def _berlekamp_massey(self, s, k=None, erasures_loc=None, erasures_eval=None, erasures_count=0):
        '''Computes and returns the errata (errors+erasures) locator polynomial (sigma) and the
        error evaluator polynomial (omega) at the same time.
        If the erasures locator is specified, we will return an errors-and-erasures locator polynomial and an errors-and-erasures evaluator polynomial, else it will compute only errors. With erasures in addition to errors, it can simultaneously decode up to v+2e <= (n-k) where v is the number of erasures and e the number of errors.
        Mathematically speaking, this is equivalent to a spectral analysis (see Blahut, "Algebraic Codes for Data Transmission", 2003, chapter 7.6 Decoding in Time Domain).
        The parameter s is the syndrome polynomial (syndromes encoded in a
        generator function) as returned by _syndromes.

        Notes:
        The error polynomial:
        E(x) = E_0 + E_1 x + ... + E_(n-1) x^(n-1)

        j_1, j_2, ..., j_s are the error positions. (There are at most s
        errors)

        Error location X_i is defined: X_i = α^(j_i)
        that is, the power of α (alpha) corresponding to the error location

        Error magnitude Y_i is defined: E_(j_i)
        that is, the coefficient in the error polynomial at position j_i

        Error locator polynomial:
        sigma(z) = Product( 1 - X_i * z, i=1..s )
        roots are the reciprocals of the error locations
        ( 1/X_1, 1/X_2, ...)

        Error evaluator polynomial omega(z) is here computed at the same time as sigma, but it can also be constructed afterwards using the syndrome and sigma (see _find_error_evaluator() method).

        It can be seen that the algorithm tries to iteratively solve for the error locator polynomial by
        solving one equation after another and updating the error locator polynomial. If it turns out that it
        cannot solve the equation at some step, then it computes the error and weights it by the last
        non-zero discriminant found, and delays the weighted result to increase the polynomial degree
        by 1. Ref: "Reed Solomon Decoder: TMS320C64x Implementation" by Jagadeesh Sankaran, December 2000, Application Report SPRA686

        The best paper I found describing the BM algorithm for errata (errors-and-erasures) evaluator computation is in "Algebraic Codes for Data Transmission", Richard E. Blahut, 2003.
        '''
        # For errors-and-erasures decoding, see: "Algebraic Codes for Data Transmission", Richard E. Blahut, 2003 and (but it's less complete): Blahut, Richard E. "Transform techniques for error control codes." IBM Journal of Research and development 23.3 (1979): 299-315. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.92.600&rep=rep1&type=pdf and also a MatLab implementation here: http://www.mathworks.com/matlabcentral/fileexchange/23567-reed-solomon-errors-and-erasures-decoder/content//RS_E_E_DEC.m
        # also see: Blahut, Richard E. "A universal Reed-Solomon decoder." IBM Journal of Research and Development 28.2 (1984): 150-158. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.2084&rep=rep1&type=pdf
        # and another good alternative book with concrete programming examples: Jiang, Yuan. A practical guide to error-control coding using Matlab. Artech House, 2010.
        n = self.n
        if not k: k = self.k

        # Initialize, depending on if we include erasures or not:
        if erasures_loc:
            sigma = [ Polynomial(erasures_loc.coefficients) ] # copy erasures_loc by creating a new Polynomial, so that we initialize the errata locator polynomial with the erasures locator polynomial.
            B = [ Polynomial(erasures_loc.coefficients) ]
            omega = [ Polynomial(erasures_eval.coefficients) ] # to compute omega (the evaluator polynomial) at the same time, we also need to initialize it with the partial erasures evaluator polynomial
            A = [ Polynomial(erasures_eval.coefficients) ] # TODO: fix the initial value of the evaluator support polynomial, because currently the final omega is not correct (it contains higher order terms that should be removed by the end of BM)
        else:
            sigma = [ Polynomial([GF2int(1)]) ] # error locator polynomial. Also called Lambda in other notations.
            B = [ Polynomial([GF2int(1)]) ] # this is the error locator support/secondary polynomial, which is a funky way to say that it's just a temporary variable that will help us construct sigma, the error locator polynomial
            omega = [ Polynomial([GF2int(1)]) ] # error evaluator polynomial. We don't need to initialize it with erasures_loc, it will still work, because Delta is computed using sigma, which itself is correctly initialized with erasures if needed.
            A = [ Polynomial([GF2int(0)]) ] # this is the error evaluator support/secondary polynomial, to help us construct omega
        L = [ 0 ] # update flag: necessary variable to check when updating is necessary and to check bounds (to avoid wrongly eliminating the higher order terms). For more infos, see https://www.cs.duke.edu/courses/spring11/cps296.3/decoding_rs.pdf
        M = [ 0 ] # optional variable to check bounds (so that we do not mistakenly overwrite the higher order terms). This is not necessary, it's only an additional safe check. For more infos, see the presentation decoding_rs.pdf by Andrew Brown in the doc folder.

