Capacity of Noisy Permutation Channels

Jennifer Tang, Yury Polyanskiy

Abstract: We establish the capacity of a class of communication channels introduced by Makur. 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 ${\frac{1}{ 2}} ( \textsf {rank}(P_{Z|X})-1)$ whenever $P_{Z|X}$ is strictly positive. This is done by establishing a converse bound matching the achievability of Makur. The two main ingredients of our proof are: 1) a sharp bound on the Kullback-Leibler divergence of a uniformly sampled vector from a type class and observed through a DMC to an iid vector; and 2) the covering $\varepsilon $ -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.

External IDs:dblp:journals/tit/TangP23