When is the zero-error capacity positive in the relay, multiple-access, broadcast and interference channels?

Abstract

Shannon determined that the zero-error capacity of a point-to-point channel whose channel p(y|x) has confusability graph G X|Y is positive if and only if there exist two inputs that are “non-adjacent”, or “non-confusable”. Equivalently, it is non-zero if and only if the independence number of G X|Y is strictly greater than 1. A multi-letter expression for the zero-error capacity of the channel with confusability graph G X|Y is known, and is given by the normalized limit as the blocklength n → 1 of the maximum independent set of the n-fold strong product of G X|Y . This is not generally computable with known methods. In this paper, we look at the zero-error capacity of four multi-user channels: the relay, the multiple-access (MAC), the broadcast (BC), and the interference (IC) channels. As a first step towards finding a multi-letter expression for the capacity of such channels, we find necessary and sufficient conditions under which the zero-error capacity is strictly positive.