Diversity of Low-Density Lattices

Abstract: The non-ergodic fading channel is a useful model for various wireless communication channels in both indoor and outdoor environments. Building on Poltyrev's work on infinite lattice constellations for the Gaussian channel, we derive a Poltyrev outage limit (POL) for lattices in presence of block fading. We prove that the diversity order of this POL is equal to the number of degrees of freedom in the channel. Further, we describe full-diversity constructions of real lattices defined by their integer-check matrix, i.e., the inverse of their generator matrix. In the first construction suited to maximum-likelihood decoding, these lattices are defined by sparse (low-density) or non-sparse integer-check matrices. Based on a special structure of the lattice binary image, a second full-diversity lattice construction is described for sparse integer-check matrices in the context of iterative probabilistic decoding. Full diversity is theoretically proved in both cases. We also propose a method to construct lattices for diversity order 4 suitable for iterative probabilistic decoding. Computer simulation results for diversity order L = 2 and L = 4 confirm that the proposed low-density lattices attain the maximal diversity order. The newly defined POL is used during this simulation to declare an outage error without decoding, which drastically improves the decoding runtime.