A New Remainder Based Decoding Algorithm for Reed-Solomon Codes

Conventional decoding techniques for decoding cyclic codes require the computation of power sum syndromes which can often account for a significant portion of the decoder computations. Since the syndromes can be computed from the remainder polynomial, the polynomial obtained by dividing the received polynomial by the code generator polynomial, it follows that this polynomial contains all the information required to decode. Thus one might hope for a decoding technique that uses the remainder polynomial directly. Berlekamp and Welch have given such an algorithm which requires the sequential testing of the parity check locations and updating of four polynomials. Whiting in his doctoral thesis has given a modification of this procedure that makes more efficient the evaluation and updating of these polynomials. The present work derives a new algorithm using only the remainder polynomial. A new key equation is derived which may be solved by the usual Euclidean algorithm. The advantages of this approach are discussed and compared to the original algorithm and a performance of the algorithm in terms of computational and circuit complexity is considered.