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

We study polarization for nonbinary channels with input alphabet of size q = 2 , r = 2, 3, . . . . Using Arıkan’s polarizing kernel H2, 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), where N is the code length and α is any constant less than 0.5.