写给编码人员的里德-所罗门编码

   t h i s
   t h a t
   c o r n

   t h i s a b c d
   t h a t b c d e
   c o r n c d e f
   t h i n d e f g

   t * * * a b * d

  t h i x a x c d
  x h i x a b c d
  t h x x a b c d

Input       101101101001011
Mask      ^ 101010000010010
Output      000111101011001

1 def qr_check_format(fmt):
2    g = 0x537 # = 0b10100110111 in python 2.6+
3    for i in range(4,-1,-1):
4       if fmt & (1 << (i+10)):
5          fmt ^= g << i
6    return fmt

encoded_format = (format<<10) ^ qr_check_format(format<<10)

10100110111 = 1 x10 + 0 x9 + 1 x8 + 0 x7 + 0 x6 + 1 x5 + 1 x4 + 0 x3 + 1 x2 + 1 x + 1 = x10 + x8 + x5 + x4 + x2 + x + 1

 1 def hamming_weight(x):
 2    weight = 0
 3    while x > 0:
 4       weight += x & 1
 5       x >>= 1
 6    return weight
 7 
 8 def qr_decode_format(fmt):
 9    best_fmt = -1
10    best_dist = 15
11    for test_fmt in range(0,32):
12       test_code = (test_fmt<<10) ^ qr_check_format(test_fmt<<10)
13       test_dist = hamming_weight(fmt ^ test_code)
14       if test_dist < best_dist:
15          best_dist = test_dist
16          best_fmt = test_fmt
17       elif test_dist == best_dist:
18          best_fmt = -1
19    return best_fmt

>>> from qr import *
>>> qr_decode_format(int("000111101011001",2))  # no errors
3
>>> qr_decode_format(int("111111101011001",2))  # 3 bit-errors
3
>>> qr_decode_format(int("111011101011001",2))  # 4 bit-errors
-1

   0101 + 0110 = 0101 - 0110 = 0101 XOR 0110 = 0011

(x2 + 1) + (x2 + x) = 2 x2 + x + 1 = 0 x2 + x + 1 = x + 1

1 def gf_add(x, y):
2     return x ^ y
3 
4 def gf_sub(x, y):
5     return x ^ y # in binary galois field, subtraction is just the same as addition (since we mod 2)

1 def cl_mul(x,y):
2     ‘‘‘Bitwise carry-less multiplication on integers‘‘‘
3     z = 0
4     i = 0
5     while (y>>i) > 0:
6         if y & (1<<i):
7             z ^= x<<i
8         i += 1
9     return z

 1 def gf_mult_noLUT(x, y, prim=0):
 2     ‘‘‘Multiplication in Galois Fields without using a precomputed look-up table (and thus it‘s slower)
 3     by using the standard carry-less multiplication + modular reduction using an irreducible prime polynomial‘‘‘
 4 
 5     ### Define bitwise carry-less operations as inner functions ###
 6     def cl_mult(x,y):
 7         ‘‘‘Bitwise carry-less multiplication on integers‘‘‘
 8         z = 0
 9         i = 0
10         while (y>>i) > 0:
11             if y & (1<<i):
12                 z ^= x<<i
13             i += 1
14         return z
15 
16     def bit_length(n):
17         ‘‘‘Compute the position of the most significant bit (1) of an integer. Equivalent to int.bit_length()‘‘‘
18         bits = 0
19         while n >> bits: bits += 1
20         return bits
21 
22     def cl_div(dividend, divisor=None):
23         ‘‘‘Bitwise carry-less long division on integers and returns the remainder‘‘‘
24         # Compute the position of the most significant bit for each integers
25         dl1 = bit_length(dividend)
26         dl2 = bit_length(divisor)
27         # If the dividend is smaller than the divisor, just exit
28         if dl1 < dl2:
29             return dividend
30         # Else, align the most significant 1 of the divisor to the most significant 1 of the dividend (by shifting the divisor)
31         for i in range(dl1-dl2,-1,-1):
32             # Check that the dividend is divisible (useless for the first iteration but important for the next ones)
33             if dividend & (1 << i+dl2-1):
34                 # If divisible, then shift the divisor to align the most significant bits and XOR (carry-less subtraction)
35                 dividend ^= divisor << i
36         return dividend
37     
38     ### Main GF multiplication routine ###
39 
40     # Multiply the gf numbers
41     result = cl_mult(x,y)
42     # Then do a modular reduction (ie, remainder from the division) with an irreducible primitive polynomial so that it stays inside GF bounds
43     if prim > 0:
44         result = cl_div(result, prim)
45 
46     return result

>>> a = 0b10001001
>>> b = 0b00101010
>>> print bin(gf_mult_noLUT(a, b, 0)) # multiplication only
0b1010001111010
>>> print bin(gf_mult_noLUT(a, b, 0x11d)) # multiplication + modular reduction
0b11000011

