Achieving the Uniform Rate Region of General Multiple Access Channels by Polar Coding

Abstract

We consider the problem of polar coding for transmission over $m$-user multiple access channels. In the proposed scheme, all users encode their messages using a polar encoder, while a joint successive cancellation decoder is deployed at the receiver. The encoding is done separately across the users and is independent of the target achievable rate, in the sense that the encoder core is the regular Ar{\i}kan's polarization matrix. For the code construction, the positions of information bits and frozen bits for each of the users are decided jointly. This is done by treating the whole polar transformation across all the $m$ users as a single polar transformation with a certain base code. We prove that the covering radius of the dominant face of the uniform rate region is upper bounded by $r = \frac{(m-1)\sqrt{m}}{L}$, where $L$ represents the length of the base code. We then prove that the proposed polar coding scheme achieves the whole uniform rate region, with small enough resolution characterized by $r$, by changing the decoding order in the joint successive cancellation decoder. The encoding and decoding complexities are $O(N \log N)$, where $N$ is the code block length, and the asymptotic block error probability of $O(2^{-N^{0.5 - \epsilon}})$ is guaranteed. Examples of achievable rates for the case of $3$-user multiple access channel are provided.