Expurgated Bounds for the Asymmetric Broadcast Channel

TL;DR

This paper develops new expurgated bounds for the asymmetric broadcast channel, analyzing ML decoders and the generalized stochastic likelihood decoder to improve error exponents for hierarchical codebooks.

Contribution

It introduces two methods of code expurgation for the asymmetric broadcast channel and compares their error exponents, also deriving expurgated exponents under the GLD.

Findings

GLD exponents are at least as tight as previous bounds

Hierarchical codebooks can achieve optimal error exponents for both users

New expurgation methods improve error bounds for broadcast channels

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.