On decoding algorithms for generalized Reed–Solomon codes

The paper is devoted to decoding algorithms for generalized Reed–Solomon codes that are based on algorithms for Reed–Solomon codes. The Gao, Sugiyama, and Berlekamp–Massey algorithms (Peterson–Gorenstein–Zierler algorithm) are given. 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 postquantum 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.