Fast Decoding of Low Density Lattice Codes

TL;DR

This paper proves convergence of Gaussian approximation-based LDLC decoders at high SNR and introduces a new decoder with linear complexity per node, maintaining error correction performance.

Contribution

It demonstrates convergence of GA-based LDLC decoders at high SNR and proposes a novel low-complexity decoder with linear per-node operations.

Findings

Proven sublinear (or faster) convergence of GA-based LDLC decoders at high SNR.

Developed a new LDLC decoder requiring only O(d) operations per variable node.

Achieved similar error correction performance with significantly reduced decoding complexity.

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.