The log-volume of optimal constant-composition codes for memoryless channels, within O(1) bits

This paper derives a tight asymptotic upper bound on the maximum volume M*_cc(n, ϵ) of length-n constant-composition codes subject to an average decoding error probability ϵ: M_bb(n, ϵ) = exp{nC - √nV Φ^- (1 - ϵ) + 1/2 log n + A_{n, ϵ} + o(1)} where Φ is the cdf of the standard normal distribution, and A_{n, ϵ} is a bounded sequence that can be explicitly identified and reduces to a constant in the nonlattice case. A lower bound is presented, differing from the upper bound by an easily computable multiplying constant. These expressions hold under certain regularity assumptions on the channel.