 1 def gf_mult_noLUT(x, y, prim=0, field_charac_full=256, carryless=True):
 2     ‘‘‘Galois Field integer multiplication using Russian Peasant Multiplication algorithm (faster than the standard multiplication + modular reduction).
 3     If prim is 0 and carryless=False, then the function produces the result for a standard integers multiplication (no carry-less arithmetics nor modular reduction).‘‘‘
 4     r = 0
 5     while y: # while y is above 0
 6         if y & 1: r = r ^ x if carryless else r + x # y is odd, then add the corresponding x to r (the sum of all x‘s corresponding to odd y‘s will give the final product). Note that since we‘re in GF(2), the addition is in fact an XOR (very important because in GF(2) the multiplication and additions are carry-less, thus it changes the result!).
 7         y = y >> 1 # equivalent to y // 2
 8         x = x << 1 # equivalent to x*2
 9         if prim > 0 and x & field_charac_full: x = x ^ prim # GF modulo: if x >= 256 then apply modular reduction using the primitive polynomial (we just subtract, but since the primitive number can be above 256 then we directly XOR).
10 
11     return r

　　10001001 * 00101010 =α74*α142=α74+ 142 =α216= 11000011

 1 gf_exp = [0] * 512 # Create list of 512 elements. In Python 2.6+, consider using bytearray
 2 gf_log = [0] * 256
 3 
 4 def init_tables(prim=0x11d):
 5     ‘‘‘Precompute the logarithm and anti-log tables for faster computation later, using the provided primitive polynomial.‘‘‘
 6     # prim is the primitive (binary) polynomial. Since it‘s a polynomial in the binary sense,
 7     # it‘s only in fact a single galois field value between 0 and 255, and not a list of gf values.
 8     global gf_exp, gf_log
 9     gf_exp = [0] * 512 # anti-log (exponential) table
10     gf_log = [0] * 256 # log table
11     # For each possible value in the galois field 2^8, we will pre-compute the logarithm and anti-logarithm (exponential) of this value
12     x = 1
13     for i in range(0, 255):
14         gf_exp[i] = x # compute anti-log for this value and store it in a table
15         gf_log[x] = i # compute log at the same time
16         x = gf_mult_noLUT(x, 2, prim)
17 
18         # If you use only generator==2 or a power of 2, you can use the following which is faster than gf_mult_noLUT():
19         #x <<= 1 # multiply by 2 (change 1 by another number y to multiply by a power of 2^y)
20         #if x & 0x100: # similar to x >= 256, but a lot faster (because 0x100 == 256)
21             #x ^= prim # substract the primary polynomial to the current value (instead of 255, so that we get a unique set made of coprime numbers), this is the core of the tables generation
22 
23     # Optimization: double the size of the anti-log table so that we don‘t need to mod 255 to
24     # stay inside the bounds (because we will mainly use this table for the multiplication of two GF numbers, no more).
25     for i in range(255, 512):
26         gf_exp[i] = gf_exp[i - 255]
27     return [gf_log, gf_exp]

1 def gf_mul(x,y):
2     if x==0 or y==0:
3         return 0
4     return gf_exp[gf_log[x] + gf_log[y]] # should be gf_exp[(gf_log[x]+gf_log[y])%255] if gf_exp wasn‘t oversized

1 def gf_div(x,y):
2     if y==0:
3         raise ZeroDivisionError()
4     if x==0:
5         return 0
6     return gf_exp[(gf_log[x] + 255 - gf_log[y]) % 255]

1 def gf_pow(x, power):
2     return gf_exp[(gf_log[x] * power) % 255]
3 
4 def gf_inverse(x):
5     return gf_exp[255 - gf_log[x]] # gf_inverse(x) == gf_div(1, x)

00000001 x4 + 00001111 x3 + 00110110 x2 + 01111000 x + 01000000 = 01 x4 + 0f x3 + 36 x2 + 78 x + 40

1 def gf_poly_scale(p,x):
2     r = [0] * len(p)
3     for i in range(0, len(p)):
4         r[i] = gf_mul(p[i], x)
5     return r

1 def gf_poly_add(p,q):
2     r = [0] * max(len(p),len(q))
3     for i in range(0,len(p)):
4         r[i+len(r)-len(p)] = p[i]
5     for i in range(0,len(q)):
6         r[i+len(r)-len(q)] ^= q[i]
7     return r

