We study finite-blocklength bounds for noisy permutation channels whose reachable output polytope may be lower-dimensional than the output simplex. Existing Gaussian achievability analyses focus on strictly positive full-rank square DMC transition matrices. The capacity result for arbitrary strictly positive DMCs is established through a weak converse, while available strong converse bounds in the lower-dimensional setting can scale with the dimension of the output simplex rather than with that of the reachable output polytope. On the achievability side, messages are placed on a simplex lattice in affine coordinates, and decoding is performed by projecting the empirical output distribution onto the reachable affine hull followed by Euclidean nearest-neighbor decoding. Writing $d$ for the affine dimension of the reachable output polytope, a geometric reduction converts decoding errors into $d(d+1)$ one-dimensional transfer events, yielding a refined Gaussian achievability lower bound based on averaged local coordinate variances and a relative volume ratio. On the converse side, a modified meta-converse, a Kullback--Leibler divergence covering, and a local binary-testing bound yield a strong converse whose blocklength-dependent term is $d\log\sqrt n$, up to a bounded additive remainder.

翻译：我们研究了噪声置换信道的有限分组长度界，其可达输出多面体的维度可能低于输出单纯形。现有的高斯可达性分析专注于严格正定满秩的方阵离散无记忆信道转移矩阵。通过弱对偶建立了任意严格正定离散无记忆信道的容量结果，而在低维场景中可用的强对偶界可能按输出单纯形的维度而非可达输出多面体的维度缩放。在可达性方面，消息被放置在仿射坐标下的单纯形格点上，解码通过将经验输出分布投影到可达仿射包上，随后进行欧几里得最近邻解码实现。记$d$为可达输出多面体的仿射维度，几何约化将解码错误转化为$d(d+1)$个一维转移事件，从而基于平均局部坐标方差和相对体积比得到改进的高斯可达性下界。在对偶性方面，通过修正元对偶、Kullback-Leibler散度覆盖以及局部二元检验界，得到一个强对偶，其与分组长度相关的项为$d\log\sqrt n$（附加有界余项）。