  We consider ensembles of channel codes that are partitioned into bins, and
focus on analysis of exact random coding error exponents associated with
optimum decoding of the index of the bin to which the transmitted codeword
belongs. Two main conclusions arise from this analysis: (i) for independent
random selection of codewords within a given type class, the random coding
exponent of optimal bin index decoding is given by the ordinary random coding
exponent function, computed at the rate of the entire code, independently of
the exponential rate of the size of the bin. (ii) for this ensemble of codes,
sub-optimal bin index decoding, that is based on ordinary maximum likelihood
(ML) decoding, is as good as the optimal bin index decoding in terms of the
random coding error exponent achieved. Finally, for the sake of completeness,
we also outline how our analysis of exact random coding exponents extends to
the hierarchical ensemble that correspond to superposition coding and optimal
decoding, where for each bin, first, a cloud center is drawn at random, and
then the codewords of this bin are drawn conditionally independently given the
cloud center. For this ensemble, conclusions (i) and (ii), mentioned above, no
longer hold necessarily in general.
