Finite-Dimensional Infinite Constellations

In the setting of a Gaussian channel without power constraints, proposed by Poltyrev in 1994, the codewords are points in an
n
-dimensional Euclidean space (an infinite constellation) and the tradeoff between their density and the error probability is considered. The normalized log density (NLD) plays the role of the communication rate, and capacity as well as error exponent bounds for this setting are known. This paper considers the infinite constellation setting in the finite block-length (dimension) regime. A simplified expression for Poltyrev's achievability bound is found and it is shown to be closely related to the sphere converse bound and to a recently proposed achievability bound based on point processes. The bounds are then analyzed asymptotically for growing
n
: for fixed NLD, the bounds turn out to be extremely tight compared to previous error exponent analysis. For fixed error probability ε, it is shown that the gap of the highest achievable NLD to the optimal NLD (Poltyrev's capacity) is approximately √{[1/(2
n
)]}
Q
^-1
(ε) , where
Q
is the standard complementary Gaussian cumulative distribution function, thus extending the channel dispersion analysis to infinite constellations. 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. Finally, the new tight bounds are compared to state-of-the-art coding schemes.