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arXiv:2306.00443 · International Conference on Communications

Efficient Near Maximum-Likelihood Reliability-Based Decoding for Short LDPC Codes

Weiyang Zhang, Chentao Yue, Yonghui Li, Branka S. Vucetic

Published Jun 9, 2024• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/icc51166.2024.10622261

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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.
This paper proposes a concatenated multi-belief ordered likelihood decoding (MB-OLD) algorithm for rate-compatible (RC) short low-density parity check (LDPC) codes, where the output log-likelihood ratios (LLRs) of belief propagation (BP) are sent to a well-designed bit-flipping decoder, which we called ordered likelihood decoder (OLD). In contrast to conventional ordered statistic decoder (OSD), the test error patterns (TEPs) sequence of OLD is generated from most likely to least likely, where the ordered reliability sequence associated with the most reliable basis (MRB) is approximated as multiple lines, and a stopping criterion (SC) is taken to reduce the decoding complexity. Furthermore, we analyze the LLR behavior of BP decoder in short block-length regimes, and propose an optimal iteration number. Based on these analyses, the output LLRs of BP within the optimal number of iterations are well combined and sent to OLD. Simulation results show that the proposed MB-OLD has the superior decoding performances in terms of error-rate and decoding complexity than its counterparts.
Low-density parity-check (LDPC) codes have been widely used in practice due to their excellent error correction capabilities. However, the error correction performance of traditional decoding algorithms for LDPC codes is limited under short block lengths. To address this problem, this paper proposes an effective scheme that contains two stages for decoding short LDPC codes. In the first stage, the traditional belief propagation (BP) iterative algorithm is used to decode an LDPC code. The decoding process terminates if a codeword is obtained in this stage. Otherwise, the second stage decoding is invoked, in which the ordered reliability bits guessing random additive noise decoding (ORBGRAND) algorithm is applied. The accumulated log-likelihood ratios in the first stage are used to order the test error patterns in the ORBGRAND algorithm. Simulation results indicate that the proposed BP-ORBGRAND scheme can outperform the traditional BP iterative algorithm with relatively low additional complexity. Compared with the decoding scheme that concatenates the BP algorithm and the order statistics decoding (OSD) algorithm, the proposed decoding scheme has similar performance and the operations involved can be efficiently implemented in the hardware platform. All these indicate that the proposed scheme is desirable for practical purposes.
To enhance the performance of short low-density parity-check (LDPC) codes, we introduce an innovative hybrid decoder that seamlessly integrates belief propagation (BP) with ordered statistics decoding (OSD). This hybrid approach initially employs BP to decode received vectors until successful or upon reaching specified conditions, either a maximum iteration limit or a trapping iteration threshold, indicating the stagnation of BP’s hard output. Upon these conditions, the decoder transitions to a pre-configured error pattern (PEP) OSD. The PEPOSD phase then pursues a valid result, flexibly utilizing either the received vectors or the soft values of BP as input, contingent upon the signal-to-noise ratio (SNR). Given the minimal error overlap between BP and OSD, simulations reveal that this hybrid decoder achieves a notable gain of 1.5dB over BP and 0.3-0.6dB over PEPOSD, while maintaining comparable complexity to BP alone.
Belief propagation (BP) combined with ordered statistics decoding (OSD) achieves near-optimal frame error rate (FER) for short LDPC codes, but the extensive re-encoding overhead of OSD limits its practicality. To address this, we propose the BPORBOSD, a low-complexity and high-performance decoder that replaces conventional OSD with the ordered reliability bits OSD (ORBOSD), where the test error patterns are produced by the ORB technique. Moreover, in contrast to conventional BP-OSD, which directly adopts the soft information from standard BP into OSD after BP fails, we employ a modified BP scheme with a normalization factor α to produce soft reliability metrics better suited for ORBOSD. We derive the decoder’s complexity and demonstrate that, in the high-SNR regime, the average decoding complexity of BPORBOSD approaches that of standard BP. Simulation results confirm that BP-ORBOSD achieves FERs close to those of conventional BPOSD and the finite-length bound while significantly reducing the average number of re-encodings.
This paper investigates decoder diversity architectures for short low-density parity-check (LDPC) codes, based on recurrent neural network (RNN) models of the belief-propagation (BP) algorithm. We propose a new approach to achieve decoder diversity in the waterfall region, by specializing BP-RNN decoders to specific classes of errors, with absorbing set support. We further combine our approach with an ordered statistics decoding (OSD) post-processing step, which effectively leverages the bit-error rate optimization deriving from the use of the binary cross-entropy loss function. We show that a single specialized BP-RNN decoder combines better than BP with the OSD post-processing step. Moreover, combining OSD post-processing with the diversity brought by the use of multiple BP-RNN decoders, provides an efficient way to bridge the gap to maximum likelihood decoding.
This article deals with Ordered Statistics Decoding (OSD) applied to the soft outputs of the Belief Propagation (BP) algorithm. We first model the weighted sum of the a posteriori LLRs across BP decoding iterations into a neuron. The neuron is then trained with the focal loss to compute for each BP decoding failure a set of accumulated Log Likelihood Ratios (LLRs) suited for OSD post-processing. Then, we propose a recursive selection procedure of LLRs sets, for multiple OSD post-processing. This selection is carried out from the sets of a posteriori LLRs calculated at each BP iteration, and from the accumulated LLRs optimized for the OSD, based on their joint probabilities of failure with OSD post-processing. An OSD is then applied to each set of LLRs belonging to the selection. In addition, we propose to reduce the OSD post-processing decoding complexity without significantly degrading its performance. Our results show that this new decoding method provides an effective way to bridge the gap to maximum likelihood decoding for short and long Low Density Parity Check (LDPC) codes.