We study volume-refined achievability and converse bounds for noisy permutation channels generated by strictly positive DMCs, allowing the reachable output polytope to have arbitrary affine dimension $d\ge 1$. The reachable output polytope may be lower-dimensional than the output simplex, whereas existing refined achievability analyses and fixed-error converses are not adapted to this intrinsic affine geometry. On the achievability side, we develop an affine-coordinate simplex-lattice construction adapted to the reachable output polytope, together with a nearest-neighbor decoder and a geometric error-reduction argument in the same coordinate space. This yields a Gaussian achievability approximation with an $o(1)$ remainder. On the converse side, we first use a meta-converse combined with a KL covering and a local testing estimate to obtain a fixed-error converse with a bounded remainder, which implies the logarithmic $ε$-capacity $d/2$. We then apply the meta-converse with a stratified Jeffreys-mixture auxiliary output distribution. Using a local Laplace approximation and a local likelihood-ratio approximation, this choice identifies the Fisher-volume term and an explicit Gaussian testing constant, yielding a constant-order converse approximation with an $o(1)$ remainder. The achievability and converse constants arise from different constructions and are not claimed to match in general.

翻译：我们研究了由严格正定离散无记忆信道生成的噪声置换信道的基于体积精化的可达性与逆界，允许可达输出多面体具有任意仿射维度$d\ge 1$。可达输出多面体的维度可能低于输出单纯形的维度，而现有的精化可达性分析与固定误差逆界尚未适应这种内蕴仿射几何结构。在可达性方面，我们提出了适应于可达输出多面体的仿射坐标单纯形-格点构造，并结合最近邻译码器以及同一坐标空间中的几何误差缩减论证。这得到了一个余项为$o(1)$的高斯可达性近似。在逆界方面，我们首先利用元逆界结合KL覆盖与局部检验估计，得到具有有界余项的固定误差逆界，由此推导出对数$\varepsilon$-容量为$d/2$。随后，我们采用具有分层Jeffreys混合辅助输出分布的元逆界，通过局部拉普拉斯近似与局部似然比近似，确定了Fisher体积项与显式高斯检验常数，从而得到余项为$o(1)$的常数阶逆近似。可达性与逆常数源于不同的构造，本文不声称其一般情形下具有匹配性。