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arXiv:2411.13876

Iterative decoding of short BCH codes and its post-processing

Guangwen Li, Xiao Yu

Published Nov 21, 2024• 0 citations
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Abstract

Effective iterative decoding of short BCH codes faces two primary challenges: identifying an appropriate parity-check matrix and accelerating decoder convergence. To address these issues, we propose a systematic scheme to derive an optimized parity-check matrix through a heuristic approach. This involves a series of binary sum and row shift operations, resulting in a low-density, quasi-regular column weight distribution with a reduced number of shortest cycles in the underlying redundant Tanner graph. For the revised normalized min-sum decoder, we concurrently integrate three types of random permutations into the alternated messages across iterations, leading to significantly faster convergence compared to existing methods. Furthermore, by utilizing the iterative trajectories of failed normalized min-sum decoding, we enhance the reliability measurement of codeword bits with the assistance of a neural network model from prior work, which accommodates more failures for the post-processing of ordered statistics decoding. Additionally, we report the types of undetected errors for the design of iterative decoders for short BCH codes, which potentially challenge efforts to approach the maximum likelihood limit. Extensive simulations demonstrate that the proposed hybrid framework achieves an attractive balance between performance, latency, and complexity.

DOI
10.48550/arxiv.2411.13876

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Effective iterative decoding of short BCH codes faces two primary challenges: identifying an appropriate parity-check matrix and accelerating decoder convergence. To address these issues, we propose a systematic scheme to derive an optimized parity-check matrix through a heuristic approach. This involves a series of binary sum and row shift operations, resulting in a low-density, quasi-regular column weight distribution with a reduced number of shortest cycles in the underlying redundant Tanner graph. For the revised normalized min-sum decoder, we concurrently integrate three types of random permutations into the alternated messages across iterations, leading to significantly faster convergence compared to existing methods. Furthermore, by utilizing the iterative trajectories of failed normalized min-sum decoding, we enhance the reliability measurement of codeword bits with the assistance of a neural network model from prior work, which accommodates more failures for the post-processing of ordered statistics decoding. Additionally, we report the types of undetected errors for the design of iterative decoders for short BCH codes, which potentially challenge efforts to approach the maximum likelihood limit. Extensive simulations demonstrate that the proposed hybrid framework achieves an attractive balance between performance, latency, and complexity.
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.
Ordered statistics decoding has been instrumental in addressing decoding failures that persist after normalized min-sum decoding in short low-density parity-check codes. Despite its benefits, the high computational complexity of effective ordered statistics decoding has limited its application in complexity-sensitive scenarios. To mitigate this issue, we propose a novel variant of the ordered statistics decoder. This approach begins with the design of a neural network model that refines the measurement of codeword bits, utilizing iterative information from normalized min-sum decoding failures. Subsequently, a fixed decoding path is established, comprising a sequence of blocks, each featuring a variety of test error patterns. The introduction of a sliding window-assisted neural model facilitates early termination of the ordered statistics decoding process along this path, aiming to balance performance and computational complexity. Comprehensive simulations and complexity analyses demonstrate that the proposed hybrid method matches state-of-the-art approaches across various metrics, particularly excelling in reducing latency.
This paper proposes a quasi-BP decoding scheme for BCH codes that preserves the parallelizable structure of belief propagation while exploiting code automorphisms and optimized redundant parity-check matrices. To eliminate the computationally expensive tanh⁡\tanhtanh and tanh⁡−1\tanh^{-1}tanh−1 functions in check node updates, we further introduce a neural-network-based variant that replaces them with a lightweight convolutional neural network trained under a triple-constraint loss function enforcing non-negativity and order consistency. Simulation results for three BCH codes demonstrate that quasi-BP decoding achieves competitive frame error rate performance, with a gap within 0.25 decibels compared with belief propagation decoding of an LDPC code of similar blocklength. The neural-network-based variant incurs negligible performance loss while enabling stable deployment with arithmetic operations on hardware accelerators. Concatenation with an ordered statistics decoding variant further bridges the gap to the maximum-likelihood bound. Hence, the proposed schemes offer a viable path toward high-throughput, low-latency decoding of BCH codes in next-generation communication systems.
As successfully adopted in standard long code scenarios, belief propagation (BP) decoding has been considered a promising universal decoding candidate for next-generation wireless communications. However, when applied to short codes, BP decoding suffers from poor error correction performance due to harmful cycle structures in the Tanner graph. In this paper, we address this issue by designing a structured, sparse parity-check matrix (ssPCM) framework, composed of multiple cycle-free parity-check row blocks (PCRBs). The resulting ssPCMs feature regular row weights and perform better than the state-of-theart 4 -cycle-free row redundant PCMs across Bose-Chaudhuri-Hocquenghem (BCH) codes of length 63.
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Abstract This paper introduces a hybrid decoding architecture that serially couples a normalized min-sum (NMS) decoder with reinforced ordered statistics decoding (OSD) to achieve near-maximum likelihood (ML) performance for short linear block codes. The framework incorporates several key innovations: a decoding information aggregation model that employs a convolutional neural network to refine bit reliability estimates for OSD using the soft-output trajectory of the NMS decoder; an adaptive decoding path for OSD, initialized by the arranged list of the most a priori likely tests algorithm and dynamically updated with empirical data; and a sliding window assisted model that enables early termination of test error patterns’ traversal, curbing complexity with minimal performance loss. For short high-rate codes, a dedicated undetected error detector identifies erroneous NMS outcomes that satisfy parity checks, ensuring they are forwarded to OSD for correction. Extensive simulations on LDPC, BCH, and RS codes demonstrate that the proposed hybrid decoder delivers a competitive trade-off, achieving near-ML frame error rate performance while maintaining advantages in throughput, latency, and complexity over state-of-the-art alternatives.