  A belief-propagation decoder for low-density lattice codes is given which
represents messages explicitly as a mixture of Gaussians functions. The key
component is an algorithm for approximating a mixture of several Gaussians with
another mixture with a smaller number of Gaussians. This Gaussian mixture
reduction algorithm iteratively reduces the number of Gaussians by minimizing
the distance between the original mixture and an approximation with one fewer
Gaussians.
  Error rates and noise thresholds of this decoder are compared with those for
the previously-proposed decoder which discretely quantizes the messages. The
error rates are indistinguishable for dimension 1000 and 10000 lattices, and
the Gaussian-mixture decoder has a 0.2 dB loss for dimension 100 lattices. The
Gaussian-mixture decoder has a loss of about 0.03 dB in the noise threshold,
which is evaluated via Monte Carlo density evolution. Further, the
Gaussian-mixture decoder uses far less storage for the messages.
