Exact Error and Erasure Exponents for the Asymmetric Broadcast Channel
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We derive exact (ensemble-tight) error and erasure exponents for the asymmetric broadcast channel given a random superposition codebook. We consider Forney's optimal decoder for both messages and the message pair for the receiver that decodes both messages. We prove 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. We propose convex optimization-based procedures to evaluate the exponents efficiently. Numerical examples are presented to illustrate the results.
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