Polar Codes for $q$-Ary Channels, $q=2^{r}$

We study polarization for nonbinary channels with input alphabet of size q=2r, r=2,3,.... Using Arikan's polarizing kernel H2, we prove that the virtual channels that arise in the process of channel evolution converge to q-ary channels with capacity 0,1,2,..., r bits. As a result of this analysis, we show that polar codes support reliable transmission over discrete memoryless channels with q-ary input for all rates below the symmetric capacity of the channel. This leads to an explicit transmission scheme for q-ary channels. The block error probability of decoding using successive cancellation behaves as exp(-Nα), where N is the code length and α is any constant less than 0.5.