Algorithms for decoding and interpolation

In this paper we consider various algorithms for decoding BCH/RS/ Goppa codes, in particular the euclidean algorithm, the Berlekamp-Massey algorithm and the Welch-Berlekamp algorithm. We focus on relationships of these algorithms with interpolation methods in system theory. We note that the problem statements in the two areas can be different: from a system theoretic point of view, rational interpolating functions with common factors between numerator and denominator are undesirable whereas common factors can be required in a decoding context. The behavioral approach was introduced by Jan C. Willems into system theory in the eighties. It proposes the family of trajectories of a system as its central focus. This makes the approach attractive for coding theorists (most naturally in the context of convolutional codes where the family of trajectories corresponds to the code). In this paper we focus on a connection between behavioral modeling and the decoding of BCH/RS/Goppa codes. In this context, the behavioral modeling approach is attractive because it naturally generates solutions with common factors. We present slight modifications of both the Berlekamp-Massey and the Welch-Berlekamp algorithm and give a derivation in terms of behavioral modeling. In particular, we derive the latter algorithm directly from Reed & Solomon's original approach. We demonstrate the similarity of the two algorithms and show that they are special instances of one general iterative behavioral modeling procedure.