On Permutation Capacity Regions of Multiple-Access Channels

Permutation networks and multiple-access channels (MACs) are objects of interest in modern information theory which find application in modeling biological storage mechanisms and wireless communication systems. In this paper, we present two variations of the recently introduced permutation adder multiple-access channel (PAMAC), and characterize their respective permutation capacity regions as an initial step towards establishing such results for general MACs. Firstly, we define the left-permutation adder MAC by interchanging the order of the adder and random permutation blocks in the original PAMAC, causing each sender's codeword to be shuffled by a different random permutation. We show that multiset coding with Bernoulli samples is a viable achievability scheme under this structural modification by extending a root stability argument from the binary PAMAC literature to general p-ary alphabets. Separately from the above, we define the permutation group-adder MAC by replacing the integer addition block in the original PAMAC with a modular addition block over $\mathbb{Z}_{p}$. Our achievability proof in this setting crucially demonstrates that time sharing strategies based on mixed-radix coding naturally generalize to an array of permutation network models beyond the original PAMAC. Lastly, using Fano's inequality and manipulations of directed graphical models, we obtain converse bounds matching our achievability results, ultimately illustrating that subtle modifications to a permutation network model may bring about significant qualitative changes in its capacity region.