  Consider the asymmetric broadcast channel with a random superposition
codebook, which may be comprised of constant composition or \iid codewords. By
applying Forney's optimal decoder for individual messages and the message pair
for the receiver that decodes both messages, exact (ensemble-tight) error and
erasure exponents are derived. It is shown that the optimal decoder designed to
decode the pair of messages achieves the optimal trade-off between the total
and undetected exponents associated with the optimal decoder for the private
message. Convex optimization-based procedures to evaluate the exponents
efficiently are proposed. Finally, numerical examples are presented to
illustrate the results.
