The noisy permutation channel is a useful abstraction introduced by Makur for point-to-point communication networks and biological storage. While the asymptotic capacity results exist for this model, the characterization of the second-order asymptotics is not available. Therefore, we analyze the converse bounds for the noisy permutation channel in the finite blocklength regime. To do this, we present a modified minimax meta-converse for noisy permutation channels by symbol relaxation. To derive the second-order asymptotics of the converse bound, we propose a way to use divergence covering in analysis. It enables the observation of the second-order asymptotics and the strong converse via Berry-Esseen type bounds. These two conclusions hold for noisy permutation channels with strictly positive matrices (entry-wise). In addition, we obtain computable bounds for the noisy permutation channel with the binary symmetric channel (BSC), including the original computable converse bound based on the modified minimax meta-converse, the asymptotic expansion derived from our subset covering technique, and the {\epsilon}-capacity result. We find that a smaller crossover probability provides a higher upper bound for a fixed finite blocklength, although the {\epsilon}-capacity is agnostic to the BSC parameter. Finally, numerical results show that the normal approximation shows remarkable precision, and our new converse bound is stronger than previous bounds.

翻译：噪声置换信道是Makur为点对点通信网络和生物存储引入的一种有用抽象模型。尽管该模型存在渐近容量结果，但其二阶渐近特性的表征尚未得到研究。为此，我们分析了有限码长条件下噪声置换信道的逆界。为实现这一目标，我们通过符号松弛方法提出了适用于噪声置换信道的修正极小极大元逆界。为推导逆界的二阶渐近特性，我们提出了一种在分析中使用散度覆盖的方法。该方法使得通过Berry-Esseen型边界观察二阶渐近特性和强逆定理成为可能。这两个结论适用于具有严格正矩阵（逐元素为正）的噪声置换信道。此外，我们获得了二进制对称信道（BSC）噪声置换信道的可计算界，包括基于修正极小极大元逆界的原始可计算逆界、通过子集覆盖技术导出的渐近展开式，以及{\epsilon}容量结果。我们发现：虽然{\epsilon}容量与BSC参数无关，但较小的交叉概率在固定有限码长条件下会给出更高的上界。最后，数值结果表明正态逼近具有显著精度，且我们提出的新逆界优于现有界。