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International Symposium on Problems of Redundancy in Information and Control Systems (REDUNDANCY)

Analysis of Binary and Ternary Message Passing Decoding for Generalized LDPC Codes

Emna Ben Yacoub, Gianluigi Liva

Published Oct 25, 2021• 2 citations
Publisher

Abstract

Binary and a ternary message passing decoding algorithms for generalized low-density parity-check codes are defined and analyzed. In all algorithms, the variable nodes exploit soft channel information. However, the exchanged messages between variable and check nodes are binary or ternary. Two types of local decoders are assumed at the check nodes, namely an optimum a-posteriori probability soft-input soft-output decoder, and a more practical bounded distance decoder. A density evolution analysis for irregular code ensembles is developed for all the introduced algorithms, which allows obtaining the scaling coefficients needed for the variable node operation. Finite-length simulation results confirm the validity of the asymptotic analysis.

DOI
10.1109/redundancy52534.2021.9606468

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IEEE Communications Letters•Oct 30, 2014•2 citations
Binary and a ternary message passing decoding algorithms for generalized low-density parity-check codes are defined and analyzed. In all algorithms, the variable nodes exploit soft channel information. However, the exchanged messages between variable and check nodes are binary or ternary. Two types of local decoders are assumed at the check nodes, namely an optimum a-posteriori probability soft-input soft-output decoder, and a more practical bounded distance decoder. A density evolution analysis for irregular code ensembles is developed for all the introduced algorithms, which allows obtaining the scaling coefficients needed for the variable node operation. Finite-length simulation results confirm the validity of the asymptotic analysis.
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.
A ternary message passing (TMP) decoding algorithm for low-density parity-check codes is developed. All messages exchanged between variable and check nodes have a ternary alphabet, and the variable nodes exploit soft information from the channel. A density evolution analysis is developed for unstructured and protograph-based ensembles. For unstructured ensembles the stability condition is derived. Optimized ensembles for TMP decoding show asymptotic gains of up to 0.6 dB with respect to ensembles optimized for binary message passing decoding. Finite length simulations of codes from TMP-optimized ensembles show gains of up to 0.5 dB under TMP compared to protograph-based codes designed for unquantized belief propagation decoding.
We consider a class of message-passing decoders for low-density parity-check (LDPC) codes whose messages are binary valued. We prove that if the channel is symmetric and all codewords are equally likely to be transmitted, an optimum decoding rule (in the sense of minimizing message error rate) should satisfy certain symmetry and isotropy conditions. Using this result, we prove that Gallager's Algorithm B achieves the optimum decoding threshold among all binary message-passing decoding algorithms for regular codes. For irregular codes, we argue that when the nodes of the message-passing decoder do not exploit knowledge of their decoding neighborhood, optimality of Gallager's Algorithm B is preserved. We also consider the problem of designing irregular LDPC codes and find a bound on the achievable rates with Gallager's Algorithm B. Using this bound, we study the case of low error-rate channels and analytically find good degree distributions for them.
Low complexity decoding algorithms are necessary to meet data rate requirements in excess of 1 Tbps. In this paper, we study one and two bit message passing algorithms for belief propagation decoding of low-density parity-check (LDPC) codes and analyze them by density evolution. The variable nodes (VNs) exploit soft information from the channel output. To decrease the data flow, the messages exchanged between check nodes (CNs) and VNs are represented by one or two bits. The newly proposed quaternary message passing (QMP) algorithm is compared asymptotically and in finite length simulations to binary message passing (BMP) and ternary message passing (TMP) for spectrally efficient communication with higher-order modulation and probabilistic amplitude shaping (PAS). To showcase the potential for high throughput forward error correction, spatially coupled LDPC codes and a target spectral efficiency (SE) of 3 bits/QAM symbol are considered. Gains of about 0.7 dB and 0.1 dB are observed compared to BMP and TMP, respectively. The gap to unquantized belief propagation (BP) decoding is reduced to about 0.75 dB. For smaller code rates, the gain of QMP compared to TMP is more pronounced and amounts to 0.24 dB in the considered example.
Low complexity decoding algorithms are necessary to meet data rate requirements in excess of 1 Tbps. In this paper, we study one and two bit message passing algorithms for belief propagation decoding of low-density parity-check (LDPC) codes and analyze them by density evolution. The variable nodes (VNs) exploit soft information from the channel output. To decrease the data flow, the messages exchanged between check nodes (CNs) and VNs are represented by one or two bits. The newly proposed quaternary message passing (QMP) algorithm is compared asymptotically and in finite length simulations to binary message passing (BMP) and ternary message passing (TMP) for spectrally efficient communication with higher-order modulation and probabilistic amplitude shaping (PAS). To showcase the potential for high throughput forward error correction, spatially coupled LDPC codes and a target spectral efficiency (SE) of 3 bits/QAM symbol are considered. Gains of about 0.7 dB and 0.1 dB are observed compared to BMP and TMP, respectively. The gap to unquantized belief propagation (BP) decoding is reduced to about 0.75 dB. For smaller code rates, the gain of QMP compared to TMP is more pronounced and amounts to 0.24 dB in the considered example.
A new low-complexity message passing algorithm is described for decoding low-density parity-check (LDPC) codes by exchanging binary messages. The algorithm computes the local maximum-likelihood binary message (LMLBM) at each symbol node, given the combination of local channel information and partial syndrome components from adjacent parity check nodes. When channel information is quantized, the locally ML messages are pre-computed and stored in a dynamic global lookup table. The proposed algorithm uses memoryless extrinsic messages so that density evolution thresholds can be directly computed. Thresholds are obtained for regular ensembles, predicting good performance on quantized binary-input additive white Gaussian noise (biAWGN) channels.