An efficient Euclidean algorithm for Reed-Solomon code to correct both errors and erasures

In this paper, an efficient Euclidean decoding algorithm is presented to solve the Berlekamp´s key equation of Reed-Solomon (RS) code for correcting erasures as well as errors by replacing the initial condition of the Euclidean algorithm with the erasure locator polynomial and the Forney syndrome polynomial. By this proposed algorithm, the errata locator polynomial and errata evaluator polynomial can be obtained simultaneously without the computations of polynomial division and field element inversion. Moreover, the whole recursive decoding produce for solving Berlekamp´s key equation could be performed with a fixed number of iterations. And, the weights used to reduce the degree of the errata evaluator polynomial at each iteration can be extracted from the coefficient of fixed degree. As a consequence, the complexity of RS decoder to correct both errors and erasures is reduced substantially. Therefore, this proposed algorithm provides more modular, regular and simple for both software and hardware implementation. An example using this proposed algorithm is given for a (255,239) RS code for correcting erasures and errors with s + 2v ≤ 16.