A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes

摘要

A generalization of the Berlekamp-Massey algorithm is presented for synthesizing minimum length linear feedback shift registers for generating prescribed multiple sequences. A more general problem is first considered, that of finding the smallest initial set of linearly dependent columns in a matrix over an arbitrary field, which includes the multisequence problem as a special case. A simple iterative algorithm, the fundamental iterative algorithm (FIA), is presented for solving this problem. The generalized algorithm is then derived through a refinement of the FIA. Application of this generalized algorithm to decoding cyclic codes up to the Hartmann-Tzeng (HT) bound and Roos bound making use of multiple syndrome sequences is considered. Conditions for guaranteeing that the connection polynomial of the shortest linear feedback shift register obtained by the algorithm will be the error-locator polynomial are determined with respect to decoding up to the HT bound and special cases of the Roos bound.