A Future Simplification of Procedure for Decoding Nonsystematic Reed-Solomon Codes Using the Berlekamp-Massey Algorithm

It is well-known that the Euclidean algorithm can be used o find the systematic errata-locator polynomial and the errata-evaluator polynomial simultaneously in Berlekamp´s key equation that is needed to decode a Reed-Solomon (RS) codes. In this paper, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with the Euclidean algorithm for efficiently decoding nonsystematic RS codes. In fact, this decoding algorithm is an appropriate modification to the algorithm developed by Shiozaki and Gao. Based on the ideas presented above, a fast algorithm described from Blahut´s classic book is derivated and proved in this paper to correct erasures as well as errors by replacing the Euclidean algorithm by the Berlekamp-Massey (BM) algorithm. These facts lead to significantly reduce the decoding complexity of the proposed RS decoder. In addition, computer simulations show that this simple and fast decoding technique reduces the decoding time when compared with existing efficient algorithms including the new Euclidean-algorithm-based decoding approach proposed in this paper.