  This paper considers error probabilities of random codes for memoryless
channels in the fixed-rate regime. Random coding is a fundamental scheme to
achieve the channel capacity and many studies have been conducted for the
asymptotics of the decoding error probability. Gallager derived the exact
asymptotics (that is, a bound with asymptotically vanishing relative error) of
the error probability below the critical rate. On the other hand, exact
asymptotics for fixed rate above the critical rate has been unknown except for
symmetric channels (in the strong sense) and strongly nonlattice channels. This
paper derives the exact asymptotics for general memoryless channels covering
all previously unsolved cases. The analysis reveals that strongly symmetric
channels and strongly nonlattice channels correspond to two extreme cases and
the expression of the asymptotics is much complicated for general channels.
