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 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(- N  α ), where  N  is the code length and α is any constant less than 0.5.
['Woomyoung Park', 'Alexander Barg']
Polar Codes for Q-Ary Channels, $q=2^{r}$