Permutation Sum-Capacity of Binary Adder Multiple-Access Channels

Propelled by recent advances in the study of point-to-point permutation channels, which stem from communication networks and biological communications applications, we analyze the permutation binary adder multiple-access channel (PAMAC) in this work. The PAMAC network model consists of d senders communicating with a single receiver through a standard binary adder multiple-access channel followed by a random permutation block that shuffles the output codeword of the multiple-access channel. We formally define an appropriate notion of permutation sum-capacity C
psum
for this model, and then establish that ${{\text{C}}_{{\text{psum }}}} = \frac{d}{2}$. To derive an achievability bound we construct d randomized encoders where input codewords are i.i.d. samples from Bernoulli distributions with carefully chosen parameters. These parameters can be perceived as roots of the probability generating function of the distribution over the output alphabet after the PAMAC's addition operation. Our achievability proof crucially uses eigenvalue perturbation results to provide quantitative estimates on the stability of the roots, which allows us to analyze decoding performance. This argument also yields an inner bound on the permutation capacity region of the PAMAC model. Finally, we also obtain a converse bound on C
psum
matching our achievability result.