The dispersion of infinite constellations
Proceedings of the IEEE International Symposium on Information Theory, St. Petersburg, Russia, July 2011

In the setting of a Gaussian channel without power constraints, proposed by Poltyrev, the codewords are points in an n-dimensional Euclidean space (an  infinite   constellation ) and their optimal density is considered. Poltyrev's “capacity” is the highest achievable normalized log density (NLD) with vanishing error probability. This capacity as well as error exponents for this setting are known. In this work we consider the optimal NLD for a fixed, nonzero error probability, as a function of the codeword length (dimension) n. We show that as n grows, the gap to capacity is inversely proportional (up to the first order) to the square-root of n where the proportion constant is given by the inverse Q-function of the allowed error probability, times the square root of 1/2. In an analogy to similar result in channel coding, the  dispersion  of  infinite   constellations  is 1/2 nat 2  per channel use. We show that this optimal convergence rate can be achieved using lattices, therefore the result holds for the maximal error probability as well. Connections to the error exponent of the power constrained Gaussian channel and to the volume-to-noise ratio as a figure of merit are discussed.