Empirical distribution of good channel codes with non-vanishing error probability (extended version)

This paper studies several properties of channel codes that approach the fundamental limits of a given (discrete or Gaussian) memoryless channel with a non-vanishing probability of error. The output distribution induced by an ǫ-capacity-achieving code is shown to be close in a strong sense to the capacity achieving output distribution. 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 actual code. It is also shown that the output distribution of a good code and the capacity achieving one cannot be distinguished with exponential reliability. The random process produced at the output of the channel is shown to satisfy the asymptotic equipartition property. Using related methods it is shown that quadratic forms and sums of q-th powers when evaluated at codewords of good AWGN codes approach the values obtained from a randomly generated Gaussian codeword. Index Terms Shannon theory, discrete memoryless channels, additive white Gaussian noise, relative entropy, empirical output statistics, asymptotic equipartition property, concentration of measure.