Capacity of Noisy Permutation Channels

We establish the capacity of a class of communication channels introduced in [2]. The n-letter input from a finite alphabet is passed through a discrete memoryless channel P
Z|X
and then the output n-letter sequence is uniformly permuted. We show that the maximal communication rate (normalized by log n) equals $\frac{1}{2}\left( {\operatorname{rank} \left( {{P_{Z\mid X}}} \right) - 1} \right)$ whenever P
Z|X
is strictly positive. This is done by establishing a converse bound matching the achievability of [2]. The two main ingredients of our proof are (1) a sharp bound on the entropy of a uniformly sampled vector from a type class and observed through a DMC; and (2) the covering ε-net of a probability simplex with Kullback-Leibler divergence as a metric. In addition to strictly positive DMC we also find the noisy permutation capacity for q-ary erasure channels, the Z-channel and others.