On decoding algorithms for generalized Reed — Solomon codes with errors and erasures

The article is devoted to the decoding algorithms for generalized Reed — Solomon codes with errors and erasures. These algorithms are based on Gao algorithm, Sugiyama algorithm, Berlekamp — Massey algorithm (Peterson — Gorenstein — Zierler algorithm). The first of these algorithms belongs to syndrome-free decoding algorithms, the others — to syndrome decoding algorithms. The relevance of these algorithms is that they are applicable for decoding Goppa codes, which are the basis of some promising post-quantum cryptosystems. These algorithms are applicable for Goppa codes over an arbitrary field, as opposed to the well-known Patterson decoding algorithm for binary Goppa codes.