Polar codes for q-ary channels, q=2^r

TL;DR

This paper extends polar coding techniques to q-ary channels with sizes as powers of two, proving convergence of virtual channels and providing an explicit scheme with favorable error decay.

Contribution

It demonstrates polarization for nonbinary channels with q=2^r using Arikan's kernel, establishing convergence to channels with capacities up to r bits and an explicit coding scheme.

Findings

Virtual channels converge to capacities 1 to r bits.

Transmission rate approaches symmetric capacity.

Decoding error probability decays as exp(-N^α) for α<0.5.

Abstract

We study polarization for nonbinary channels with input alphabet of size q=2^r,r=2,3,... Using Arikan's polarizing kernel H_2, we prove that the virtual channels that arise in the process of polarization converge to q-ary channels with capacity 1,2,...,r bits, and that the total transmission rate approaches the symmetric capacity of the channel. This leads to an explicit transmission scheme for q-ary channels. The error probability of decoding using successive cancellation behaves as exp(-N^\alpha), where N is the code length and {\alpha} is any constant less than 0.5.