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

Shannon’s analysis of the fundamental capacity limits for memoryless communication channels 65 years ago has been refined over time. Until recently, the sharpest general result was: The maximum volume M∗ avg(n, ǫ) of length-n codes subject to an average decoding error probability ǫ is of the form exp{nC − √ nV Q(ǫ) + O(log n)} as n → ∞, where C is the Shannon capacity, V the channel dispersion, or second-order coding rate, and Q is the tail probability of the normal distribution. This paper derives a sharper expression: A ǫ + o(1) ≤ logM∗ avg(n, ǫ)− [nC − √ nV Q(ǫ) + 1 2 logn] ≤ Aǫ + o(1) where the constants A ǫ and Aǫ are explicitly identified. This expression holds under certain regularity assumptions on the channel. The gap Aǫ − Aǫ is one nat for weakly symmetric channels in the Cover-Thomas sense, and typically a few nats for other symmetric channels and for the Z channel. The derivation is based on strong large-deviations analysis and refined central limit asymptotics. The result is derived in a general setting, allowing abstract alphabets and cost constraints. In the absence of cost constraints, a random coding scheme that achieves the lower bound is presented. While generally not optimal, Shannon’s random codes (drawn from the capacity-achieving input distribution) are third-order optimal in the absence of cost constraints.