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In this paper, a new approach for decoding low-rate Reed-Solomon codes beyond half the minimum distance is considered and analyzed. The maximum error correcting radius coincides with the error correcting radius of the Sudan algorithm published in 1997. However, unlike the Sudan Algorithm, the approach described here is not a list decoding algorithm, and is not based on polynomial interpolation. The algorithm in this paper is rather syndrome based, like classical algebraic decoding algorithms. The computational complexity of the new algorithm is of the same order as the complexity of the well-known Berlekamp-Massey algorithm. To decode errors beyond half the minimum distance, the new decoder is allowed to fail for some high-weight error patterns with a very small probability.