Decoding algebraic-geometric codes using Grobner bases

Developments in coding theory have seen two ideas for generalizing Reed-Solomon codes to obtain better codes: multidimensional cyclic (MDC) coder, and algebraic-geometric (AG) codes. In MDC codes, polynomials in several variables take the place of univariate polynomials. In AG codes, the concept of a polynomial in one variable is generalized to a rational function on an algebraic curve. Grobner bases have proved to be a useful tool both in the theory of multivariate polynomials, and in computations involving them. Therefore it is natural that they play an central role in decoding algorithms for MDC and AG codes. We present two decoding algorithms based on Grobner bases which generalize well-known algorithms for decoding Reed-Solomon codes