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An Efficient Ordered Likelihood Decoder for Rate-Compatible Short LDPC codes

Chunjie Li, Ke Zhang, Ye Wang, Jian Jiao, Xiao Ma, Qinyu Zhang

Published Dec 8, 2024• 4 citations
Publisher

Abstract

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.

DOI
10.1109/globecom52923.2024.10901442

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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.
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.
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.
In this work, we present an improvement on low complexity serial concatenation of belief propagation (BP) - order statistic decoding (OSD) algorithm for low-density parity-check (LDPC) codes. We introduce a more reliable method to reconstruct ordered information sequence in terms of the accumulated log-likelihood ratio (LLR) transitions of variable nodes. We give a general expression for both the existing BP-OSD algorithm and our proposed new method according to a structure similar to an infinite impulse response (IIR) filter. This improved algorithm achieves noticeable performance gains with only modest increase in computation complexity.
A low-density parity check (LDPC) code with the belief propagation (BP) or the log-likelihood ratio belief propagation (LLR-BP) can achieve good bit error rate (BER) performance approaching the Shannon limit. When a parity check matrix of the LDPC code has the cycle, the BP and LLR-BP decoding algorithms achieve approximate maximum a posterior probability (MAP) decoding. Although the decoding algorithms are approximate MAP decoding, LDPC codes can achieve very good BER. For the short and middle length LDPC codes, BER and block error rate (BLER) performances are affected by cycle largely. In each iteration, the magnitudes of a posterior LLRs of some bits oscillate owing to cycles. The oscillation is the dominant error factor in the high Eb/Noregion for short and middle length LDPC codes. In this paper, we extend the definition of oscillation to extrinsic LLR (ex-LLR) derived in the bit node process and propose the modified LLR-BP and the modified UMP-BP decoding algorithms. To reduce effects of oscillating ex-LLRs on decoding, for oscillating ex-LLRs, we add the previous ex-LLR to the current ex-LLR. From the computer simulation, we show that for short and middle length LDPC codes, with a simple modification, our proposed decoding algorithms can improve the conventional LLR-BP and UMP-BP decoding algorithms. In particular, we show that the modified UMP-BP decoding algorithm with low complexity can achieve better BER and BLER than the conventional LLR-BP decoding algorithm
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.