Point-to-point permutation channels are useful models of communication networks and biological storage mechanisms and have received theoretical attention in recent years. Propelled by relevant advances in this area, we analyze the permutation adder multiple-access channel (PAMAC) in this work. In the PAMAC network model, $d$ senders communicate with a single receiver by transmitting $p$-ary codewords through an adder multiple-access channel whose output is subsequently shuffled by a random permutation block. We define a suitable notion of permutation capacity region $\mathcal{C}_\mathsf{perm}$ for this model, and establish that $\mathcal{C}_\mathsf{perm}$ is the simplex consisting of all rate $d$-tuples that sum to $d(p - 1) / 2$ or less. We achieve this sum-rate by encoding messages as i.i.d. samples from categorical distributions with carefully chosen parameters, and we derive an inner bound on $\mathcal{C}_\mathsf{perm}$ by extending the concept of time sharing to the permutation channel setting. Our proof notably illuminates various connections between mixed-radix numerical systems and coding schemes for multiple-access channels. Furthermore, we derive an alternative inner bound on $\mathcal{C}_\mathsf{perm}$ for the binary PAMAC by analyzing the root stability of the probability generating function of the adder's output distribution. Using eigenvalue perturbation results, we obtain error bounds on the spectrum of the probability generating function's companion matrix, providing quantitative estimates of decoding performance. Finally, we obtain a converse bound on $\mathcal{C}_\mathsf{perm}$ matching our achievability result.

翻译：点对点置换信道是通信网络和生物存储机制的有用模型，近年来受到理论界的关注。受该领域相关进展的推动，本文分析了置换加法多址接入信道（PAMAC）。在PAMAC网络模型中，$d$个发送者通过加法多址接入信道传输$p$进制码字与单个接收者通信，该信道的输出随后被随机置换块打乱。我们为此模型定义了置换容量区域$\mathcal{C}_\mathsf{perm}$的适当概念，并证明$\mathcal{C}_\mathsf{perm}$是一个单纯形，由所有速率$d$元组构成，这些元组之和等于或小于$d(p - 1) / 2$。我们通过将消息编码为具有精心选择参数的分类分布的独立同分布样本来实现该和速率，并通过将时分复用的概念扩展到置换信道场景，推导出$\mathcal{C}_\mathsf{perm}$的一个内界。我们的证明特别阐明了混合进制数系与多址接入信道编码方案之间的多种联系。此外，通过分析加法器输出分布的概率生成函数的根稳定性，我们推导了二进制PAMAC下$\mathcal{C}_\mathsf{perm}$的另一种内界。利用特征值扰动结果，我们得到了概率生成函数伴随矩阵谱的误差界，提供了解码性能的定量估计。最后，我们得到了与可达性结果匹配的$\mathcal{C}_\mathsf{perm}$的逆界。