 1 def gf_poly_mul(p,q):
 2     ‘‘‘Multiply two polynomials, inside Galois Field‘‘‘
 3     # Pre-allocate the result array
 4     r = [0] * (len(p)+len(q)-1)
 5     # Compute the polynomial multiplication (just like the outer product of two vectors,
 6     # we multiply each coefficients of p with all coefficients of q)
 7     for j in range(0, len(q)):
 8         for i in range(0, len(p)):
 9             r[i+j] ^= gf_mul(p[i], q[j]) # equivalent to: r[i + j] = gf_add(r[i+j], gf_mul(p[i], q[j]))
10                                                          # -- you can see it‘s your usual polynomial multiplication
11     return r

1 def gf_poly_eval(poly, x):
2     ‘‘‘Evaluates a polynomial in GF(2^p) given the value for x. This is based on Horner‘s scheme for maximum efficiency.‘‘‘
3     y = poly[0]
4     for i in range(1, len(poly)):
5         y = gf_mul(y, x) ^ poly[i]
6     return y

1 class ReedSolomonError(Exception):
2     pass

1 raise ReedSolomonError("Some error message")

1 try:
2     raise ReedSolomonError("Some error message")
3 except ReedSolomonError, e:
4     pass # do something here

1 def rs_generator_poly(nsym):
2     ‘‘‘Generate an irreducible generator polynomial (necessary to encode a message into Reed-Solomon)‘‘‘
3     g = [1]
4     for i in range(0, nsym):
5         g = gf_poly_mul(g, [1, gf_pow(2, i)])
6     return g

 1 def gf_poly_div(dividend, divisor):
 2     ‘‘‘Fast polynomial division by using Extended Synthetic Division and optimized for GF(2^p) computations
 3     (doesn‘t work with standard polynomials outside of this galois field, see the Wikipedia article for generic algorithm).‘‘‘
 4     # CAUTION: this function expects polynomials to follow the opposite convention at decoding:
 5     # the terms must go from the biggest to lowest degree (while most other functions here expect
 6     # a list from lowest to biggest degree). eg: 1 + 2x + 5x^2 = [5, 2, 1], NOT [1, 2, 5]
 7 
 8     msg_out = list(dividend) # Copy the dividend
 9     #normalizer = divisor[0] # precomputing for performance
10     for i in range(0, len(dividend) - (len(divisor)-1)):
11         #msg_out[i] /= normalizer # for general polynomial division (when polynomials are non-monic), the usual way of using
12                                   # synthetic division is to divide the divisor g(x) with its leading coefficient, but not needed here.
13         coef = msg_out[i] # precaching
14         if coef != 0: # log(0) is undefined, so we need to avoid that case explicitly (and it‘s also a good optimization).
15             for j in range(1, len(divisor)): # in synthetic division, we always skip the first coefficient of the divisior,
16                                               # because it‘s only used to normalize the dividend coefficient
17                 if divisor[j] != 0: # log(0) is undefined
18                     msg_out[i + j] ^= gf_mul(divisor[j], coef) # equivalent to the more mathematically correct
19                                                                # (but xoring directly is faster): msg_out[i + j] += -divisor[j] * coef
20 
21     # The resulting msg_out contains both the quotient and the remainder, the remainder being the size of the divisor
22     # (the remainder has necessarily the same degree as the divisor -- not length but degree == length-1 -- since it‘s
23     # what we couldn‘t divide from the dividend), so we compute the index where this separation is, and return the quotient and remainder.
24     separator = -(len(divisor)-1)
25     return msg_out[:separator], msg_out[separator:] # return quotient, remainder.

 1 def rs_encode_msg(msg_in, nsym):
 2     ‘‘‘Reed-Solomon main encoding function‘‘‘
 3     gen = rs_generator_poly(nsym)
 4 
 5     # Pad the message, then divide it by the irreducible generator polynomial
 6     _, remainder = gf_poly_div(msg_in + [0] * (len(gen)-1), gen)
 7     # The remainder is our RS code! Just append it to our original message to get our full codeword (this represents a polynomial of max 256 terms)
 8     msg_out = msg_in + remainder
 9     # Return the codeword
10     return msg_out