        # 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.
        # 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 (that we do not use in this lib but it may be implemented in the future), which set to 0 the coefficients corresponding to erasures, leaving only the coefficients corresponding to errors.
        synd_shift = 0
        if len(s) > (n-k): synd_shift = len(s) - (n-k)

        # Polynomial constants:
        ONE = Polynomial(z0=GF2int(1))
        ZERO = Polynomial(z0=GF2int(0))
        Z = Polynomial(z1=GF2int(1)) # used to shift polynomials, simply multiply your poly * Z to shift

        # Precaching
        s2 = ONE + s

        # Iteratively compute the polynomials n-k-erasures_count times. The last ones will be correct (since the algorithm refines the error/errata locator polynomial iteratively depending on the discrepancy, which is kind of a difference-from-correctness measure).
        for l in _range(0, n-k-erasures_count): # skip the first erasures_count iterations because we already computed the partial errata locator polynomial (by initializing with the erasures locator polynomial)
            K = erasures_count+l+synd_shift # skip the FIRST erasures_count iterations (not the last iterations, that's very important!)

            # Goal for each iteration: Compute sigma[l+1] and omega[l+1] such that
            # (1 + s)*sigma[l] == omega[l] in mod z^(K)

            # For this particular loop iteration, we have sigma[l] and omega[l],
            # and are computing sigma[l+1] and omega[l+1]

            # First find Delta, the non-zero coefficient of z^(K) in
            # (1 + s) * sigma[l]
            # Note that adding 1 to the syndrome s is not really necessary, you can do as well without.
            # This delta is valid for l (this iteration) only
            Delta = ( s2 * sigma[l] ).get_coefficient(K) # Delta is also known as the Discrepancy, and is always a scalar (not a polynomial).
            # Make it a polynomial of degree 0, just for ease of computation with polynomials sigma and omega.
            Delta = Polynomial(x0=Delta)

            # Can now compute sigma[l+1] and omega[l+1] from
            # sigma[l], omega[l], B[l], A[l], and Delta
            sigma.append( sigma[l] - Delta * Z * B[l] )
            omega.append( omega[l] - Delta * Z * A[l] )

            # Now compute the next support polynomials B and A
            # There are two ways to do this
            # This is based on a messy case analysis on the degrees of the four polynomials sigma, omega, A and B in order to minimize the degrees of A and B. For more infos, see https://www.cs.duke.edu/courses/spring10/cps296.3/decoding_rs_scribe.pdf
            # In fact it ensures that the degree of the final polynomials aren't too large.
            if Delta == ZERO or 2*L[l] > K+erasures_count \
                or (2*L[l] == K+erasures_count and M[l] == 0):
            #if Delta == ZERO or len(sigma[l+1]) <= len(sigma[l]): # another way to compute when to update, and it doesn't require to maintain the update flag L
                # Rule A
                B.append( Z * B[l] )
                A.append( Z * A[l] )
                L.append( L[l] )
                M.append( M[l] )

            elif (Delta != ZERO and 2*L[l] < K+erasures_count) \
                or (2*L[l] == K+erasures_count and M[l] != 0):
            # elif Delta != ZERO and len(sigma[l+1]) > len(sigma[l]): # another way to compute when to update, and it doesn't require to maintain the update flag L
                # Rule B
                B.append( sigma[l] // Delta )
                A.append( omega[l] // Delta )
                L.append( K - L[l] ) # the update flag L is tricky: in Blahut's schema, it's mandatory to use `L = K - L - erasures_count` (and indeed in a previous draft of this function, if you forgot to do `- erasures_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 `- erasures_count` to update L on your own implementation and see which one works OK without producing wrong decoding failures.
                M.append( 1 - M[l] )

            else:
                raise Exception("Code shouldn't have gotten here")

        # Hack to fix the simultaneous computation of omega, the errata evaluator polynomial: because A (the errata evaluator support polynomial) is not correctly initialized (I could not find any info in academic papers). So at the end, we get the correct errata evaluator polynomial omega + some higher order terms that should not be present, but since we know that sigma is always correct and the maximum degree should be the same as omega, we can fix omega by truncating too high order terms.
        if omega[-1].degree > sigma[-1].degree: omega[-1] = Polynomial(omega[-1].coefficients[-(sigma[-1].degree+1):])

        # Debuglines, uncomment to show the result of every iterations
        #print "SIGMA BM"
        #for i,x in enumerate(sigma):
            #print i, ":", x

        # Return the last result of the iterations (since BM compute iteratively, the last iteration being correct - it may already be before, but we're not sure)
        return sigma[-1], omega[-1]