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arXiv:1902.03575 · IEEE Transactions on Communications

Binary Message Passing Decoding of Product-Like Codes

Alireza Sheikh, Alexandre Graell i Amat, Gianluigi Liva

Published Sep 11, 2019• 49 citations
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Abstract

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.

DOI
10.1109/tcomm.2019.2940180

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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 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.
The performance of generalized low-density parity-check (GLDPC) codes under binary and ternary message passing decoding (BMP/TMP) is analyzed from a density evolution (DE) perspective. At the check nodes, two types of local decoders are considered, namely optimum a-posteriori probability (APP) soft-input soft-output decoding, and bounded distance decoding (BDD). The purpose is to shed light on the performance loss incurred by BMP and TMP decoding of GLDPC codes with respect to unquantized belief propagation (BP) decoding. A DE analysis for irregular code ensembles is developed for all the algorithms, which allows obtaining the scaling coefficients needed for the variable node operation of BMP and TMP decoders. The stability analysis for the case of bounded distance decoding at the check nodes is derived. The asymptotic DE analysis is confirmed by the finite-length simulation results. For the codes analyzed in this paper, which rely on extended Hamming component codes, the study shows that under BMP decoding, BDD at the check nodes yields almost the same performance as optimum APP check node processing, while under TMP decoding the loss incurred by the sub-optimum BDD at check nodes is within 0.7 dB, when compared with APP decoding at the check nodes.
In this paper, we propose a binary message-passing algorithm for decoding low-density parity-check (LDPC) codes. The algorithm substantially improves the performance of purely hard-decision iterative algorithms with a small increase in the memory requirements and the computational complexity. We associate a reliability value to each nonzero element of the code's parity-check matrix, and differentially modify this value in each iteration based on the sum of the extrinsic binary messages from the check nodes. For the tested random and finitegeometry LDPC codes, the proposed algorithm can perform as close as about 1 dB and 0.5 dB to belief propagation (BP) at the error rates of interest, respectively. This is while, unlike BP, the algorithm does not require the estimation of channel signal to noise ratio. Low memory and computational requirements and binary message-passing make the proposed algorithm attractive for high-speed low-power applications.