Message variance convergence condition for generalizations of LDLC lattices

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.