Permutation Capacity Region of Adder Multiple-Access Channels

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.