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IEEE Access

A BRB Algorithm for Non-Binary LDPC Decoding With Single-Symbol Extrinsic Information Modification

Xiangcheng Li, Ziyun Ruan, Dongri Ban, Yihui Huang, Youming Sun

Published Jan 1, 2024• 0 citations
Publisher

Abstract

In the weighted bit-reliability based (wBRB) Non-binary LDPC decoding algorithm, using the minimum bit-reliability as the reliability of the entire symbol for information updating significantly decreases the decoding performance in cases of a large order of Galois field or small column weight. The improved algorithm——MwBRB increases the decoding performance by introducing three kinds of extrinsic information transfer but suffers from high decoding complexity. To address these problems, a BRB Non-binary LDPC decoding algorithm based on single-symbol extrinsic information modification is proposed. At the variable node$V_{j}$, using a symbol-related reliability calculation method, the maximum reliability information and its corresponding most possible Galois field symbol are obtained, and then transferred to the check node$C_{i}$; at the check node$C_{i}$, only the most possible Galois field symbol is used for extrinsic information processing, and its reliability of the extrinsic information is represented by the minimum reliability of other Galois field symbols. Finally, at the check node$C_{i}$, the extrinsic information is processed by based on the Hamming distance coefficient in terms of bits. Simulation and performance analysis show that the proposed algorithm reduces the complexity by over 50% compared to the MwBRB algorithm, and maintains consistent or slightly superior decoding performance. Additionally, the proposed algorithm exhibits similar decoding performance to LDPC codes constructed by different methods and shows better robustness.

