On Decoding Complexity of Reed-Solomon Codes on the Packet Erasure Channel

Abstract

We study the finite-length complexity of the Berlekamp-Massey algorithm (BM) and of the Gaussian elimination algorithm (GE) for decoding a Reed-Solomon (RS) code of length n on the packet erasure channel (PEC). In particular, we are interested in the dependency of the complexity on the packet size g. We show that, for large packet sizes, the complexity of both algorithms is O(g) and the constants hidden by the O-notation are comparable. As an example, we consider a RS code from the Digital Video Broadcasting - Handhelds (DVB-H) standard and we show that, although the asymptotic complexity of the GE is O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) and the one of the BM is O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), the finite-length complexity of both algorithms is comparable already for small/moderate packet sizes, in particular for all packet sizes considered within the standard, making the GE not inferior to the BM, in this context.