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Neural Network-Based Iterative Decoding, Code Design, and Erasure Correction of Low-Density Parity-Check Codes

Xin Wei Xiao

Published Jan 1, 2021• 0 citations
OpenAlex

Abstract

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.

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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 this paper, we introduce a method of using quantized neural networks (QNN) to design finite alphabet message passing decoders (FAID) for Low-Density Parity Check (LDPC) codes. Specifically, we construct a neural network with low precision activations to optimize a FAID over Additive White Gaussian Noise Channel (AWGNC). The low precision activations cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We introduce straight-through estimators (STE) to avoid this problem, by replacing zero derivatives of quantized activations with surrogate gradients in the chain rules. We present a systematic approach to train such networks while minimizing the bit error rate, which is a widely used and accurate metric to measure the performance of iterative decoders. Examples and simulations show that by training a QNN, a FAID with 3-bit of message and 4-bit of channel output can be obtained, which performs better than the more complex floating-point minsum decoding algorithm. This methodology is promising in the sense that it facilitates designing low-precision FAID for LDPC codes while maintaining good error performance in a flexible and efficient manner.
Protecting logical information in the form of a classical bit or a quantum bit (qubit) is an essential step in ensuring fault-tolerant classical or quantum computation. Error correction codes and their decoders perform this step by adding redundant information that aids the decoder to recover or protect the logical information even in the presence of noise. Low-density parity-check (LDPC) codes have been one of the most popular error correction candidates in modern communication and data storage systems. Similarly, their quantum analogues, quantum LDPC codes are being actively pursued as excellent prospects for error correction in future fault-tolerant quantum systems due to their asymptotically non-zero rates, sparse parity check matrices, and efficient iterative decoding algorithms. This dissertation deals with failure configurations, known as \emph{trapping sets} of classical and quantum LDPC codes when decoded with iterative message passing decoding algorithms, and the \emph{error floor phenomenon} - the degradation of logical error rate performance at low physical noise regime. The study of quantum trapping sets will enable the construction of better quantum LDPC codes and also help in modifying iterative quantum decoders to achieve higher fault-tolerant thresholds and lower error floors. Towards this goal, the dissertation also presents iterative decoders for classical and quantum LDPC codes using the \emph{deep neural network framework}, novel iterative decoding algorithms, and a decoder-aware \emph{expansion-contraction method} for error floor estimation. In this dissertation, we first establish a systematic methodology by which one can identify and classify \emph{quantum trapping sets} (QTSs) according to their topological structure and decoder used. For this purpose, we leverage the known harmful configurations in the Tanner graph, called \emph{trapping sets} (TSs), from the classical error correction world. The conventional definition of a trapping set of classical LDPC codes is generalized to address the syndrome decoding scenario for quantum LDPC codes. Furthermore, we show that the knowledge of QTSs can be used to design better quantum LDPC codes and decoders. In the context of the development of novel decoders, we extend the stochastic resonance based decoders to quantum LDPC codes, propose iteration-varying message passing decoders with their message update rules learned by neural networks tuned for low logical error rate, and present a syndrome based generalized belief propagation algorithm for tackling convergence failure of iterative decoders due to the presence of short cycles. Our analysis of TSs of a layered decoding architecture clearly reveals the dependence of the harmfulness of TSs (classical or quantum) on the iterative decoder, and thus on the error floor estimates. We present a computationally efficient method for estimating error floors of LDPC codes over the binary symmetric channel without any prior knowledge of its trapping sets. The sub-graph expansion-contraction method is a general procedure for TS characterization, which lists all harmful error patterns up to a given weight for the LDPC code and decoder. Based on this decoder-aware trapping set characterization for LDPC codes, we propose a model-driven deep neural network (DNN) framework that unfolds the decoding iterations, to design the \emph{decoder diversity of finite alphabet iterative decoders (FAIDs)}. Our decoder diversity DNN-FAID delivers excellent waterfall performance along with a low error floor.
