On the calculation of the minimax-converse of the channel coding problem

TL;DR

This paper investigates the properties of a minimax-converse for channel coding, proving the existence of saddle points for both forms and providing a polynomial-time algorithm for computing the bounds in discrete memoryless channels.

Contribution

It proves the existence of saddle points for the second form of the minimax-converse and develops a polynomial-time algorithm for its computation in DMCs.

Findings

Proved saddle point existence for the second converse form.

Developed a polynomial-time algorithm for saddle point computation in DMCs.

Provided a new method for analyzing channel coding bounds.

Abstract

A minimax-converse has been suggested for the general channel coding problem by Polyanskiy etal. This converse comes in two flavors. The first flavor is generally used for the analysis of the coding problem with non-vanishing error probability and provides an upper bound on the rate given the error probability. The second flavor fixes the rate and provides a lower bound on the error probability. Both converses are given as a min-max optimization problem of an appropriate binary hypothesis testing problem. The properties of the first converse were studies by Polyanskiy and a saddle point was proved. In this paper we study the properties of the second form and prove that it also admits a saddle point. Moreover, an algorithm for the computation of the saddle point, and hence the bound, is developed. In the DMC case, the algorithm runs in a polynomial time.