DOI
10.1109/access.2024.3487644

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In the weighted bit-reliability based (wBRB) Non-binary LDPC decoding algorithm, using the minimum bit-reliability as the reliability of the entire symbol for information updating significantly decreases the decoding performance in cases of a large order of Galois field or small column weight. The improved algorithm——MwBRB increases the decoding performance by introducing three kinds of extrinsic information transfer but suffers from high decoding complexity. To address these problems, a BRB Non-binary LDPC decoding algorithm based on single-symbol extrinsic information modification is proposed. At the variable nodeVjV_{j}Vj​, using a symbol-related reliability calculation method, the maximum reliability information and its corresponding most possible Galois field symbol are obtained, and then transferred to the check nodeCiC_{i}Ci​; at the check nodeCiC_{i}Ci​, only the most possible Galois field symbol is used for extrinsic information processing, and its reliability of the extrinsic information is represented by the minimum reliability of other Galois field symbols. Finally, at the check nodeCiC_{i}Ci​, the extrinsic information is processed by based on the Hamming distance coefficient in terms of bits. Simulation and performance analysis show that the proposed algorithm reduces the complexity by over 50% compared to the MwBRB algorithm, and maintains consistent or slightly superior decoding performance. Additionally, the proposed algorithm exhibits similar decoding performance to LDPC codes constructed by different methods and shows better robustness.
A modified bit-reliability based decoding algorithm is presented based on a recent work by Huang et al. For the presented algorithm, only one Galois field element is passed and exchanged along the edges of the Tanner graph. At variable nodes, full messages rather than extrinsic messages are processed to further reduce the computational complexity. At check nodes, only hard reliability is considered and the main operation is to compute the check-sum and send one extrinsic symbol back to variable nodes. Simulation results show that, even with lower complexity and less memory consumption, the presented algorithm still can perform very closely to the original wBRB algorithm with low quantization bits (3 ~ 4 bits) when decoding the majority-logic decodable nonbinary LDPC codes. For codes constructed in high order fields, the presented algorithm can even outperform the original wBRB algorithm.
This paper presents bit-reliability based majority-logic decoding (MLgD) algorithms for non-binary LDPC codes. The proposed algorithms pass only one Galois field element and its reliability along each edge of the Tanner graph of a non-binary LDPC code. Since their reliability updates are in terms of bits rather than symbols, they are more efficient than traditional MLgD based decoding algorithms. By weighting the soft reliability of the extrinsic information-sums based on their hard reliability, the proposed algorithms can achieve good error performance for non-binary LDPC codes with various column weights. Moreover, their computational complexity and memory consumption are remarkably reduced compared with existing MLgD based decoding algorithms. As a result, they provide effective tradeoffs between error performance and complexity for decoding of non-binary LDPC codes.
The main challenge for hardware implementation of non-binary LDPC decoding is the high computational complexity and large memory requirement. To address this challenge, five new low complexity LDPC decoding algorithms are proposed in this paper. The proposed algorithms are developed specifically towards the low complexity, yet effective, decoding of the NB LDPC codes. The proposed decoding algorithms update, iteratively, the hard decision received vector to search for the valid codeword in the vector space of Galois field (GF){(GF)}(GF) . The selection criterion for least reliable symbol positions is based on the information from the failed checks and the reliability information from the Galois field structure as well as from the received channel soft information. To choose the correct value for the candidate symbol, two methods are used. The first method is based on the prediction of the error symbol from the set of Galois field symbols which maximize an objective function. In the second method, individual bits are flipped based on the reliability information obtained from the channel. Algorithms 1 and 2 flip a single symbol per iteration whilst the other three algorithms 3, 4 and 5 flip multiple symbols in each iteration. The proposed voting based Algorithms 1, 2 and 5 first short list the unreliable positions using a majority voting scheme and then choose the candidate symbol value from the set of the symbols in GF(q){GF(q)}GF(q) while not violating the field order q{q}q . These methods simplify the decoding complexity in terms of computation and memory. Results and analysis of these algorithms show an appealing tradeoff between computational complexity and bit error rate performance for NB LDPC codes.
As an extension of binary low-density parity-check (LDPC) codes, non-binary LDPC (NB-LDPC) codes show significantly better performance when the code length is moderate or small. Recently, enhanced iterative hard reliability based (EIHRB) decoding algorithm is proposed to reduce the computation complexity. However, the EIHRB algorithm suffers a lot from significant performance degradation when the column weight is small. In this paper, a symbol reliability based (SRB) decoding algorithm, which also performs well when the column weight is low, is proposed for NB-LDPC decoding to improve the decoding performance. With the same maximum iteration number, around 0.38 dB extra coding gain is achieved. Furthermore, the corresponding efficient decoder architecture is proposed. Comparison results have shown that the proposed SRB algorithm can not only achieve good coding gain, but the cost for hardware implementation is reasonable.
The bit reliability based (BRB) and weighted bit reliability based (wBRB) algorithms are non-binary low-density parity-check (LDPC) code decoding algorithms with an excellent tradeoff between computational complexity and performance. However, the performance of these algorithms needs further improvement. We apply deep learning to these algorithms. Weights are assigned to each edge of the Tanner graphs of the non-binary LDPC codes in the proposed algorithms. We demonstrate the effectiveness of applying deep learning to the BRB and wBRB algorithms in terms of implementation and performance. The proposed algorithms achieve an approximately 0.3 dB higher bit error rate performance than the original algorithms in the high SNR region. The increase in computational complexity and memory consumption does not significantly change the implementation of the algorithms.
Message-passing decoders typically perform well for nonbinary low-density parity-check (NB-LDPC) codes with large computational complexity. As another type of simplified decoders, symbol-reliability-based decoders further reduce the computational complexity. However, the previously proposed algorithms suffer severe error performance degradation for NB-LDPC codes with low column weights. In this paper, a weighted bit-reliability based (wBRB) decoder for NB-LDPC codes is developed and implemented with efficient layered partial-parallel structure. It not only balances the tradeoff between complexity and error performance, but also reduces the memory usage significantly. Furthermore, to enhance the performance of the wBRB decoder, a full bit-reliability-based (FBRB) decoder is proposed. The FBRB decoder is derived based on the binary matrix representation of the nonzero entries in the parity-check matrix. Since more bit-reliability values are passed through the edges of the Tanner graph, the FBRB decoder can achieve better error performance and faster convergence rate than the wBRB decoder. Both of the decoders are implemented on a Xilinx Virtex-5 XC5VLX155T FPGA device for a (403,226) code over GF(25). The results shows that they achieve 118.98 and 95.73 Mbps throughput with 15 iterations, respectively.