 1 def rs_encode_msg(msg_in, nsym):
 2     ‘‘‘Reed-Solomon main encoding function, using polynomial division (algorithm Extended Synthetic Division)‘‘‘
 3     if (len(msg_in) + nsym) > 255: raise ValueError("Message is too long (%i when max is 255)" % (len(msg_in)+nsym))
 4     gen = rs_generator_poly(nsym)
 5     # Init msg_out with the values inside msg_in and pad with len(gen)-1 bytes (which is the number of ecc symbols).
 6     msg_out = [0] * (len(msg_in) + len(gen)-1)
 7     # Initializing the Synthetic Division with the dividend (= input message polynomial)
 8     msg_out[:len(msg_in)] = msg_in
 9 
10     # Synthetic division main loop
11     for i in range(len(msg_in)):
12         # Note that it‘s msg_out here, not msg_in. Thus, we reuse the updated value at each iteration
13         # (this is how Synthetic Division works: instead of storing in a temporary register the intermediate values,
14         # we directly commit them to the output).
15         coef = msg_out[i]
16 
17         # log(0) is undefined, so we need to manually check for this case. There‘s no need to check
18         # the divisor here because we know it can‘t be 0 since we generated it.
19         if coef != 0:
20             # in synthetic division, we always skip the first coefficient of the divisior, because it‘s only used to normalize the dividend coefficient (which is here useless since the divisor, the generator polynomial, is always monic)
21             for j in range(1, len(gen)):
22                 msg_out[i+j] ^= gf_mul(gen[j], coef) # equivalent to msg_out[i+j] += gf_mul(gen[j], coef)
23 
24     # At this point, the Extended Synthetic Divison is done, msg_out contains the quotient in msg_out[:len(msg_in)]
25     # and the remainder in msg_out[len(msg_in):]. Here for RS encoding, we don‘t need the quotient but only the remainder
26     # (which represents the RS code), so we can just overwrite the quotient with the input message, so that we get
27     # our complete codeword composed of the message + code.
28     msg_out[:len(msg_in)] = msg_in
29 
30     return msg_out

>>> msg_in = [ 0x40, 0xd2, 0x75, 0x47, 0x76, 0x17, 0x32, 0x06,
...            0x27, 0x26, 0x96, 0xc6, 0xc6, 0x96, 0x70, 0xec ]
>>> msg = rs_encode_msg(msg_in, 10)
>>> for i in range(0,len(msg)):
...    print(hex(msg[i]), end=‘ ‘)
... 
0x40 0xd2 0x75 0x47 0x76 0x17 0x32 0x6 0x27 0x26 0x96 0xc6 0xc6 0x96 0x70 0xec
0xbc 0x2a 0x90 0x13 0x6b 0xaf 0xef 0xfd 0x4b 0xe0

 1 def rs_calc_syndromes(msg, nsym):
 2     ‘‘‘Given the received codeword msg and the number of error correcting symbols (nsym), computes the syndromes polynomial.
 3     Mathematically, it‘s essentially equivalent to a Fourrier Transform (Chien search being the inverse).
 4     ‘‘‘
 5     # Note the "[0] +" : we add a 0 coefficient for the lowest degree (the constant). This effectively shifts the syndrome, and will shift every computations depending on the syndromes (such as the errors locator polynomial, errors evaluator polynomial, etc. but not the errors positions).
 6     # This is not necessary, you can adapt subsequent computations to start from 0 instead of skipping the first iteration (ie, the often seen range(1, n-k+1)),
 7     synd = [0] * nsym
 8     for i in range(0, nsym):
 9         synd[i] = gf_poly_eval(msg, gf_pow(2,i))
10     return [0] + synd # pad with one 0 for mathematical precision (else we can end up with weird calculations sometimes)

>>> synd = rs_calc_syndromes(msg, 10)
>>> print(synd)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0] # not corrupted message = all 0 syndromes
>>> msg[0] = 0  # deliberately damage the message
>>> synd = rs_calc_syndromes(msg, 10)
>>> print(synd)
[64, 192, 93, 231, 52, 92, 228, 49, 83, 245] # when corrupted, the syndromes will be non zero

1 def rs_check(msg, nsym):
2     ‘‘‘Returns true if the message + ecc has no error of false otherwise (may not always catch a wrong decoding or a wrong message, particularly if there are too many errors -- above the Singleton bound --, but it usually does)‘‘‘
3     return ( max(rs_calc_syndromes(msg, nsym)) == 0 )

 1 def rs_find_errata_locator(e_pos):
 2     ‘‘‘Compute the erasures/errors/errata locator polynomial from the erasures/errors/errata positions
 3        (the positions must be relative to the x coefficient, eg: "hello worldxxxxxxxxx" is tampered to "h_ll_ worldxxxxxxxxx"
 4        with xxxxxxxxx being the ecc of length n-k=9, here the string positions are [1, 4], but the coefficients are reversed
 5        since the ecc characters are placed as the first coefficients of the polynomial, thus the coefficients of the
 6        erased characters are n-1 - [1, 4] = [18, 15] = erasures_loc to be specified as an argument.‘‘‘
 7 
 8     e_loc = [1] # just to init because we will multiply, so it must be 1 so that the multiplication starts correctly without nulling any term
 9     # erasures_loc = product(1 - x*alpha**i) for i in erasures_pos and where alpha is the alpha chosen to evaluate polynomials.
10     for i in e_pos:
11         e_loc = gf_poly_mul( e_loc, gf_poly_add([1], [gf_pow(2, i), 0]) )
12     return e_loc

