Strong Converse Theorems for Multimessage Networks with Tight Cut-Set Bound

TL;DR

This paper proves the strong converse theorem for multimessage networks with tight cut-set bounds under discrete memoryless and Gaussian models, establishing that error probabilities tend to one outside the capacity region.

Contribution

It extends the strong converse theorem to general multimessage networks with tight cut-set bounds, including Gaussian models with power constraints, and derives new results for various relay channels.

Findings

Strong converse holds for networks with tight cut-set bounds.

Error probabilities approach one outside the capacity region.

Results include new bounds for Gaussian MAC and relay channels.

Abstract

This paper considers a multimessage network where each node may send a message to any other node in the network. Under the discrete memoryless model, we prove the strong converse theorem for any network whose cut-set bound is tight, i.e., achievable. Our result implies that for any fixed rate vector that resides outside the capacity region, the average error probabilities of any sequence of length-$n$ codes operated at the rate vector must tend to $1$ as $n$ approaches infinity. The proof is based on the method of types and is inspired by the work of Csisz\'{a}r and K\"{o}rner in 1982 which fully characterized the reliability function of any discrete memoryless channel (DMC) with feedback for rates above capacity. In addition, we generalize the strong converse theorem to the Gaussian model where each node is subject to an almost-sure power constraint. Important consequences of our results are new strong converses for the Gaussian multiple access channel (MAC) with feedback and the following relay channels under both models: The degraded relay channel (RC), the RC with orthogonal sender components, and the general RC with feedback.