On optimal constellations for BICM at low SNR

In this paper we study the problem of finding capacity-maximizing constellations in BICM for asymptotically low signal-to-noise ratios (SNRs). We base our analysis on the so-called Hadamard transform and on a linear approximation of the BICM capacity for asymptotically low SNRs. We fully characterize the set of constellations, input distributions, and binary labelings that achieve Shannon´s limit E_b/N₀ = -1.59 dB. For equiprobable input distributions, a constellation achieves this limit if and only if it is a linear projection of a hypercube.