Multi-Bit Relaying over a Tandem of Channels

TL;DR

This paper investigates error exponents for relaying messages over two channels in tandem, extending beyond 1-bit scenarios, and identifies conditions under which 1-hop and 2-hop exponents are equal or differ.

Contribution

It generalizes previous 1-bit relay results to multi-bit messages and explores conditions affecting the equivalence of 1-hop and 2-hop error exponents.

Findings

1-hop and 2-hop exponents coincide under certain conditions.

The exponents can differ when assumptions are relaxed.

Extensions include relaxing pairwise reversibility and channel similarity.

Abstract

We study error exponents for the problem of relaying a message over a tandem of two channels sharing the same transition law, in particular moving beyond the 1-bit setting studied in recent related works. Our main results show that the 1-hop and 2-hop exponents coincide in both of the following settings: (i) the number of messages is fixed, and the channel law satisfies a condition called pairwise reversibility, or (ii) the channel is arbitrary, and a zero-rate limit is taken from above. In addition, we provide various extensions of our results that relax the assumptions of pairwise reversibility and/or the two channels having identical transition laws, and we provide an example for which the 2-hop exponent is strictly below the 1-hop exponent.