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We consider a broadcast channel with a degraded message set, in which asingle transmitter sends a common message to two receivers and a privatemessage to one of the receivers only. The main goal of this work is to find newlower bounds to the error exponents of the strong user, the one that shoulddecode both messages, and of the weak user, that should decode only the commonmessage. Unlike previous works, where suboptimal decoders where used, theexponents we derive in this work pertain to optimal decoding and depend on bothrates. We take two different approaches.  The first approach is based, in part, on variations of Gallager-type boundingtechniques that were presented in a much earlier work on error exponents forerasure/list decoding. The resulting lower bounds are quite simple tounderstand and to compute.  The second approach is based on a technique that is rooted in statisticalphysics, and it is exponentially tight from the initial step and onward. Thistechnique is based on analyzing the statistics of certain enumerators.Numerical results show that the bounds obtained by this technique are tighterthan those obtained by the first approach and previous results. The derivation,however, is more complex than the first approach and the retrieved exponentsare harder to compute.

Error Exponents for Broadcast Channels with Degraded Message Sets