Expurgated Bounds for the Asymmetric Broadcast Channel

Abstract— This work contains two main contributions concerning the expurgation of hierarchical ensembles for the asymmetric broadcast channel. The first is an analysis of the optimal
maximum likelihood (ML) decoders for the weak and strong
user. Two different methods of code expurgation will be used,
that will provide two competing error exponents. The second is
the derivation of expurgated exponents under the generalized
stochastic likelihood decoder (GLD). We prove that the GLD
exponents are at least as tight as the maximum between the
random coding error exponents derived in an earlier work
by Averbuch and Merhav (2017) and one of our ML–based
expurgated exponents. By that, we actually prove the existence
of hierarchical codebooks that achieve the best of the random
coding exponent and the expurgated exponent simultaneously for
both users.