The Log-Volume of Optimal Codes for Memoryless Channels, Up to a Few Nats

This paper derives a tight asymptotic upper bound on the maximum volume M∗(n, ) of length-n codes for memoryless channels subject to an average decoding error probability : M(n, ) = exp{nC − √nV Φ−1( ) + 1 2 log n + An, + o(1)} where C is Shannon capacity, V is channel dispersion, Φ is the tail probability of the normal distribution, and An, is a bounded sequence that can be explicitly identified and reduces to a constant in the nonlattice case. A matching lower bound is presented, differing from the upper bound by a small multiplying constant. These expressions hold under certain regularity assumptions on the channel.