n = 255
F.<alpha> = GF(n+1, 'a', modulus='primitive')
assert alpha.multiplicative_order() == n
R.<s> = F['s']

# example of searching for roots in this polynomial ring
# print((s**2 * (s-1)).roots())
# consult ?s.roots for more help

k = 223
t = n - k
generator_poly = prod([s - alpha**j for j in range(1, t+1)])
received_bytes = b'O\x04\xc0A\xd1\xceY\x05\xb44\xda\xf6\xe5F_V\xd1N\xf9\xc2\xe2\x01l\xc2\xbb\xf0w:\x01\x8bw\xaa\x00\x01\x02\x03\x04\x05\x06\x07\x08\t\n\x0b\x0c\xab\x0e\x0f\x10\x11\x12\x13\x14\x15\x16\x17\x18\x19\x1a\x1b\x8b\x1d\x1e\x1f !"#$%&\'()*\xa3,-./0123456789C;<=>?@ABCDEFGH#JKLMNOPQRSTUVW\x03YZ[\\]^_`abcdef+hijklmnopqrstuKwxyz{|}~\x7f\x80\x81\x82\x83\x84\xab\x86\x87\x88\x89\x8a\x8b\x8c\x8d\x8e\x8f\x90\x91\x92\x93\x8b\x95\x96\x97\x98\x99\x9a\x9b\x9c\x9d\x9e\x9f\xa0\xa1\xa2\xb3\xa4\xa5\xa6\xa7\xa8\xa9\xaa\xab\xac\xad\xae\xaf\xb0\xb1\xb3\xb3\xb4\xb5\xb6\xb7\xb8\xb9\xba\xbb\xbc\xbd\xbe\xbf\xc0\xc1\xc2\xc3\xc4\xc5\xc6\xc7\xc8\xc9\xca\xcb\xcc\xcd\xce\xcf\xd0\xd1\xd2\xd3\xd4\xd5\xd6\xd7\xd8\xd9\xda\xdb\xdc\xdd\xde'

# produce the error locations and error values in this message

# Given parameters
n = 255
F.<alpha> = GF(n+1, 'a', modulus='primitive')
assert alpha.multiplicative_order() == n
R.<s> = F['s']

# example of searching for roots in this polynomial ring
# print((s**2 * (s-1)).roots())
# consult ?s.roots for more help

# Calculate the number of errors to correct
k = 223
t = n - k

# Generate the generator polynomial for Reed-Solomon code
generator_poly = prod([s - alpha**j for j in range(1, t+1)])

# Example received message
received_bytes = b'O\x04\xc0A\xd1\xceY\x05\xb44\xda\xf6\xe5F_V\xd1N\xf9\xc2\xe2\x01l\xc2\xbb\xf0w:\x01\x8bw\xaa\x00\x01\x02\x03\x04\x05\x06\x07\x08\t\n\x0b\x0c\xab\x0e\x0f\x10\x11\x12\x13\x14\x15\x16\x17\x18\x19\x1a\x1b\x8b\x1d\x1e\x1f !"#$%&\'()*\xa3,-./0123456789C;<=>?@ABCDEFGH#JKLMNOPQRSTUVW\x03YZ[\\]^_`abcdef+hijklmnopqrstuKwxyz{|}~\x7f\x80\x81\x82\x83\x84\xab\x86\x87\x88\x89\x8a\x8b\x8c\x8d\x8e\x8f\x90\x91\x92\x93\x8b\x95\x96\x97\x98\x99\x9a\x9b\x9c\x9d\x9e\x9f\xa0\xa1\xa2\xb3\xa4\xa5\xa6\xa7\xa8\xa9\xaa\xab\xac\xad\xae\xaf\xb0\xb1\xb3\xb3\xb4\xb5\xb6\xb7\xb8\xb9\xba\xbb\xbc\xbd\xbe\xbf\xc0\xc1\xc2\xc3\xc4\xc5\xc6\xc7\xc8\xc9\xca\xcb\xcc\xcd\xce\xcf\xd0\xd1\xd2\xd3\xd4\xd5\xd6\xd7\xd8\xd9\xda\xdb\xdc\xdd\xde'

# Converting receiving bytes to a polynomial
received_polynomial = sum([Integer(received_bytes[i]) * s**i for i in range(len(received_bytes))])

# Calculating syndromes by evaluating the received polynomial at powers of alpha
syndromes = [received_polynomial(alpha**i) for i in range(1, n-k+1)]

# Finding error locator polynomial using Berlekamp-Massey algorithm
err_locator = berlekamp_massey(syndromes, F)

# Finding error locations by calculating the roots of the error locator polynomial
err_locations = err_locator.roots()

# Finding error values by evaluating error evaluator polynomial at error locations
err_values = [received_polynomial(alpha**(-i)) for i in err_locations]

# Correcting errors in received polynomial
correct_polynomial = received_polynomial - sum([err_values[i] * s**(-err_locations[i]) for i in range(len(err_locations))])

# Converting corrected polynomial to bytes
correct_bytes = bytes([int(correct_polynomial.coefficient(s, i)) for i in range(len(received_bytes))])

# Printing error locations and values:
print("Error Locations:", err_locations)
print("Error Values:", err_values)
print("Corrected Message:", correct_bytes)