The Optimal Density of Infinite Constellations for the Gaussian Channel

The setting of a Gaussian channel without power constraints is considered. In this setting the codewords are points in an n-dimensional Euclidean space (an infinite constellation). The channel coding analog of the number of codewords is the density of the constellation points, and the analog of the communication rate is the normalized log density (NLD). The highest achievable NLD with vanishing error probability (which can be thought of as the capacity) is known, as well as error exponents for the setting. In this work we are interested in t he optimal NLD for communication when a fixed, nonzero error probability is allowed. In classi cal channel coding the gap to capacity is characterized by the channel dispersion (and cannot be derived from error exponent theory). In the unconstrained setting, we show that as the codeword length (dimension) n grows, the gap to the highest achievable NLD is inversely proportional (to the first order) to the square root of the block length. We give an explicit expression for the proportion constant, which is given by the inverse Q-function of the allowed error probability, times the square root of 1 . In an analogy to a similar result in classical channel coding, it follows tha t the dispersion of infinite constellations is given by 1 nat 2 per channel use. We show that this optimal convergence rate can be achieved using lattices, therefore the result holds for the maximal e rror 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.