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Abstract-A general class of polynomial remainder codes is considered. These codes are very flexible in rate and length and include Reed-Solomon codes as a special case. In general, the code symbols of such codes are polynomials of different degree, which leads to two different notions of weights and of distances.The notion of an error locator polynomial is generalized to such codes. A key equation is proposed, from which the error locator polynomial can be computed by means of a gcd algorithm. From the error locator polynomial, the transmitted message can be recovered in two different ways, which may be new even when specialized to Reed-Solomon codes.

The paper introduces the simultaneous partialinverse problem (SPI) for polynomials and develops its application to decoding interleaved Reed-Solomon codes beyond half the minimum distance. While closely related both to standard key equations and to well-known Padé approximation problems, the SPI problem stands out in several respects. First, the SPI problem has a unique solution (up to a scale factor), which satisfies a natural degree bound. Second, the SPI problem can be transformed (monomialized) into an equivalent SPI problem where all moduli are monomials. Third, the SPI problem can be solved by an efficient algorithm of the Berlekamp-Massey type. Fourth, decoding interleaved Reed-Solomon codes (or subfieldevaluation codes) beyond half the minimum distance can be analyzed in terms of a partial-inverse condition for the error pattern: if that condition is satisfied, then the (true) error locator polynomial is the unique solution of a standard key equation and can be computed in many different ways, including the wellknown multi-sequence Berlekamp-Massey algorithm and the SPI algorithm of this paper. Two of the best performance bounds from the literature (the Schmidt-Sidorenko-Bossert bound and the Roth-Vontobel bound) are generalized to hold for the partialinverse condition and thus to apply to several different decoding algorithms.

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Abstract-We propose a new algorithm for decoding ReedSolomon codes (up to half the minimum distance) and for computing inverses in F [x]/m(x). The proposed algorithm is similar in spirit and structure to the Berlekamp-Massey algorithm, but it works naturally for general m(x).