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arXiv:0710.1589

Fast Reliability-based Algorithm of Finding Minimum-weight Codewords for LDPC Codes

Guangwen Li, Guangzeng Feng

Published Oct 8, 2007• 0 citations
arXivHTMLPDF

Abstract

Despite the NP hardness of acquiring minimum distance $d_m$ for linear codes theoretically, in this paper we propose one experimental method of finding minimum-weight codewords, the weight of which is equal to $d_m$ for LDPC codes. One existing syndrome decoding method, called serial belief propagation (BP) with ordered statistic decoding (OSD), is adapted to serve our purpose. We hold the conjecture that among many candidate error patterns in OSD reprocessing, modulo 2 addition of the lightest error pattern with one of the left error patterns may generate a light codeword. When the decoding syndrome changes to all-zero state, the lightest error pattern reduces to all-zero, the lightest non-zero error pattern is a valid codeword to update lightest codeword list. Given sufficient codewords sending, the survived lightest codewords are likely to be the target. Compared with existing techniques, our method demonstrates its efficiency in the simulation of several interested LDPC codes.

DOI
10.48550/arxiv.0710.1589

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Despite the NP hardness of acquiring minimum distance dmd_mdm​ for linear codes theoretically, in this paper we propose one experimental method of finding minimum-weight codewords, the weight of which is equal to dmd_mdm​ for LDPC codes. One existing syndrome decoding method, called serial belief propagation (BP) with ordered statistic decoding (OSD), is adapted to serve our purpose. We hold the conjecture that among many candidate error patterns in OSD reprocessing, modulo 2 addition of the lightest error pattern with one of the left error patterns may generate a light codeword. When the decoding syndrome changes to all-zero state, the lightest error pattern reduces to all-zero, the lightest non-zero error pattern is a valid codeword to update lightest codeword list. Given sufficient codewords sending, the survived lightest codewords are likely to be the target. Compared with existing techniques, our method demonstrates its efficiency in the simulation of several interested LDPC codes.
Aiming at bridge gap between maximum likelihood decoding (MLD) and belief propagation (BP) decoding for short or medium LDPC codes, we present one modified ordered statistics decoding (OSD) reprocessing method, which is serially concatenated with standard BP. Meanwhile, two novel points in implementation are addressed. For one thing, the new definition of bit reliability proposed enhances error-correcting performance via centralizing more decoding errors of standard BP within the scope of order-p OSD where p ≤ 2. For another, one way of reducing computational complexity is put forward in selecting optimal error pattern among all candidates. Simulation results demonstrate our approach could achieve good tradeoff between reprocessing performance and computational complexity. Techniques on further improvement are also discussed briefly. Reliability-based syndrome decoding of LDPC codes 1 I.
Low-density parity-check (LDPC) codes in their broader-sense definition are linear codes whose parity-check matrices have fewer 1s than 0s. Finding their minimum distance is therefore in general an NP-hard problem. We propose a randomized algorithm called nearest nonzero codeword search (NNCS) approach to tackle this problem for iteratively decodable LDPC codes. The principle of the NNCS approach is to search codewords locally around the all-zero codeword perturbed by minimal noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum-Hamming- weight codeword whose Hamming weight is equal to the minimum distance of the linear code. This approach has its roots in Berrou et al.'s error-impulse method and a form of Fossorier's list decoding for LDPC codes.
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In this paper, we propose an efficient decoding algorithm for short low-density parity check (LDPC) codes by carefully combining the belief propagation (BP) decoding and ordered statistics decoding (OSD) algorithms. Specifically, a modified BP (mBP) algorithm is applied for a certain number of iterations prior to OSD to enhance the reliability of the received message, where an offset parameter is utilized in mBP to control the weight of the extrinsic information in message passing. By carefully selecting the offset parameter and the number of mBP iterations, the number of errors in the most reliable positions (MRPs) in OSD can be reduced by mBP, thereby significantly improving the overall decoding performance of error rate and complexity. Simulation results show that the proposed algorithm can approach the maximum-likelihood decoding (MLD) for short LDPC codes with only a slight increase in complexity compared to BP and a significant decrease compared to OSD. Specifically, the order- (m -1) decoding of the proposed algorithm can achieve the performance of the order-m OSD.
For short low-density parity-check codes,due to the message dependency,there exists a performance gap between the belief propagation (BP) and the maximum likelihood decoding (MLD) algorithms.Aiming at bridging the gap at the cost of some complexity,we put forward a beam search (BS) postprocessing scheme,based on the fact that LDPC codes commonly have sparsely distributed codewords and only a small portion of the codeword bits are erroneous in a BP decoding failure.In the implementation of BS to search for the error pattern,when evaluating the intermediate individuals given a limited beam width,we take advantage of the information about the decoding syndrome and the reliability metric of each bit fully.Simulation results show that the proposed method achieves a good tradeoff between performance and complexity.
Abstract Aimed at bridging the gap between the maximum likelihood and the suboptimal decoding schemes for short or medium LDPC codes, a generalised ordered statistic decoding (OSD), in the form of syndrome decoding, is presented in the paper. The OSD is invoked once when a nonzero decoding syndrome is returned for the preceded belief propagation (BP) or min‐sum (MS) variants. With respect to the OSD with an accumulative reliability metric for the BP in the logarithmic likelihood ratio (LLR) domain, we extend the metric to the BP in the probability domain. Furthermore, the OSD with the metric specific for the MS variants, shows an appealing trade‐off between performance and complexity. Meanwhile, when determining the optimal error pattern estimation, an alternative designed to reduce the number of the required real additions proves to be effective. Copyright © 2008 John Wiley & Sons, Ltd.