Recent research has investigated refined bounds for transmission over
memoryless
channels. This paper derives the following general expression for the volume
of
optimal length-$n$ codes for memoryless channels with average error
probability
$0 < epsilon < 1/2$:
$M(n,epsilon) = exp{nC - sqrt{nV} ,Q^{-1}(epsilon) + frac{1}{2} log n +
A_n + o(1)}$
where $C$ is the Shannon capacity, $V$ the channel dispersion, or second-order
coding rate, and $A_n$ is a bounded sequence that can be explicitly
identified.
The derivation is based on strong large deviations analysis and is rooted in
Laplace's method. The family of codes that achieve
the optimal log-volume --- up to a few nats --- is characterized. While not
optimal, Shannon's random codes are third-order optimal.