The interpolation step of Guruswami and Sudan's list decoding of Reed-Solomon codes poses the problem of finding the minimal polynomial of an ideal with respect to a certain monomial order. An efficient algorithm that solves the problem is presented based on the theory of Gröbner bases of modules. In a special case, this algorithm reduces to a simple Berlekamp-Massey-like decoding algorithm.It is easy to identify a set of generators of I v,m,l , from which we compute a Gröbner basis. First we present a natural set of generators for the ideal I v,m .Proposition 3 As an ideal of F[x, y],where η = n j=1 (x − α j ). PROOF. Let

List decoding of Hermitian codes is reformulated to allow an efficient and simple algorithm for the interpolation step. The algorithm is developed using the theory of Gröbner bases of modules. The computational complexity of the algorithm seems comparable to previously known algorithms achieving the same task, and the algorithm is better suited for hardware implementation.2000 Mathematics Subject Classification. 94B35,11T71.

We present a unique decoding algorithm of algebraic geometry codes on plane curves, Hermitian codes in particular, from an interpolation point of view. The algorithm successfully corrects errors of weight up to half of the order bound on the minimum distance of the AG code. The decoding algorithm is the first to combine some features of the interpolation based list decoding with the performance of the syndrome decoding with majority voting scheme. The regular structure of the algorithm allows a straightforward parallel implementation.

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A unique decoding algorithm for general AG codes, namely multipoint evaluation codes on algebraic curves, is presented. It is a natural generalization of the previous decoding algorithm which was only for one-point AG codes. As such, it retains the same advantages of fast speed and regular structure with the previous algorithm. Compared with other known decoding algorithms for general AG codes, it is much simpler in its description and implementation.Comment: 11 pages,submitted to the IEEE Transactions on Information Theory; added a citation in the section III-