  The intuitive sphere-packing argument is used to obtain
analytically-tractable closed-form approximations for achievable information
rates of coded modulation transmission systems, for which only
analytically-intractable expressions are available in the literature. These
approximations provide a number of insights, possess useful properties and
facilitate design/optimization of such systems. They apply to constellations of
various cardinalities (including large ones), are simple yet reasonably
accurate over the whole signal-to-noise ratio range, and compare favorably to
the achieved rates of recent state-of-the art experiments.
