  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
$\epsilon$-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.
