Polarization for arbitrary discrete memoryless channels

TL;DR

This paper extends the concept of channel polarization from binary to arbitrary discrete memoryless channels, enabling polar coding for a broader class of channels and input alphabet sizes, including prime, composite, and randomized cases.

Contribution

It generalizes polarization to all discrete memoryless channels, including those with composite input sizes, and introduces randomized methods without increasing complexity.

Findings

Polarization is achievable for prime input alphabet sizes.

Channels with composite input sizes can be polarized via decomposition.

Error probability behavior extends to arbitrary discrete channels.

Abstract

Channel polarization, originally proposed for binary-input channels, is generalized to arbitrary discrete memoryless channels. Specifically, it is shown that when the input alphabet size is a prime number, a similar construction to that for the binary case leads to polarization. This method can be extended to channels of composite input alphabet sizes by decomposing such channels into a set of channels with prime input alphabet sizes. It is also shown that all discrete memoryless channels can be polarized by randomized constructions. The introduction of randomness does not change the order of complexity of polar code construction, encoding, and decoding. A previous result on the error probability behavior of polar codes is also extended to the case of arbitrary discrete memoryless channels. The generalization of polarization to channels with arbitrary finite input alphabet sizes leads to polar-coding methods for approaching the true (as opposed to symmetric) channel capacity of arbitrary channels with discrete or continuous input alphabets.