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

We study polarization for nonbinary channels with input alphabet of size
q
=2
^r
,
r
=2,3,.... Using Arıkan'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.