In this paper, we analyze the performance of neural belief propagation (BP) decoding on the additive white Gaussian noise (AWGN) channel, compared to the traditional BP algorithm. Previous investigations have shown that assigning pre-trained weights to BP messages can significantly improve the decoding performance in case of high-density parity-check (HDPC) codes, by reducing the negative impact of short cycles. These weights are trained by a neural network whose structure matches the trellis of the decoder. Specifically, we show that medium-density paritycheck (MDPC) codes decoded with neural BP algorithm can achieve lower bit error rate versus HDPC codes with the same codeword length and the same code rate.
In this paper, we analyze the performance of neural belief propagation (BP) decoding on the additive white Gaussian noise (AWGN) channel, compared to the traditional BP algorithm. Previous investigations have shown that assigning pre-trained weights to BP messages can significantly improve the decoding performance in case of high-density parity-check (HDPC) codes, by reducing the negative impact of short cycles. These weights are trained by a neural network whose structure matches the trellis of the decoder. Specifically, we show that medium-density paritycheck (MDPC) codes decoded with neural BP algorithm can achieve lower bit error rate versus HDPC codes with the same codeword length and the same code rate.
Summary The simplicity of decoding is one of the most important characteristics of the low density parity check (LDPC) codes. Belief propagation (BP) decoding algorithm is a well‐known decoding algorithm for LDPC codes. Most LDPC codes with long lengths have short cycles in their Tanner graphs, which reduce the performance of the BP algorithm. In this paper, we present 2 methods to improve the BP decoding algorithm for LDPC codes. In these methods, the calculation of the variable nodes is controlled by using “multiplicative correction factor” and “additive correction factor.” These factors are obtained for 2 separate channels, namely additive white Gaussian noise (AWGN) and binary symmetric channel (BSC), as 2 functions of code and channel parameters. Moreover, we use the BP‐based method in the calculation of the check nodes, which reduces the required resources. Simulation results show the proposed algorithm has better performance and lower decoding error as compared to BP and similar methods like normalized‐BP and offset‐BP algorithms.
Neural network decoding algorithms are recently introduced by Nachmani et al. to decode high-density parity-check (HDPC) codes. In contrast with iterative decoding algorithms such as sum-product or min-sum algorithms in which the weight of each edge is set to 1, in the neural network decoding algorithms, the weight of every edge depends on its impact in the transmitted codeword. In this paper, we provide a novel feed-forward neural network lattice decoding algorithm suitable to decode lattices constructed based on Construction A, whose underlying codes have HDPC matrices. We first establish the concept of feed-forward neural network for HDPC codes and improve their decoding algorithms compared to Nachmani et al. We then apply our proposed decoder for a Construction A lattice with HDPC underlying code, for which the well-known iterative decoding algorithms show poor performances. The main advantage of our proposed algorithm is that instead of assigning and training weights for all edges, which turns out to be time-consuming especially for high-density parity-check matrices, we concentrate on edges which are present in most of 4-cycles and removing them gives a girth -6 Tanner graph. This approach, by slight modifications using updated LLRs instead of initial ones, simultaneously accelerates the training process and improves the error performance of our proposed decoding algorithm.
Neural network decoding algorithms are recently introduced by Nachmani et al. to decode high-density parity-check (HDPC) codes. In contrast with iterative decoding algorithms such as sum-product or min-sum algorithms in which the weight of each edge is set to 1, in the neural network decoding algorithms, the weight of every edge depends on its impact in the transmitted codeword. In this paper, we provide a novel \emph{feed-forward neural network lattice decoding algorithm} suitable to decode lattices constructed based on Construction A, whose underlying codes have HDPC matrices. We first establish the concept of feed-forward neural network for HDPC codes and improve their decoding algorithms compared to Nachmani et al. We then apply our proposed decoder for a Construction A lattice with HDPC underlying code, for which the well-known iterative decoding algorithms show poor performances. The main advantage of our proposed algorithm is that instead of assigning and training weights for all edges, which turns out to be time-consuming especially for high-density parity-check matrices, we concentrate on edges which are present in most of 4-cycles and removing them gives a girth-6 Tanner graph. This approach, by slight modifications using updated LLRs instead of initial ones, simultaneously accelerates the training process and improves the error performance of our proposed decoding algorithm.
