  Shannon's analysis of the fundamental capacity limits for memoryless
communication channels has been refined over time. In this paper, the maximum
volume $M_\avg^*(n,\epsilon)$ of length-$n$ codes subject to an average
decoding error probability $\epsilon$ is shown to satisfy the following tight
asymptotic lower and upper bounds as $n \to \infty$: \[ \underline{A}_\epsilon
+ o(1) \le \log M_\avg^*(n,\epsilon) - [nC - \sqrt{nV_\epsilon}
\,Q^{-1}(\epsilon) + \frac{1}{2} \log n] \le \overline{A}_\epsilon + o(1) \]
where $C$ is the Shannon capacity, $V_\epsilon$ the $\epsilon$-channel
dispersion, or second-order coding rate, $Q$ the tail probability of the normal
distribution, and the constants $\underline{A}_\epsilon$ and
$\overline{A}_\epsilon$ are explicitly identified. This expression holds under
mild regularity assumptions on the channel, including nonsingularity. The gap
$\overline{A}_\epsilon - \underline{A}_\epsilon$ is one nat for weakly
symmetric channels in the Cover-Thomas sense, and typically a few nats for
other symmetric channels, for the binary symmetric channel, and for the $Z$
channel. The derivation is based on strong large-deviations analysis and
refined central limit asymptotics. A random coding scheme that achieves the
lower bound is presented. The codewords are drawn from a capacity-achieving
input distribution modified by an $O(1/\sqrt{n})$ correction term.
