A linear program for the finite block length converse of Polyanskiy-Poor-Verd\'u via non-signalling codes

TL;DR

This paper introduces a linear programming approach to analyze non-signalling codes, providing an alternative proof for the finite block length converse in channel coding and characterizing when zero-error capacity is achievable.

Contribution

It presents a linear program formulation for non-signalling codes, offering new insights and proofs for finite block length converses and capacity conditions.

Findings

Non-signalling codes achieve the finite block length converse.

The linear program characterizes optimal performance of these codes.

Zero-error capacity is linked to the channel's dispersion being zero.

Abstract

Motivated by recent work on entanglement-assisted codes for sending messages over classical channels, the larger, easily characterised class of non-signalling codes is defined. Analysing the optimal performance of these codes yields an alternative proof of the finite block length converse of Polyanskiy, Poor and Verd\'u, and shows that they achieve this converse. This provides an explicit formulation of the converse as a linear program which has some useful features. For discrete memoryless channels, it is shown that non-signalling codes attain the channel capacity with zero error probability if and only if the dispersion of the channel is zero.