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

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.