Low density lattice codes (LDLCs) are high-dimensional lattices with a sparse inverse generator matrix that can be decoded efficiently using iterative decoding. In the iterative LDLC decoder, the messages are Gaussian mixtures, and for any implementation, the Gaussian mixtures must be approximated. This paper describes a parametric LDLC decoding algorithm, where internally at the variable node, infinite Gaussian mixtures are approximated with three or two Gaussians, while the messages between nodes are single Gaussians. Strengths of the algorithm include its simplicity and suitability for analysis. Analysis is performed by evaluating the Kullback-Leibler divergence between the true messages and the three/two Gaussian approximation. The approximation using three or two Gaussians is more accurate than previously proposed approximations. Also, noise thresholds for the proposed LDLC decoder are presented, and the proposed decoder reduces the noise thresholds 0.05 dB compared with previous parametric decoders. The numerical results show that for n = 100 and n = 1000, the two-Gaussian approximation is the same as the full-complexity decoder. But when the dimension is n = 10 000, a three-Gaussian approximation is needed.

For low-density lattice codes (LDLC), there is a condition for convergence of the messages under belief-propagation decoding, specifically a condition on the inverse generator matrix for the variances in the Gaussian mixture to converge. It offers guidance on the design of Latin square LDLC lattices. This paper revisits this condition, and then describes two other constructions, a modified Latin square construction and a triangular array code construction. We illustrate how the condition can be applied, demonstrating the validity of the condition for more general LDLC lattices.