In this paper, we propose a new approach to design finite alphabet iterative decoders (FAIDs) for Low-Density Parity Check (LDPC) codes over binary symmetric channel (BSC) via recurrent quantized neural networks (RQNN). We focus on the linear FAID class and use RQNNs to optimize the message update look-up tables by jointly training their message levels and RQNN parameters. Existing neural networks for channel coding work well over Additive White Gaussian Noise Channel (AWGNC) but are inefficient over BSC due to the finite channel values of BSC fed into neural networks. We propose the bit error rate (BER) as the loss function to train the RQNNs over BSC. The low precision activations in the RQNN and quantization in the BER cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We leverage straight-through estimators as surrogate gradients to tackle this issue and provide a joint training scheme. We show that the framework is flexible for various code lengths and column weights. Specifically, in high column weight case, it automatically designs low precision linear FAIDs with superior performance, lower complexity, and faster convergence than the floating-point belief propagation algorithms in waterfall region.
Decoder diversity is a powerful error correction framework in which a collection of decoders collaboratively correct a set of error patterns otherwise uncorrectable by any individual decoder. In this paper, we propose a new approach to design the decoder diversity of finite alphabet iterative decoders (FAIDs) for Low-Density Parity Check (LDPC) codes over the binary symmetric channel (BSC), for the purpose of lowering the error floor while guaranteeing the waterfall performance. The proposed decoder diversity is achieved by training a recurrent quantized neural network (RQNN) to learn/design FAIDs. We demonstrated for the first time that a machine-learned decoder can surpass in performance a man-made decoder of the same complexity. As RQNNs can model a broad class of FAIDs, they are capable of learning an arbitrary FAID. To provide sufficient knowledge of the error floor to the RQNN, the training sets are constructed by sampling from the set of most problematic error patterns - trapping sets. In contrast to the existing methods that use the cross-entropy function as the loss function, we introduce a frame-error-rate (FER) based loss function to train the RQNN with the objective of correcting specific error patterns rather than reducing the bit error rate (BER). The examples and simulation results show that the RQNN-aided decoder diversity increases the error correction capability of LDPC codes and lowers the error floor.
Decoder diversity is a powerful error correction framework in which a collection of decoders collaboratively correct a set of error patterns otherwise uncorrectable by any individual decoder. In this paper, we propose a new approach to design the decoder diversity of finite alphabet iterative decoders (FAIDs) for Low-Density Parity Check (LDPC) codes over the binary symmetric channel (BSC), for the purpose of lowering the error floor while guaranteeing the waterfall performance. The proposed decoder diversity is achieved by training a recurrent quantized neural network (RQNN) to learn/design FAIDs. We demonstrated for the first time that a machine-learned decoder can surpass in performance a man-made decoder of the same complexity. As RQNNs can model a broad class of FAIDs, they are capable of learning an arbitrary FAID. To provide sufficient knowledge of the error floor to the RQNN, the training sets are constructed by sampling from the set of most problematic error patterns - trapping sets. In contrast to the existing methods that use the cross-entropy function as the loss function, we introduce a frame-error-rate (FER) based loss function to train the RQNN with the objective of correcting specific error patterns rather than reducing the bit error rate (BER). The examples and simulation results show that the RQNN-aided decoder diversity increases the error correction capability of LDPC codes and lowers the error floor.
This thesis investigates the optimization of Low-Density Parity-Check (LDPC) decoding through machine learning techniques, with a specific focus on 5G communication systems. Traditional LDPC decoding algorithms face a critical trade-off: Belief Propagation (BP) offers excellent error-correction performance but with high computational complexity, while Min-Sum (MS) reduces complexity at the cost of degraded performance. To address this limitation, a hybrid neural network architecture is proposed that incorporates learnable normalization and offset weights into the message-passing framework of LDPC decoders. This approach integrates multiplicative (normalization) and additive (offset) weights into the Tanner graph structure, allowing the decoder to adapt to specific channel conditions. A novel weighted binary cross-entropy loss function is developed that prioritizes entire block recovery by assigning different importance to information and parity bits during training. Experimental results demonstrate that ML-assisted decoder consistently outperforms conventional Min-Sum decoding, approaching the error-correction capability of Belief Propagation while maintaining comparable computational efficiency to Min-Sum. Comprehensive evaluations across different weight configurations (scalar and vector-based) reveal that increased parametrization generally improves decoding performance. Analysis of the impact of iteration counts shows that the optimal number of decoding iterations varies between different code configurations with studied Basegraph 1 case requiring 15 iterations for optimal performance while studied Basegraph 2 case benefits significantly from 20 iterations. The proposed ML-assisted LDPC decoder represents a significant advancement for 5G systems, offering improved error correction at lower signal strengths while balancing latency and computational requirements. This research contributes to the ongoing evolution of wireless technology by demonstrating how machine learning can effectively enhance traditional decoding algorithms, providing a practical pathway toward more efficient and adaptive error correction in modern communication systems.