A soft-decision decoding computes a first syndrome polynomial in accordance with a received word, computes a second syndrome polynomial by multiplying the first syndrome polynomial by a locator polynomial based on locations of erasures within the received word, finds a basis and private solution to an affine space of polynomials that solve key equations based on the second syndrome polynomial, determines a weak set of a locations of symbols in the received word with confidence below a certain confidence level, computes a matrix from the basis, the private solution and the weak set, determines sub-matrices in the matrix whose rank is equal to a rank of the matrix, determines error locator polynomial (ELP) candidates from the sub-matrices, the basis, and the private solution, and corrects the received word using a selected one of the ELP candidates.