Fast erasure-and-error decoding of any one-point AG codes up to the Feng-Rao bound

Fast decoding methods of algebraic-geometric (AG) codes have been proposed as applications of Sakata´s (1988) algorithm (the multidimensional Berlekamp-Massey algorithm). To improve the probability of correct decoding, it is desirable to devise an efficient decoding algorithm which can correct both errors and erasures. Skorobogatov and Vladut (1990) were the pioneers of erasure-and-error decoding of AG codes. Extending their error-only decoding method, Feng and Rao (see Proceedings of 1993 IEEE Information Theory Workshop, Shizuoka, Japan, June, 1993) gave an erasure-and-error decoding method which can correct t errors and τ erasures such that 2t+τ<d_FR. We propose a fast erasure-and-error decoding method based on a unification of our error-only decoding method and an algorithm for finding a minimal polynomial vector set of a vector of multidimensional arrays. Our main concern is how to find the unknown syndrome values and the error locations in addition to the given erasure locations more efficiently than the Feng-Rao´s scheme based on matrix calculations