  We address the problem of bounding below the probability of error under
maximum likelihood decoding of a binary code with a known distance distribution
used on a binary symmetric channel. An improved upper bound is given for the
maximum attainable exponent of this probability (the reliability function of
the channel). In particular, we prove that the ``random coding exponent'' is
the true value of the channel reliability for code rate $R$ in some interval
immediately below the critical rate of the channel. An analogous result is
obtained for the Gaussian channel.
