r) + 1)
for i, errorLocation in enumerate(errorLocations):
errorValue = errorValues[i]
e[errorLocation] = errorValue
c = GF.addPoly(r, e)
if verbose:
print()
print('Error polynomial:')
print('e(X) = ' + GF.polyToString(e))
print()
print('Code vector:')
print('c = r + e = ' + str(c))
return c.astype(int)
def decode(self, r, verbose = True):
"""Decode received polynomial r(X) using the Euclidean Algorithm.
(slide 26)
Args:
r: received polynomial
verbose: print the algorithm steps.
"""
GF = self.GF()
if verbose:
print()
print('Decode the received polynomial:')
print('r(X) = ' + self.GF().polyToString(r))
S = self.S(r, verbose)
if verbose:
print('The Euclidean algorithm is applied by constructing the ' + \
'following table:')
ri, ti = GF.HCF(X(2 * self.t()), S, verbose)
lamb = GF.monicMultiplier(ti)
errorLocationPoly = GF.multPoly(lamb, ti);
errorLocationPolyDerivative = GF.derivePoly(errorLocationPoly)
errorEvalutationPoly = GF.multPoly(lamb, ri);
if verbose:
print(u'An element \u03BB \u2208 GF(' + str(GF.q()) + ') is conveniently selected to multiply')
print(u't_i(X) by, in order to convert it into a monic polynomial.')
print(u'This value of \u03BB is \u03BB = ' + GF.elementToString(lamb))
print('Therefore:')
print()
print('Error location polynomial:')
print(u'\u03C3(X) = \u03BB * t_i(X) = ' + GF.elementToString(lamb) + '(' + GF.polyToString(ti) + ')')
print(' = ' + GF.polyToString(errorLocationPoly) )
print()
print(u'\u03C3\'(X) = ' + GF.polyToString(errorLocationPolyDerivative))
print()
print('Error evaluation polynomial:')
print(u'W(X) = -\u03BB * r_i(X) = ' + GF.elementToString(lamb) + ' * ' + GF.polyToString(ri))
print(' = ' + GF.polyToString(errorEvalutationPoly) )
print()
print(u'Performing Chien search in the error location polynomial \u03C3(X):')
print()
errorLocations = []
for i, root in enumerate(GF.roots(errorLocationPoly)):
j = GF.elementToExp(GF.elementFromExp(-GF.elementToExp(root)))
errorLocations.append(j)
if verbose:
print(u'\u03B1^(-j_' + str(i+1) + ') = ' + GF.elementToString(root) + \
'\t-> j_' + str(i+1) + ' = ' + str(j))
if verbose:
print('\nError values:')
errorValues = []
for i, errorLocation in enumerate(errorLocations):
alpha = GF.elementFromExp(-errorLocation)
res_W = GF.substituteElementIntoPoly(errorEvalutationPoly, alpha)
res_o = GF.substituteElementIntoPoly(errorLocationPolyDerivative, alpha)
errorValue = GF.divElements(res_W, res_o)
errorValues.append(errorValue)
if verbose:
print('e_j' + str(i+1) + ' = W(' + GF.elementToString(alpha) + \
u') / \u03C3\'(' + GF.elementToString(alpha) + ') = ' + \
GF.elementToString(res_W) + \
' / ' + GF.elementToString(res_o) + ' = ' + \
GF.elementToString(errorValue))
e = np.zeros(degree(r)+1)
for i, errorLocation in enumerate(errorLocations):
errorValue = errorValues[i]
e[errorLocation] = errorValue
c = GF.addPoly(r, e)
if verbose:
print()
print('Error polynomial:')
print('e(X) = ' + GF.polyToString(e))
print()
print('Code vector:')
print('c = r + e = ' + str(c))
return c.astype(int)