Multilevel polarization for nonbinary codes and parallel channels

We study polarization for nonbiliary channels with input alphabet of size q = 2^r, r = 2, 3, ... . Using Ankan´s successive cancellation decoding, 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. The multilevel polarization arising in this transmission represents a new phenomenon which is well suited to the description of dependent parallel channels that are subject to fading according to the order of their indices.