On Multisequence Shift Register Synthesis and Generalized-Minimum-Distance Decoding of Reed-Solomon Codes

Abstract

In this paper, it is shown that the problem of generalized-minimum-distance (GMD) decoding of Reed-Solomon (RS) codes can be reduced to the problem of multisequence shift register synthesis, and a simple algorithm is presented that yields a solution for this problem by finding, for k = 1, 2, . , the shortest linear feedback shift register that can generate each of the first k sequences of a special kind of multisequence. The algorithm is based on the well-known Berlekamp-Massey algorithm for a single-sequence problem and is only a little more complex than it. Also presented is a GMD decoding algorithm for RS codes which employs the proposed multisequence shift register synthesis algorithm and whose complexity is less than 3 nd + 8 d 2 for the code length n and the minimum distance d. This GMD decoding algorithm provides an alternative to algorithms based on the Welch-Berlekamp algorithm.