Berlekamp-Massey Algorithm: Euclid in Disguise

Abstract

In this paper we take a fresh look at the well-known Berlekamp-Massey (BM) algorithm for decoding of Reed-Solomon (RS) and Bose-Chaudhuri–Hocquenghem (BCH) codes. RS and BCH codes are a very important family of cyclic codes, and are included in most elementary courses on code theory. One of the most important tools in decoding of RS and BCH codes was developed by Berlekamp, and later formulated as an algorithm for synthesizing short LFSR-s by Massey and is now known as the Berlekamp-Massey (BM) algorithm. An alternative algorithm for decoding such codes is the extended Euclid algorithm. We present another viewpoint to the BM algorithm which is simpler than the Massey formulation, and mirrors the extended Euclid algorithm. This presentation may replace the common treatment of the BM algorithm in elementary courses with a simpler and more rigorous presentation. Moreover, this approach enables to improve the HW implementation of BCH decoding. This is promising as BCH codes are gaining renewed interest lately in latency sensitive applications. Another advantage of the new approach is that it provides a simple derivation of erasure decoding.