 1 def rs_find_error_evaluator(synd, err_loc, nsym):
 2     ‘‘‘Compute the error (or erasures if you supply sigma=erasures locator polynomial, or errata) evaluator polynomial Omega
 3        from the syndrome and the error/erasures/errata locator Sigma.‘‘‘
 4 
 5     # Omega(x) = [ Synd(x) * Error_loc(x) ] mod x^(n-k+1)
 6     _, remainder = gf_poly_div( gf_poly_mul(synd, err_loc), ([1] + [0]*(nsym+1)) ) # first multiply syndromes * errata_locator, then do a
 7                                                                                    # polynomial division to truncate the polynomial to the
 8                                                                                    # required length
 9 
10     # Faster way that is equivalent
11     #remainder = gf_poly_mul(synd, err_loc) # first multiply the syndromes with the errata locator polynomial
12     #remainder = remainder[len(remainder)-(nsym+1):] # then slice the list to truncate it (which represents the polynomial), which
13                                                                           # is equivalent to dividing by a polynomial of the length we want
14 
15     return remainder

 1 def rs_correct_errata(msg_in, synd, err_pos): # err_pos is a list of the positions of the errors/erasures/errata
 2     ‘‘‘Forney algorithm, computes the values (error magnitude) to correct the input message.‘‘‘
 3     # calculate errata locator polynomial to correct both errors and erasures (by combining the errors positions given by the error locator polynomial found by BM with the erasures positions given by caller)
 4     coef_pos = [len(msg_in) - 1 - p for p in err_pos] # need to convert the positions to coefficients degrees for the errata locator algo to work (eg: instead of [0, 1, 2] it will become [len(msg)-1, len(msg)-2, len(msg) -3])
 5     err_loc = rs_find_errata_locator(coef_pos)
 6     # calculate errata evaluator polynomial (often called Omega or Gamma in academic papers)
 7     err_eval = rs_find_error_evaluator(synd[::-1], err_loc, len(err_loc)-1)[::-1]
 8 
 9     # Second part of Chien search to get the error location polynomial X from the error positions in err_pos (the roots of the error locator polynomial, ie, where it evaluates to 0)
10     X = [] # will store the position of the errors
11     for i in range(0, len(coef_pos)):
12         l = 255 - coef_pos[i]
13         X.append( gf_pow(2, -l) )
14 
15     # Forney algorithm: compute the magnitudes
16     E = [0] * (len(msg_in)) # will store the values that need to be corrected (substracted) to the message containing errors. This is sometimes called the error magnitude polynomial.
17     Xlength = len(X)
18     for i, Xi in enumerate(X):
19 
20         Xi_inv = gf_inverse(Xi)
21 
22         # Compute the formal derivative of the error locator polynomial (see Blahut, Algebraic codes for data transmission, pp 196-197).
23         # the formal derivative of the errata locator is used as the denominator of the Forney Algorithm, which simply says that the ith error value is given by error_evaluator(gf_inverse(Xi)) / error_locator_derivative(gf_inverse(Xi)). See Blahut, Algebraic codes for data transmission, pp 196-197.
24         err_loc_prime_tmp = []
25         for j in range(0, Xlength):
26             if j != i:
27                 err_loc_prime_tmp.append( gf_sub(1, gf_mul(Xi_inv, X[j])) )
28         # compute the product, which is the denominator of the Forney algorithm (errata locator derivative)
29         err_loc_prime = 1
30         for coef in err_loc_prime_tmp:
31             err_loc_prime = gf_mul(err_loc_prime, coef)
32         # equivalent to: err_loc_prime = functools.reduce(gf_mul, err_loc_prime_tmp, 1)
33 
34         # Compute y (evaluation of the errata evaluator polynomial)
35         # This is a more faithful translation of the theoretical equation contrary to the old forney method. Here it is an exact reproduction:
36         # Yl = omega(Xl.inverse()) / prod(1 - Xj*Xl.inverse()) for j in len(X)
37         y = gf_poly_eval(err_eval[::-1], Xi_inv) # numerator of the Forney algorithm (errata evaluator evaluated)
38         y = gf_mul(gf_pow(Xi, 1), y)
39         
40         # Compute the magnitude
41         magnitude = gf_div(y, err_loc_prime) # magnitude value of the error, calculated by the Forney algorithm (an equation in fact): dividing the errata evaluator with the errata locator derivative gives us the errata magnitude (ie, value to repair) the ith symbol
42         E[err_pos[i]] = magnitude # store the magnitude for this error into the magnitude polynomial
43 
44     # Apply the correction of values to get our message corrected! (note that the ecc bytes also gets corrected!)
45     # (this isn‘t the Forney algorithm, we just apply the result of decoding here)
46     msg_in = gf_poly_add(msg_in, E) # equivalent to Ci = Ri - Ei where Ci is the correct message, Ri the received (senseword) message, and Ei the errata magnitudes (minus is replaced by XOR since it‘s equivalent in GF(2^p)). So in fact here we substract from the received message the errors magnitude, which logically corrects the value to what it should be.
47     return msg_in

