We establish the capacity of a class of communication channels introduced in [1]. 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 $1/2 (rank(P_{Z|X})-1)$ whenever $P_{Z|X}$ is strictly positive. This is done by establishing a converse bound matching the achievability of [1]. 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 $\epsilon$-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.

翻译：本文建立了文献[1]中引入的一类通信信道的容量。有限字母表上的$n$字母输入经过离散无记忆信道$P_{Z|X}$后，输出$n$字母序列被均匀置换。我们证明当$P_{Z|X}$严格正时，最大通信速率（以$\log n$归一化）等于$1/2 (rank(P_{Z|X})-1)$。这一结果通过建立与文献[1]可达性相匹配的逆界得以实现。证明的两个关键要素是：(1) 对从类型类中均匀采样并通过DMC观测的向量熵的精确界；(2) 以Kullback-Leibler散度为度量的概率单纯形的覆盖$\epsilon$-网。除严格正DMC外，我们还得到了$q$元擦除信道、Z信道及其他信道的带噪置换容量。