Performance of lattice codes over the Gaussian channel

We derive an upper bound on the error probability of lattice codes combined with Quadrature Amplitude Modulation (qam) over the additive white Gaussian noise channel. This bound depends on a lattice figure of merit and is readily put in exponential form by using Chernoff bound. An interesting lower bound is derived by a similar reasoning. We also examine the estimation of the average information rate based upon the continuous approximation of the average power normalized to two dimensions, and suggest to improve it by using the sphere packing idea. Examples of performance evaluation are given for a few lattices. Finally, we present upper and lower bounds on the best fundamental coding gains per dimension (due to both density and thickness) for an arbitrarily large number of dimensions. It is shown in the Appendix that, as the Ungerboeck codes, the lattice codes do not shape the signal power spectrum.