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IEEE Open Journal of the Communications Society

Efficient Decoder Design for Low-Density Lattice Codes From the Lattice Viewpoint

Xuebo Wang, Wai Ho Mow

Published Jan 1, 2023• 0 citations
Publisher

Abstract

Low-density lattice codes (LDLCs) achieve near-capacity performance on additive white Gaussian noise (AWGN) channels. The$M$-Gaussian decoder is the state-of-the-art message passing decoder for LDLCs in terms of the error performance. However, this decoder has complexity$O(M^{d-1})$with messages represented by Gaussian mixtures, where$d$is the degree of an LDLC and$M$is the number of Gaussian functions for approximating each check node message. In this paper, we establish the correspondence between Gaussian functions for approximating a variable node message and points of a certain lattice. Based on this lattice viewpoint, the problem of approximating a variable node message is formulated as a lattice point enumeration (LPE) problem. Then, an LPE decoder with linear complexity$O(d)$is proposed. Our simulation results validate that the LPE decoder achieves almost the same error performance as the$M$-Gaussian decoder.

DOI
10.1109/ojcoms.2023.3305480

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Low density lattice codes (LDLC) are a family of lattice codes that can be decoded efficiently using a message-passing algorithm. In the original LDLC decoder, the message exchanged between variable nodes and check nodes are continuous functions, which must be approximated in practice. A promising method is Gaussian approximation (GA), where the messages are approximated by Gaussian functions. However, current GA-based decoders share two weaknesses: firstly, the convergence of these approximate decoders is unproven; secondly, the best known decoder requires O(2d)O(2^d)O(2d) operations at each variable node, where ddd is the degree of LDLC. It means that existing decoders are very slow for long codes with large ddd. The contribution of this paper is twofold: firstly, we prove that all GA-based LDLC decoders converge sublinearly (or faster) in the high signal-to-noise ratio (SNR) region; secondly, we propose a novel GA-based LDLC decoder which requires only O(d)O(d)O(d) operations at each variable node. Simulation results confirm that the error correcting performance of proposed decoder is the same as the best known decoder, but with a much lower decoding complexity.
Low density lattice codes (LDLC) can be decoded efficiently using iterative decoding, and approach the capacity of the AWGN channel. In the iterative LDLC decoder the messages are Gaussian mixtures. In any implementation, the Gaussian mixtures must be approximated. This work describes a three/two Gaussian parametric LDLC decoder. Internally at the variable node the periodic Gaussian mixtures are approximated with three or two Gaussians, while the messages between nodes are single Gaussians. The proposed approximation is more accurate than the previously-proposed approximation. Numerical results shows that for moderate dimension, e.g. n = 1, 000, the two Gaussian approximation is sufficient for accurate performance. But for large dimension, e.g. n = 10, 000, three Gaussians are needed.
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Multilevel LDPC Lattices With Efficient Encoding and Decoding and a Generalization of Construction D′\text{D}'D′
Lattice codes are elegant and powerful structures that not only achieve the capacity of the additive white Gaussian noise (AWGN) channel but are also a key ingredient to many multiterminal schemes that exploit linearity properties. However, constructing lattice codes that can realize these benefits with low complexity is still a challenging problem. In this paper, efficient encoding and decoding algorithms are proposed for multilevel binary low-density parity-check (LDPC) lattices constructed via Construction D' whose complexity is linear in the total number of coded bits. Moreover, a generalization of Construction D' is proposed, which relaxes some of the nesting constraints on the parity-check matrices of the component codes, leading to a simpler and improved design. Based on this construction, low-complexity multilevel LDPC lattices are designed whose performance under multistage decoding is comparable to that of polar lattices and close to that of low-density lattice codes on the power-unconstrained AWGN channel.