Polar Codes for Arbitrary DMCs and Arbitrary MACs

TL;DR

This paper extends polar codes to arbitrary discrete memoryless channels and multiple access channels by using algebraic structures, achieving low error probabilities and efficient encoding/decoding, with some limitations in capacity region coverage.

Contribution

It introduces a method to construct polar codes for arbitrary channels and MACs using quasigroup and Abelian group structures, expanding the applicability of polar coding.

Findings

Achieves low block error probability with $o(2^{-N^{1/2-\epsilon}})$ decay

Provides encoding and decoding complexity of $O(N\log N)$

Characterizes channels where symmetric capacity region is preserved

Abstract

Polar codes are constructed for arbitrary channels by imposing an arbitrary quasigroup structure on the input alphabet. Just as with "usual" polar codes, the block error probability under successive cancellation decoding is $o(2^{-N^{1/2-\epsilon}})$, where $N$ is the block length. Encoding and decoding for these codes can be implemented with a complexity of $O(N\log N)$. It is shown that the same technique can be used to construct polar codes for arbitrary multiple access channels (MAC) by using an appropriate Abelian group structure. Although the symmetric sum capacity is achieved by this coding scheme, some points in the symmetric capacity region may not be achieved. In the case where the channel is a combination of linear channels, we provide a necessary and sufficient condition characterizing the channels whose symmetric capacity region is preserved by the polarization process. We also provide a sufficient condition for having a maximal loss in the dominant face.