>>> msg[0] = 0
>>> synd = rs_calc_syndromes(msg, 10)
>>> msg = rs_correct_errata(msg, synd, [0]) # [0] is the list of the erasures locations, here it‘s the first character, at position 0
>>> print(hex(msg[0]))
0x40

 1 def rs_find_error_locator(synd, nsym, erase_loc=None, erase_count=0):
 2     ‘‘‘Find error/errata locator and evaluator polynomials with Berlekamp-Massey algorithm‘‘‘
 3     # The idea is that BM will iteratively estimate the error locator polynomial.
 4     # To do this, it will compute a Discrepancy term called Delta, which will tell us if the error locator polynomial needs an update or not
 5     # (hence why it‘s called discrepancy: it tells us when we are getting off board from the correct value).
 6 
 7     # Init the polynomials
 8     if erase_loc: # if the erasure locator polynomial is supplied, we init with its value, so that we include erasures in the final locator polynomial
 9         err_loc = list(erase_loc)
10         old_loc = list(erase_loc)
11     else:
12         err_loc = [1] # This is the main variable we want to fill, also called Sigma in other notations or more formally the errors/errata locator polynomial.
13         old_loc = [1] # BM is an iterative algorithm, and we need the errata locator polynomial of the previous iteration in order to update other necessary variables.
14     #L = 0 # update flag variable, not needed here because we use an alternative equivalent way of checking if update is needed (but using the flag could potentially be faster depending on if using length(list) is taking linear time in your language, here in Python it‘s constant so it‘s as fast.
15 
16     # Fix the syndrome shifting: when computing the syndrome, some implementations may prepend a 0 coefficient for the lowest degree term (the constant). This is a case of syndrome shifting, thus the syndrome will be bigger than the number of ecc symbols (I don‘t know what purpose serves this shifting). If that‘s the case, then we need to account for the syndrome shifting when we use the syndrome such as inside BM, by skipping those prepended coefficients.
17     # Another way to detect the shifting is to detect the 0 coefficients: by definition, a syndrome does not contain any 0 coefficient (except if there are no errors/erasures, in this case they are all 0). This however doesn‘t work with the modified Forney syndrome, which set to 0 the coefficients corresponding to erasures, leaving only the coefficients corresponding to errors.
18     synd_shift = 0
19     if len(synd) > nsym: synd_shift = len(synd) - nsym
20 
21     for i in range(0, nsym-erase_count): # generally: nsym-erase_count == len(synd), except when you input a partial erase_loc and using the full syndrome instead of the Forney syndrome, in which case nsym-erase_count is more correct (len(synd) will fail badly with IndexError).
22         if erase_loc: # if an erasures locator polynomial was provided to init the errors locator polynomial, then we must skip the FIRST erase_count iterations (not the last iterations, this is very important!)
23             K = erase_count+i+synd_shift
24         else: # if erasures locator is not provided, then either there‘s no erasures to account or we use the Forney syndromes, so we don‘t need to use erase_count nor erase_loc (the erasures have been trimmed out of the Forney syndromes).
25             K = i+synd_shift
26 
27         # Compute the discrepancy Delta
28         # Here is the close-to-the-books operation to compute the discrepancy Delta: it‘s a simple polynomial multiplication of error locator with the syndromes, and then we get the Kth element.
29         #delta = gf_poly_mul(err_loc[::-1], synd)[K] # theoretically it should be gf_poly_add(synd[::-1], [1])[::-1] instead of just synd, but it seems it‘s not absolutely necessary to correctly decode.
30         # But this can be optimized: since we only need the Kth element, we don‘t need to compute the polynomial multiplication for any other element but the Kth. Thus to optimize, we compute the polymul only at the item we need, skipping the rest (avoiding a nested loop, thus we are linear time instead of quadratic).
31         # This optimization is actually described in several figures of the book "Algebraic codes for data transmission", Blahut, Richard E., 2003, Cambridge university press.
32         delta = synd[K]
33         for j in range(1, len(err_loc)):
34             delta ^= gf_mul(err_loc[-(j+1)], synd[K - j]) # delta is also called discrepancy. Here we do a partial polynomial multiplication (ie, we compute the polynomial multiplication only for the term of degree K). Should be equivalent to brownanrs.polynomial.mul_at().
35         #print "delta", K, delta, list(gf_poly_mul(err_loc[::-1], synd)) # debugline
36 
37         # Shift polynomials to compute the next degree
38         old_loc = old_loc + [0]
39 
40         # Iteratively estimate the errata locator and evaluator polynomials
41         if delta != 0: # Update only if there‘s a discrepancy
42             if len(old_loc) > len(err_loc): # Rule B (rule A is implicitly defined because rule A just says that we skip any modification for this iteration)
43             #if 2*L <= K+erase_count: # equivalent to len(old_loc) > len(err_loc), as long as L is correctly computed
44                 # Computing errata locator polynomial Sigma
45                 new_loc = gf_poly_scale(old_loc, delta)
46                 old_loc = gf_poly_scale(err_loc, gf_inverse(delta)) # effectively we are doing err_loc * 1/delta = err_loc // delta
47                 err_loc = new_loc
48                 # Update the update flag
49                 #L = K - L # the update flag L is tricky: in Blahut‘s schema, it‘s mandatory to use `L = K - L - erase_count` (and indeed in a previous draft of this function, if you forgot to do `- erase_count` it would lead to correcting only 2*(errors+erasures) <= (n-k) instead of 2*errors+erasures <= (n-k)), but in this latest draft, this will lead to a wrong decoding in some cases where it should correctly decode! Thus you should try with and without `- erase_count` to update L on your own implementation and see which one works OK without producing wrong decoding failures.
50 
51             # Update with the discrepancy
52             err_loc = gf_poly_add(err_loc, gf_poly_scale(old_loc, delta))
53 
54     # Check if the result is correct, that there‘s not too many errors to correct
55     while len(err_loc) and err_loc[0] == 0: del err_loc[0] # drop leading 0s, else errs will not be of the correct size
56     errs = len(err_loc) - 1
57     if (errs-erase_count) * 2 + erase_count > nsym:
58         raise ReedSolomonError("Too many errors to correct")    # too many errors to correct
59 
60     return err_loc

