The BM Algorithm and the BMS Algorithm

Abstract

We present a brief overview of the Berlekamp-Massey (BM) algorithm and of the Berlekamp-Massey-Sakata (BMS) algorithm, with an emphasis on a close relationship between them, which can be summarized as follows: BM algorithm + Grobner basis = BMS algorithm. Furthermore, we describe various forms of these algorithms, all of which are related to fast decoding of algebraic codes including algebraic-geometric (AG) codes. As a byproduct, we obtain a recursive BMS algorithm which is an extension of Blahut’s recursive BM algorithm. Throughout this paper, we recall the assertions of R.E. Blahut on the importance of fast decoding for good codes, particularly as described in his books, which have spurred recent developments of novel decoding algorithms of AG codes.