Low-density parity-check (LDPC) codes have been one of the most popular error correction candidates in modern communication and data storage systems for the past three decades due to their capacity-approaching error correction capability under the Belief Propagation (BP) decoding algorithms. In this dissertation, we investigate three topics related to efficient and effective decoding algorithms and the code design of LDPC codes. In the first part of this dissertation, we propose a methodology with the help of artificial intelligence to design Finite Alphabet Iterative Decoders (FAID) for LDPC codes, which are not only of great theoretical but also practical importance. This model-driven neural network (NN) framework unfolds the decoding iterations, introduces quantized neural networks (QNN) to model the FAIDs, and proposes a new cost function, namely, bit error rate (BER), to address the inefficiency in discrete-output channel scenarios. The low precision activations in the QNN and quantization in the cost functions cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We design and leverage straight-through estimators as surrogate gradients to tackle this issue and provide a joint training scheme. These QNN-aided FAIDs, are capable of surpassing the floating-point BP algorithms, with much lower complexity and faster convergence. The QNN-aided approach is general, as it can be applied to both continuous and discrete-output channels. In the second part of this dissertation, we propose a new code construction of LDPC codes based on the conventional parity-check matrices of Reed-Solomon (RS) codes. We investigate the girth and cycle structure and introduce an efficient searching algorithm to reduce short cycles in their Tanner graphs and to construct LDPC codes with a girth of at least eight. We use masking to further reduce the number of short cycles and to adjust the column and row weights of the parity-check matrices of the constructed codes for performance improvement. Our designs are systematic and flexible, which balance the requirements of good waterfall performance and low error-floor while maintaining a structure that facilitates implementation. In the third part of this dissertation, we investigate binary quasi-cyclic (QC) LDPC codes and their capability to correct phased bursts of erasures. We prove that the peeling algorithm when applied to such parity-check matrices of column weight two can correct all erasures that can be corrected by a maximum likelihood (ML) decoder. We show that by modifying the parity-check matrices of column weight two and globally coupling them, the erasure correcting capability can be further enhanced. For the Additive White Gaussian Noise (AWGN) channel, we design QC LDPC codes with parity-check matrices of column weight three or more that can correct phased bursts of erasures and perform well over the AWGN channel. Our analysis of QC-LDPC codes with parity-check matrices of column weight two is rather comprehensive as we determine, for all such codes, their dimensions, minimum Hamming distances, and their capabilities to correct phased bursts.
In long wave communication scenarios, channel interferences lead to noise correlation. Since traditional belief propagation (BP) decoding algorithm is designed based on additive white Gaussian noise (AWGN), a performance cost will appear when it comes to correlated noise. In this paper, a new LDPC decoding scheme that applies to correlated Gaussian noise is proposed. We design a BP-gated neural network-BP structure to carry on two rounds of BP decoding with the second round based on optimized noise. By adopting gated neurons in typical NN, CNN, training performance is improved. Simulation shows that compared with BP decoding algorithm, the new decoding scheme outperforms traditional method by 0.5~0.61 dB when bit error rate is 10-6. This decoding scheme also works on pectinate noise with performance gain of 1.5 dB.
With the rapid development of deep learning research and physical layer communication applications, deep learning-based channel coding has gradually become a research hotspot. In this paper, we propose a LDPC decoding network based on neural belief propagation (NBP) decoding. By introducing weights sharing mechanism, the weights vary across different iterations. The number of parameters is reduced, which significantly lowers memory requirements. In addition, we optimize the loss function to better train the model, achieving a lower bit error rate (BER) performance. Experimental results show that the proposed decoder yields significant performance improvements with respect to NBP, and significantly reducing the number of learnable weights.