Empirical Distribution of Good Channel Codes with Non-Vanishing Error Probability
This paper studies several properties of channel codes that approach the fundamental limits of a given memoryless channel with a non-vanishing probability of error. The output distribution induced by an ?"-capacity-achieving code is shown to be close in a strong sense to the capacity achieving output distribution (for DMC and AWGN). Relying on the concentration of measure (isoperimetry) property enjoyed by the latter, it is shown that regular (Lipschitz) functions of channel outputs can be precisely estimated and turn out to be essentially non-random and independent of the used code. It is also shown that the binary hypothesis testing between the output distribution of a good code and the capacity achieving one cannot be performed with exponential reliability.