Error Exponents for Broadcast Channels with Degraded Message Sets

TL;DR

This paper derives new lower bounds on error exponents for a broadcast channel with degraded message sets, using optimal decoding and two different analytical approaches, improving upon previous suboptimal bounds.

Contribution

It introduces two novel methods for lower bounding error exponents with optimal decoding, one based on Gallager techniques and another rooted in statistical physics, with the latter providing tighter bounds.

Findings

The second approach yields tighter bounds than the first and previous results.

The bounds are dependent on both message rates and decoding strategies.

Numerical results confirm the effectiveness of the physics-based method.

Abstract

We consider a broadcast channel with a degraded message set, in which a single transmitter sends a common message to two receivers and a private message to one of the receivers only. The main goal of this work is to find new lower bounds to the error exponents of the strong user, the one that should decode both messages, and of the weak user, that should decode only the common message. Unlike previous works, where suboptimal decoders where used, the exponents we derive in this work pertain to optimal decoding and depend on both rates. We take two different approaches. The first approach is based, in part, on variations of Gallager-type bounding techniques that were presented in a much earlier work on error exponents for erasure/list decoding. The resulting lower bounds are quite simple to understand and to compute. The second approach is based on a technique that is rooted in statistical physics, and it is exponentially tight from the initial step and onward. This technique is based on analyzing the statistics of certain enumerators. Numerical results show that the bounds obtained by this technique are tighter than those obtained by the first approach and previous results. The derivation, however, is more complex than the first approach and the retrieved exponents are harder to compute.