The Log-Volume of Optimal Codes for Memoryless Channels, Asymptotically Within a Few Nats

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, ∈) of length-n codes subject to an average decoding error probability ∈ is shown to satisfy the following tight asymptotic lower and upper bounds as n → ∞: A
_∈
+ o(1) ≤ log M*
_avg
(n, ∈) - [nC - √nV
_∈
Q
^-1
(∈)+ (1/2)log n] ≤ A
_∈
+ o(1), where C is the Shannon capacity, V
_∈
is the ∈-channel dispersion, or secondorder coding rate, Q is the tail probability of the normal distribution, and the constants AE and AE are explicitly identified. This expression holds under mild regularity assumptions on the channel, including nonsingularity. The gap A
_∈
- A
_∈
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 capacityachieving input distribution modified by an O(1/√n) correction term.