Expurgated infinite constellations at finite dimensions

We revisit the setting of a Gaussian channel without power constraints, proposed by Poltyrev, where the codewords are points in Euclidean space and their density is considered instead of the communication rate. We refine the expurgation technique (proposed by Poltyrev for the derivation of the error exponent) to the finite dimensions case and obtain a finite-dimensional achievability bound. While the expurgation exponent improves upon the random coding exponent only for certain rates (below a rate known as δ_ex), we show that for finite dimensions the expurgation technique is useful for a broader range of rates. In addition, we present precise asymptotical analysis of the expurgation bound and find the sub-exponential terms, which turn out to be non-negligible.