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

We study polarization for nonbinary channels with input alphabet of size q = 2^r, r = 2,3,.... Using Arikan´s polarizing kernel H₂, we prove that 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.