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

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.