Polar Codes for $q$-Ary Channels, $q=2^{r}$
Woomyoung Park, Alexander Barg · IEEE Transactions on Information Theory · 2012
We study polarization for nonbinary channels with input alphabet of sizeq=2r,r=2,3,.... Using Arıkan's polarizing kernelH2, we prove that the virtual channels that arise in the process of channel evolution converge toq-ary channels with capacity 0,1,2,...,rbits. As a result of this analysis, we show that polar codes support reliable transmission over discrete memoryless channels withq-ary input for all rates below the symmetric capacity of the channel. This leads to an explicit transmission scheme forq-ary channels. The block error probability of decoding using successive cancellation behaves as exp(-Nα), whereNis the code length and α is any constant less than 0.5.