1 S OFT -D ECISION D ECODING OF L INEAR B LOCK C ODES U SING P REPROCESSING AND D IVERSIFICATION Yingquan Wu and Christoforos Hadjicostis Abstract— Order-w reprocessing is a suboptimal soft-decision decoding approach for binary linear block codes in which up to w bits are systematically flipped on the so-called most reliable (information) basis (MRB). At each iteration a candidate codeword is recoded and in the end, the most likely codeword is picked from the candidate list. In this paper, we first incorporate two preprocessing rules into order-w reprocessing: i) a test error pattern with Hamming weight up to w +1 is discarded without further computing its actual likelihood if it results in more than θ bit errors within the MRB and the t most reliable bits of the redundancy part; ii) a test error pattern with Hamming weight w +2 is discarded if it results in more than 1 bit errors among the τ most reliable bits of the redundancy part, where θ, t, and τ are parameters to be determined. We show also that, with appropriate choice of parameters, the proposed order-w reprocessing with preprocessing requires comparable complexity to order-w reprocessing but achieves asymptotically the performance of order-(w + 2) reprocessing. To complement the MRB, we also employ iterative recoding on a second basis for practical SNRs and systematically extend this approach to a multi-basis order-w reprocessing scheme for high SNRs. We show that the proposed multi-basis scheme significantly enlarges the error-correction radius, a commonly used measure of performance at high SNRs, over the original (single-basis) order-w reprocessing. As a by-product, we also precisely characterize the asymptotic performance of the well- known Chase and GMD decoding algorithms. Our simulation results show that the proposed algorithm successfully decodes the (192, 96) Reed-Solomon concatenated code and the (256, 147) extended BCH code in near optimal manner (within 0.01 dB at a block-error-rate of 10 -5 ) with affordable computational cost. Index Terms— Soft-decision decoding, most reliable basis, multi-basis, preprocessing, asymptotic performance, binary linear block codes, I. I NTRODUCTION Maximum-likelihood (ML) soft-decision decoding provides substantial performance gain in comparison to bounded- distance (hard-decision) decoding. However, ML soft-decision decoding has been shown to be NP-hard whereas polynomial- complexity bounded-distance decoding algorithms exist for many known codes. The problem of finding computationally efficient and practically implementable soft-decision decoding algorithms has been investigated extensively and remains an open and challenging problem, particularly for long block codes. In [8], Forney provided for the first time a suboptimal soft-decision decoding approach that utilizes the concept of generalized minimum distance (GMD) decoding based on channel erasure information. In [5], Chase presented another suboptimal algorithm which uses channel reliability infor- mation to search for codewords by successively applying bounded-distance decoding to candidate test error patterns cor- responding to certain least reliable bit positions. At practical signal-to-noise ratios (SNRs), the methods in [8], [5] provide mild gains at manageable computational costs; however, these methods rely on an efficient hard-decision bounded-distance decoder which may not be available for many codes. An active research direction for the soft-decision decoding of general binary linear block codes is iterative recoding based on the so-called most reliable basis (MRB), which can be obtained by applying a greedy search algorithm to the decreasing reliability order of the received word. At each iteration of iterative recoding, a test error pattern (TEP) is added to the information message associated with the MRB and then a candidate codeword is recoded using the systematic generator matrix associated with the MRB. This was first suggested by Dorsch in [7] whose approach was based on flipping TEPs that are iterated in ascending reliability order. In [9], Fossorier and Lin proposed a simple order-w reprocessing scheme that systematically flips up to w bits over the MRB. This method was shown to be asymptotically optimal when w ≈ min{⌈ dmin 4 ⌉,K}, where K denotes the information di- mension and d min denotes the minimum Hamming distance of the code. In [13], Fossorier presented a multi-basis algorithm that does not require a reprocessing order larger than (w − 1) in its main phase and that is able to correct any error pattern of at most w errors in the K + t most reliable positions (the parameter t satisfies 0 <t ≤ min{K, N − K} where N denotes the code length and K the information dimension). In [22], Wu and Hadjicostis proved that Dorsch’s proposed