Permutation Capacity Region of Adder Multiple-Access Channels

TL;DR

This work analyzes the permutation-affected adder multi-access channel (PAMAC), where d senders transmit over a p-ary adder MAC followed by a random permutation. It establishes an exact characterization of the permutation capacity region: C_perm = { R ∈ R_+^d : ∑_i R_i ≤ d(p−1)/2 }, proven via complementary achievability and converse arguments. The authors develop three achievability schemes—binary root-stability, time sharing, and a general p-ary approach based on mixed-radix time sharing—along with two matching outer bounds, yielding a tight sum-rate bound and explicit region for all d and p. A binary-specific inner bound leverages the root structure of the adder’s output distribution and spectral perturbation (via Bauer-Fike), while the general achievability extends to arbitrary alphabets with a time-sharing bijection and least-squares decoding. The results illuminate fundamental limits of permutation channels in MAC settings and offer insight into coding strategies for unordered multiuser networks and related storage systems.

Abstract

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.

Figures (6)

• Figure 1: Illustration of a $d$-user PAMAC communication system.
• Figure 2: Visualization of our encoding scheme in the case $d = 2, \left| \mathcal{M}_1 \right| = 3, \left| \mathcal{M}_2 \right| = 4$. The $[0, 1]$ number line represents the Bernoulli parameter space. Each tick annotated with $\theta_{i,\ell}$ under the number line is a message in $\mathcal{M}_i$. Note the padding around each sender's subinterval.
• Figure 3: Visual interpretation of our time sharing bijection in the case $d = 2$, $m_1 = 3$, and $m_2 = 4$ (hence, $\left| \mathcal{M}_1 \right| = 9$ and $\left| \mathcal{M}_2 \right| = 16$).
• Figure 4: Visualization of the lattice $\mathcal{L}_i \subset \mathcal{S}_{p-1}$ in the case $i < d$, $p = 3$, and $m_i = 7$. The green surface is $\mathcal{S}_2$ and the blue dots are $\mathcal{L}_i$.
• Figure 5: Visualization of $\mathbf{C}$ in the case $d = 2$ and $p = 3$. Zeros are omitted for clarity. The two block rows are $\mathbf{C}_1 \in \mathbb{B}^{5 \times 6}$ and $\mathbf{C}_2 \in \mathbb{B}^{2 \times 6}$ respectively. Each block column is a group of columns with the same $c$ index. The number of block columns is $d = 2$. The width of each block column is $p = 3$. The underlined $1$s were generated by the corresponding cases in \ref{['eq:c1-matrix']}.

Theorems & Definitions (33)

• Theorem 1: Binary Achievability Using Root Stability
• Proposition 1: Binary Adder MAC Output Distribution Makur2019
• proof
• Theorem 2: Binary Achievability Using Time Sharing
• Theorem 3: General Achievability
• Theorem 4: General Converse
• Corollary 1: Binary Converse
• Theorem 5: General Permutation Capacity Region
• Corollary 2: Binary Permutation Capacity Region
• Corollary 3: General Permutation Sum-Capacity