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arXiv:1806.05524

Fast Decoding of Low Density Lattice Codes

Shuiyin Liu, Yi Hong, Emanuele Viterbo, Alessia Marelli, Rino Micheloni

Published Jun 14, 2018• 1 citation
arXivHTMLPDF

Abstract

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 $O(2^d)$ operations at each variable node, where $d$ is the degree of LDLC. It means that existing decoders are very slow for long codes with large $d$. 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 $O(d)$ 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.

DOI
10.48550/arxiv.1806.05524

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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 (LDLCs) have been shown to approach the AWGN channel capacity and can be decoded by a message passing decoder. The messages passed between check nodes and variable nodes are probability density functions, which are approximated by Gaussian mixtures in existing LDLC decoders. However, in the best-known M-Gaussian decoder, the Gaussian mixtures for approximating variable node messages still contain unnecessary Gaussian components, resulting in exponential decoding complexity in the LDLC degree d. In this paper, an efficient LDLC decoder is proposed from the lattice viewpoint, which only finds out important Gaussian components for approximating the variable node messages. The decoding complexity for the proposed decoder is linear in d. Numerical results show that the proposed decoder can achieve significant complexity saving without degrading the performance compared with the M-Gaussian decoder. In particular, the runtime saving is 75.3% when the code length is 10000 and d is 7.
Low-density lattice codes (LDLCs) achieve near-capacity performance on additive white Gaussian noise (AWGN) channels. TheMMM-Gaussian decoder is the state-of-the-art message passing decoder for LDLCs in terms of the error performance. However, this decoder has complexityO(Md−1)O(M^{d-1})O(Md−1)with messages represented by Gaussian mixtures, wheredddis the degree of an LDLC andMMMis 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 complexityO(d)O(d)O(d)is proposed. Our simulation results validate that the LPE decoder achieves almost the same error performance as theMMM-Gaussian decoder.
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.
Gottesman-Kitaev-Preskill (GKP) codes provide a family of promising schemes for encoding discrete quantum information (qudits) into infinite-dimensional bosonic modes based on mathematical lattices. While such codes, when concatenated with discrete-variable codes, are relatively well studied, the construction and decoding of native GKP codes has largely remained open due to the computationally hard problems encountered. To address this challenge, we advocate a strategy of co-designing the decoder and the quantum error-correcting code itself by constructing lattices for which decoding is feasible: The requirement of efficient decoding effectively determines the quantum error-correcting code. This construction is built on classical low-density lattice codes (LDLCs), a lattice analogue of low-density parity-check codes, here lifted to families of GKP codes. Concretely, we introduce quantum versions of classical, randomly constructed LDLCs. We show that after suitable dimensionality reduction these codes have code properties comparable to or better than concatenated GKP-surface codes of equal number of modes. However, the GKP-LDLCs constructed here do not have a strictly sparse parity check matrix, which motivates our study of the performance of natively analog message-passing decoders originally developed for LDLCs when applied to concatenated GKP-LDPC codes. We show that the fully analog, linear-time decoder achieves performances close to state-of-the-art hybrid qubit-analog decoders. To facilitate future research on the structure and performance of general GKP codes, the relevant source code will be released in open-source Julia packages LatticeDecoder.jl and SymplecticGKP.jl.
The recent work of Sommer, Feder and Shalvi presented a new family of codes called low density lattice codes (LDLC) that can be decoded efficiently and approach the capacity of the AWGN channel. A linear time iterative decoding scheme which is based on a message-passing formulation on a factor graph is given. In the current work we report our theoretical findings regarding the relation between the LDLC decoder and belief propagation. We show that the LDLC decoder is an instance of non-parametric belief propagation and further connect it to the Gaussian belief propagation algorithm. Our new results enable borrowing knowledge from the non-parametric and Gaussian belief propagation domains into the LDLC domain. Specifically, we give more general convergence conditions for convergence of the LDLC decoder (under the same assumptions of the original LDLC convergence analysis). We discuss how to extend the LDLC decoder from Latin square to full rank, non-square matrices. We propose an efficient construction of sparse generator matrix and its matching decoder. We report preliminary experimental results which show our decoder has comparable symbol to error rate compared to the original LDLC decoder.
presented a new family of codes called low density lattice codes (LDLC) that can be decoded efficiently and approach the capacity of the AWGN channel. A linear time iterative decoding scheme which is based on a message-passing formulation on a factor graph is given. In the current work we report our theoretical findings regarding the relation between the LDLC decoder and belief propagation. We show that the LDLC decoder is an instance of non-parametric belief propagation and further connect it to the Gaussian belief propagation algorithm. Our new results enable borrowing knowledge from the non-parametric and Gaussian belief propagation domains into the LDLC domain. Specifically, we give more general convergence conditions for convergence of the LDLC decoder (under the same assumptions of the original LDLC convergence analysis). We discuss how to extend the LDLC decoder from Latin square to full rank, non-square matrices. We propose an efficient construction of sparse generator matrix and its matching decoder. We report preliminary experimental results which show our decoder has comparable symbol to error rate compared to the original LDLC decoder. I.