Efficient Decoding Of Reed-Solomon Codes Over Z q Based On Remainder Polynomials

– In 1983, Welch, L.R. and Berlekamp, E.R. introduced a new key equation based on rational interpolation for algebraically decoding Reed-Solomon (RS) codes over finite fields. A key feature of their so-called remainder decoding approach is that the computation of the syndrome vector corresponding to a given received word is not needed. For long codes, this implies significant computational savings. We extend the remainder decoding approach to RS codes over Zq where q is a prime power, focussing on double-error correction. This requires a characterization of the set of minimal solutions to the key equation over Zq[[X]] as well as a modified Welch-Berlekamp-type algorithm for solving it. Key-Words: – Remainder Decoding, Reed-Solomon codes, Welch-Berlekamp key equation.