Spectral Thinning in GLD Lattices

This paper deals with generalized low-density (GLD) lattices which have been recently shown to be an excellent family of lattices for communication over the Gaussian channel. Under iterative decoding, numerical results for GLD lattices show an error rate per lattice coordinate extremely close to Poltyrev theoretical limit of an infinite constellation. These results are found for GLD ensembles with degree-2 variable nodes. In the error floor region, the probability of error per lattice coordinate decreases when the lattice dimension grows. This phenomenon is similar to spectral thinning in parallel concatenated convolutional codes (Turbo codes) also known as interleaving gain. The present work aims to give the theoretical explanation of this error floor decay as a function of the lattice dimension. Namely, we show how spectral thinning applies to codes on graphs associated with GLD lattices. This stands on the proof that the number of cycles of small length in random bipartite graphs follows a Poisson distribution. Our theorem on cycles is an adaptation of a theorem by Béla Bollobás to irregular bipartite graphs.