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

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^α), where N is the code length and α is any constant less than 0.5.

This version appears under a new title. Complete proofs of polarization have been added. The preliminary draft (v2) was published in Proc. 49th Allerton Conference, 2011