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arXiv:2610.21187 · IEEE Communications Letters

Four-Level Quantized Soft Information Aided Hard-Decision Decoder for Product-Like Codes

Xiaowei Wu, Min Qiu, Yueying Zhan, Lei Yang

Published Jan 1, 2026• 0 citations
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Abstract

We propose a novel low-complexity soft-aided hard-decision decoding scheme for product-like codes, termed iterative bounded-distance decoding (iBDD) with graded scaled reliability and bit-flipping (GSRF). This scheme operates on four-level quantized soft information. It incorporates a graded scaled reliability rule to assign decoding confidence to the bounded-distance decoding result, and employs a bit-flipping rule based on bit-level and component-code-level reliability. A selective re-decoding rule is designed to further facilitate the error performance while keeping the number of BDD attempts in reasonably low levels. The decoding parameter design of iBDD-GSRF is investigated for product codes and staircase codes, respectively. Simulation results on both codes verify that iBDD-GSRF achieves a gain of more than 0.4 dB over the conventional iBDD, comparable to several existing unquantized soft-aided decoders.

DOI
10.1109/lcomm.2026.3737455

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We propose a novel low-complexity soft-aided hard-decision decoding scheme for product-like codes, termed iterative bounded-distance decoding (iBDD) with graded scaled reliability and bit-flipping (GSRF). This scheme operates on four-level quantized soft information. It incorporates a graded scaled reliability rule to assign decoding confidence to the bounded-distance decoding result, and employs a bit-flipping rule based on bit-level and component-code-level reliability. A selective re-decoding rule is designed to further facilitate the error performance while keeping the number of BDD attempts in reasonably low levels. The decoding parameter design of iBDD-GSRF is investigated for product codes and staircase codes, respectively. Simulation results on both codes verify that iBDD-GSRF achieves a gain of more than 0.4 dB over the conventional iBDD, comparable to several existing unquantized soft-aided decoders.
We propose a novel soft-aided hard-decision decoding scheme for product codes based on two-bit quantized reliability markers. In particular, one bit is used for a two-level tolerance miscorrection check to validate the bounded distance decoding (BDD) result, while the other identifies bits to be randomly flipped upon BDD failure and miscorrection. List decoding is incorporated to further facilitate the error performance. Density evolution is performed for the design of the quantization thresholds. Compared to iterative BDD, the proposed scheme requires only moderate increases in memory and computational overhead. Simulation results verify that our scheme outperforms several unquantized soft-aided decoders, while maintaining small memory costs and low BDD attempts.
We propose a novel binary message passing decoding algorithm for product-like codes based on bounded distance decoding (BDD) of the component codes. The algorithm, dubbed iterative BDD with scaled reliability (iBDD-SR), exploits the channel reliabilities and is therefore soft in nature. However, the messages exchanged by the component decoders are binary (hard) messages, which significantly reduces the decoder data flow. The exchanged binary messages are obtained by combining the channel reliability with the BDD decoder output reliabilities, properly conveyed by a scaling factor applied to the BDD decisions. We perform a density evolution analysis for generalized low-density parity-check (GLDPC) code ensembles and spatially coupled GLDPC code ensembles, from which the scaling factors of the iBDD-SR for product and staircase codes, respectively, can be obtained. For the white additive Gaussian noise channel, we show performance gains up to 0.29 dB and 0.31 dB for product and staircase codes compared to conventional iterative BDD (iBDD) with the same decoder data flow. Furthermore, we show that iBDD-SR approaches the performance of ideal iBDD that prevents miscorrections.
Product codes (PCs) are typically decoded using iterative bounded distance decoding (iBDD) to ensure a low decoding complexity. To obtain further performance gain, a soft-aided decoding algorithm, termed the iBDD with scaled reliability (iBDD-SR), was proposed for PCs. In this paper, we propose an enhanced iBDD-SR by introducing threshold and memory when passing messages between the component decoders. The resulting algorithm is referred to as the threshold-based binary message passing (TB-BMP) with memory. In the proposed decoding algorithm, the soft reliability of the BDD output at the current half-iteration is a weighted sum of the BDD output, the channel reliability, and the content of the memory unit, where the content of the memory unit at the current half-iteration is related to the selected threshold and the BDD output at last half-iteration. Due to the existence of memory, the Bayesian network is used to model the decoding process of the TB-BMP. Based on the Bayesian network, we derive the density evolution (DE) equations for the TB-BMP under the constraint of extrinsic message passing (EMP). The analytical results of the DE analysis can be used to guide the selection of the parameters of the TB-BMP decoder. Extensive simulation results show that the TB-BMP decoder outperforms the iBDD-SR over the binary-input additive white Gaussian noise (Bi-AWGN) channels. In particular, for a PC based on a two-error-correcting extended Bose-Chaudhuri-Hocquenghem (BCH) code of length 256, the TB-BMP decoder performs about 0.28 dB better than the iBDD-SR at a bit error rate (BER) of 10-7.
We present a simple soft-decision decoding algorithm that modifies Sipser and Spielman's (see IEEE Trans, Inform. Theory, vol.42, p.1710-22, Nov. 1996) hard-decision sequential "bit-flipping" algorithm for decoding expander codes. The algorithm incorporates symbol reliability information and a simple "taboo" function that avoids repeated flipping of the same bit. The two algorithms have comparable simplicity, but simulations show that the soft-decision algorithm results in both improved performance and-because fewer decoding iterations are necessary-improved speed.
In this study, a new method for improving hard-decision bit-flipping (BF) decoding is proposed for low-density parity-check codes. The flipping criterion is derived theoretically from the soft-decision belief-propagation decoding algorithm. The proposed BF decoding algorithm is channel independent and is improved by introducing a more efficient method for computing the reliability of the parity checks. Extensive simulations are provided to demonstrate the efficiency of the proposed algorithm. Simulation results show that the proposed algorithm can achieve about 1.6 dB coding gain improvement at BER=10−5 while reducing up to 53% iterations for decoding and maintaining low-decoding complexity, compared with the conventional weighted BF algorithm.