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arXiv:2604.04066

Quasi-BP for BCH Codes and its Optimization

Guangwen Li

Published Apr 5, 2026• 0 citations
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Abstract

This paper proposes a quasi-belief propagation decoder for BCH codes that systematically integrates domain knowledge--specifically, channel noise variance, the cyclic property of the codes, and the deliberate redundancy in their parity-check matrices--to enable efficient iterative decoding. We rigorously formalize this parallelizable decoder within an information-theoretic framework by tracking mutual information evolution through the constituent variable and check decoders, thereby validating the use of scattered EXIT charts as a tool for optimizing the decoder's parameters. At each iteration, an input dilation operation expands the set of messages, while a subsequent merging operation accelerates mutual information growth, ensuring rapid convergence. The proposed decoder achieves decoding performance approaching that of LDPC codes with comparable rate and blocklength, effectively pioneering the feasible deployment of BP-like decoding for high-density parity-check codes. The generality and robustness of the scheme are demonstrated through extensive simulations across codes of varying rates and blocklengths.

DOI
10.48550/arxiv.2604.04066

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This paper proposes a quasi-belief propagation decoder for BCH codes that systematically integrates domain knowledge--specifically, channel noise variance, the cyclic property of the codes, and the deliberate redundancy in their parity-check matrices--to enable efficient iterative decoding. We rigorously formalize this parallelizable decoder within an information-theoretic framework by tracking mutual information evolution through the constituent variable and check decoders, thereby validating the use of scattered EXIT charts as a tool for optimizing the decoder's parameters. At each iteration, an input dilation operation expands the set of messages, while a subsequent merging operation accelerates mutual information growth, ensuring rapid convergence. The proposed decoder achieves decoding performance approaching that of LDPC codes with comparable rate and blocklength, effectively pioneering the feasible deployment of BP-like decoding for high-density parity-check codes. The generality and robustness of the scheme are demonstrated through extensive simulations across codes of varying rates and blocklengths.
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.
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.
At the heart of modern coding theory lies the fact that low-density parity-check (LDPC) codes can be efficiently decoded by belief propagation (BP). The BP is an inference algorithm which operates on a graphical model of a code, and lends itself to low-complexity and high-speed implementations, making it the algorithm of choice in many applications. It has unprecedentedly good error rate performance, so good that when decoded by the BP, LDPC codes approach theoretical limits of channel capacity. However, this capacity approaching property holds only in the asymptotic limit of code length, while codes of practical lengths suffer abrupt performance degradation in the low noise regime known as the error floor phenomenon. Our study of error floor has led to an interesting and surprising finding that it is possible to design iterative decoders which are much simpler yet better than belief propagation! These decoders do not propagate beliefs but a rather different kind of messages that reflect the local structure of the code graph. This has opened a plethora of exciting theoretical problems and applications. This paper introduces this new paradigm.
In this paper, we propose a class of finite alphabet iterative decoder (FAID), called mutual information-maximizing quantized belief propagation (MIM-QBP) decoder, for decoding regular low-density parity-check (LDPC) codes. Our decoder follows the reconstruction-calculation-quantization (RCQ) decoding architecture that is widely used in FAIDs. We present the first complete and systematic design framework for the RCQ parameters, and prove that our design with sufficient precision at node update is able to maximize the mutual information between coded bits and exchanged messages. Simulation results show that the MIM-QBP decoder can always considerably outperform the state-of-the-art mutual information-maximizing FAIDs that adopt two-input single-output lookup tables for decoding. Furthermore, with only 3 bits being used for each exchanged message, the MIM-QBP decoder can outperform the floating-point belief propagation decoder at the high signal-to-noise ratio regions when testing on high-rate LDPC codes with a maximum of 10 and 30 iterations.
This paper proposes a two-stage iterative decoding scheme named matrix transform belief propagation scheme (MT-BP) based on conventional belief propagation (BP) algorithm for lowdensity parity-check (LDPC) codes.By introducing information measurement, location vectors and parity check matrix transform, the proposed method can establish additional parity check equations and take more information bits of variable nodes into consideration compared with conventional BP algorithm.The simulations are performed on random regular LDPC codes and algebraic quasi-cyclic codes (QC-LDPC), the experimental results show MT-BP method can efficiently reduce the bit error rate (BER).
In this paper, we propose an optimized belief propagation (OBP) based progressive edge-growth (PEG) method for constructing quasi-cyclic low density parity check (QC-LDPC) codes. In this proposed method, Tanner graphs are built by progressively appending the check nodes rather than the variable nodes. Moreover, this OBP-PEG method considers three new constraint conditions to select the QC-LDPC code sets constructed by the PEG method. Compared with the PEG-based QC-LDPC decoders, the proposed OBP-PEG decoders decrease the number of the input ports of check-node processer by up to 25%, accelerates the convergence speed by up to 11.7% in layered decoding process, and improves the success probability by up to ten times in constructing fast-convergence QC-LDPC codes.