 1 def rs_find_errors(err_loc, nmess): # nmess is len(msg_in)
 2     ‘‘‘Find the roots (ie, where evaluation = zero) of error polynomial by brute-force trial, this is a sort of Chien‘s search
 3     (but less efficient, Chien‘s search is a way to evaluate the polynomial such that each evaluation only takes constant time).‘‘‘
 4     errs = len(err_loc) - 1
 5     err_pos = []
 6     for i in range(nmess): # normally we should try all 2^8 possible values, but here we optimize to just check the interesting symbols
 7         if gf_poly_eval(err_loc, gf_pow(2, i)) == 0: # It‘s a 0? Bingo, it‘s a root of the error locator polynomial,
 8                                                                                    # in other terms this is the location of an error
 9             err_pos.append(nmess - 1 - i)
10     # Sanity check: the number of errors/errata positions found should be exactly the same as the length of the errata locator polynomial
11     if len(err_pos) != errs:
12         # couldn‘t find error locations
13         raise ReedSolomonError("Too many (or few) errors found by Chien Search for the errata locator polynomial!")
14     return err_pos

>>> print(hex(msg[10]))
0x96
>>> msg[0] = 6
>>> msg[10] = 7
>>> msg[20] = 8
>>> synd = rs_calc_syndromes(msg, 10)
>>> err_loc = rs_find_error_locator(synd, nsym)
>>> pos = rs_find_errors(err_loc[::-1], len(msg)) # find the errors locations
>>> print(pos)
[20, 10, 0]
>>> msg = rs_correct_errata(msg, synd, pos)
>>> print(hex(msg[10]))
0x96

 1 def rs_forney_syndromes(synd, pos, nmess):
 2     # Compute Forney syndromes, which computes a modified syndromes to compute only errors (erasures are trimmed out). Do not confuse this with Forney algorithm, which allows to correct the message based on the location of errors.
 3     erase_pos_reversed = [nmess-1-p for p in pos] # prepare the coefficient degree positions (instead of the erasures positions)
 4 
 5     # Optimized method, all operations are inlined
 6     fsynd = list(synd[1:])      # make a copy and trim the first coefficient which is always 0 by definition
 7     for i in range(0, len(pos)):
 8         x = gf_pow(2, erase_pos_reversed[i])
 9         for j in range(0, len(fsynd) - 1):
10             fsynd[j] = gf_mul(fsynd[j], x) ^ fsynd[j + 1]
11 
12     # Equivalent, theoretical way of computing the modified Forney syndromes: fsynd = (erase_loc * synd) % x^(n-k)
13     # See Shao, H. M., Truong, T. K., Deutsch, L. J., & Reed, I. S. (1986, April). A single chip VLSI Reed-Solomon decoder. In Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP‘86. (Vol. 11, pp. 2151-2154). IEEE.ISO 690
14     #erase_loc = rs_find_errata_locator(erase_pos_reversed, generator=generator) # computing the erasures locator polynomial
15     #fsynd = gf_poly_mul(erase_loc[::-1], synd[1:]) # then multiply with the syndrome to get the untrimmed forney syndrome
16     #fsynd = fsynd[len(pos):] # then trim the first erase_pos coefficients which are useless. Seems to be not necessary, but this reduces the computation time later in BM (thus it‘s an optimization).
17 
18     return fsynd

