We consider two recent asymptotics: moderate deviations and exact asymptotics.
In the former, we consider the optimal error performance of the sequence of
codes with rates increasing to the capacity and prove that for a particular
regime of increase, the error probability vanishes sub-exponentially fast with a
rate related to the dispersion of the channel. In the latter, we concentrate on
the sub-exponential factors of the well-known exponentially decaying bounds on
the error probability and overview our recent refinements of the sphere-packing
and random-coding bounds. The order of sub-exponential factors of these
refinements almost coincide in general, and are equal for the case of symmetric
channels. We conclude with a discussion on the unification of the existing
asymptotic regimes.