Riunione Annuale GTTI 2008 – Sessione su Trasmissione Numerica 1 Abstract— In this paper, we elaborate on a new technique we have recently proposed for application of soft decoding algorithms based on belief propagation to classic binary cyclic codes, like BCH codes. Such linear block codes are widely used in practical applications and their soft-decision decoding still represents an open issue. We show that the proposed technique can be easily extended also to RS codes, by considering their binary expansion. In the case of RS codes, very good error correction performance can be achieved by using the recently proposed adaptive belief propagation algorithm that, however, requires heavy computations on the parity check matrix during decoder iterations. We show that, by adopting a simplified parity-check matrix adaptation principle in our technique, performance can be improved without increasing the decoding complexity. I. INTRODUCTION Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes represent classic families of linear error correcting codes, characterized by very good error correction capability and low complexity encoding and decoding. For these reasons, they have been included in many telecommunication standards and practical applications. Encoding and decoding of BCH and RS codes can be accomplished through very simple circuits that implement operations over finite fields. However, classic decoding techniques rely on hard-decision decoders, while the use of channel measurements in soft-decision decoders can improve significantly the error correction capability [1]. Decoding algorithms based on the belief propagation (BP) principle represent the state of the art in forward error correction, and their application to low-density parity-check (LDPC) codes permits to approach the channel capacity [2]. However, in order to achieve this result, BP decoding needs a parity-check matrix characterized by: i) low density of 1 symbols, ii) absence of short cycles in the associated Tanner graph and iii) optimized (regular or irregular) row and column weight distributions. Such properties are rarely ensured by parity-check matrices of classic codes. For example, it can be shown that (n = 2 m – 1, k, d)-BCH codes, where n is the codeword length and k the number of information bits, with rate greater than or equal to 1/2 and 3 ≤ m ≤ 8, cannot have 4- cycle-free Tanner graphs [3]. For these reasons, many alternative solutions have been proposed in the literature for effectively applying BP decoders to generic linear block codes, binary cyclic codes, or specific classes of cyclic codes [4]-[10]. All these techniques aim at finding, through different approaches, a graph representation for the code suited to BP decoding. The approach proposed in [4] exploits the extended parity- check matrix (EPCM) in order to obtain a regular Tanner graph. In [5] and [6], instead, the generalized parity-check matrix (GPCM) is adopted to reduce the number of short cycles. Such approach has been further investigated in [7], where an algorithm is presented that achieves a 4-cycle-free representation. All techniques based on GPCMs, however, require the introduction of auxiliary bits that do not correspond to transmitted bits and, hence, may cause performance degradation. In [8], it is demonstrated that Vardy’s technique can be used to find sparse parity-check matrices for RS codes. Clever techniques for applying belief- propagation decoding to RS and more general codes have been also proposed in [9] and [10]. The rationale of these methods lies in adapting the parity-check matrix at each iteration, through linear combinations of rows, in such a way that the least reliable bits correspond to unitary weight columns in the code parity-check matrix. Indeed, such approach is able to provide good error correction performance, but it requires to continuously perform Gaussian elimination on the parity-check matrix during decoding iterations, that can represent a problem in practical implementations. In this paper, we describe an alternative approach we have recently proposed, based on “reduced” and “spread” parity- check matrices [11], and show how it is able to achieve good performance in the case of BCH codes. Then, we extend the application of the proposed method to RS codes, by considering the binary expansion of these codes, Low Complexity Soft-Decision Decoding of BCH and RS Codes based on Belief Propagation Marco Baldi, Giovanni Cancellieri and Franco Chiaraluce Università Politecnica delle Marche Facoltà di Ingegneria DEIT Via Brecce Bianche I-60131 Ancona, Italy {m.baldi, g.cancellieri, f.chiaraluce}@univpm.it