 1 def rs_correct_msg(msg_in, nsym, erase_pos=None):
 2     ‘‘‘Reed-Solomon main decoding function‘‘‘
 3     if len(msg_in) > 255: # can‘t decode, message is too big
 4         raise ValueError("Message is too long (%i when max is 255)" % len(msg_in))
 5 
 6     msg_out = list(msg_in)     # copy of message
 7     # erasures: set them to null bytes for easier decoding (but this is not necessary, they will be corrected anyway, but debugging will be easier with null bytes because the error locator polynomial values will only depend on the errors locations, not their values)
 8     if erase_pos is None:
 9         erase_pos = []
10     else:
11         for e_pos in erase_pos:
12             msg_out[e_pos] = 0
13     # check if there are too many erasures to correct (beyond the Singleton bound)
14     if len(erase_pos) > nsym: raise ReedSolomonError("Too many erasures to correct")
15     # prepare the syndrome polynomial using only errors (ie: errors = characters that were either replaced by null byte
16     # or changed to another character, but we don‘t know their positions)
17     synd = rs_calc_syndromes(msg_out, nsym)
18     # check if there‘s any error/erasure in the input codeword. If not (all syndromes coefficients are 0), then just return the message as-is.
19     if max(synd) == 0:
20         return msg_out[:-nsym], msg_out[-nsym:]  # no errors
21 
22     # compute the Forney syndromes, which hide the erasures from the original syndrome (so that BM will just have to deal with errors, not erasures)
23     fsynd = rs_forney_syndromes(synd, erase_pos, len(msg_out))
24     # compute the error locator polynomial using Berlekamp-Massey
25     err_loc = rs_find_error_locator(fsynd, nsym, erase_count=len(erase_pos))
26     # locate the message errors using Chien search (or brute-force search)
27     err_pos = rs_find_errors(err_loc[::-1] , len(msg_out))
28     if err_pos is None:
29         raise ReedSolomonError("Could not locate error")    # error location failed
30 
31     # Find errors values and apply them to correct the message
32     # compute errata evaluator and errata magnitude polynomials, then correct errors and erasures
33     msg_out = rs_correct_errata(msg_out, synd, (erase_pos + err_pos)) # note that we here use the original syndrome, not the forney syndrome
34                                                                                                                                   # (because we will correct both errors and erasures, so we need the full syndrome)
35     # check if the final message is fully repaired
36     synd = rs_calc_syndromes(msg_out, nsym)
37     if max(synd) > 0:
38         raise ReedSolomonError("Could not correct message")     # message could not be repaired
39     # return the successfully decoded message
40     return msg_out[:-nsym], msg_out[-nsym:] # also return the corrected ecc block so that the user can check()

 1 # Configuration of the parameters and input message
 2 prim = 0x11d
 3 n = 20 # set the size you want, it must be > k, the remaining n-k symbols will be the ECC code (more is better)
 4 k = 11 # k = len(message)
 5 message = "hello world" # input message
 6 
 7 # Initializing the log/antilog tables
 8 init_tables(prim)
 9 
10 # Encoding the input message
11 mesecc = rs_encode_msg([ord(x) for x in message], n-k)
12 print("Original: %s" % mesecc)
13 
14 # Tampering 6 characters of the message (over 9 ecc symbols, so we are above the Singleton Bound)
15 mesecc[0] = 0
16 mesecc[1] = 2
17 mesecc[2] = 2
18 mesecc[3] = 2
19 mesecc[4] = 2
20 mesecc[5] = 2
21 print("Corrupted: %s" % mesecc)
22 
23 # Decoding/repairing the corrupted message, by providing the locations of a few erasures, we get below the Singleton Bound
24 # Remember that the Singleton Bound is: 2*e+v <= (n-k)
25 corrected_message, corrected_ecc = rs_correct_msg(mesecc, n-k, erase_pos=[0, 1, 2])
26 print("Repaired: %s" % (corrected_message+corrected_ecc))
27 print(‘‘.join([chr(x) for x in corrected_message]))

Original:  [104, 101, 108, 108, 111,  32, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
Corrupted: [  0,   2,   2,   2,   2,   2, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
Repaired:  [104, 101, 108, 108, 111,  32, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
hello world