Multilevel polarization of polar codes over arbitrary discrete memoryless channels

It is shown that the original construction of polar codes suffices to achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes. It is shown that in general, channel polarization happens in several, rather than only two, levels so that the synthesized channels are either useless, perfect or "partially perfect". Given a coset decomposition of the input alphabet, there exists a corresponding partially perfect channel whose outputs uniquely determine the coset where the channel input symbol belongs to. By a slight modification of the encoding and decoding rules, it is shown that perfect transmission of certain information letters over partially perfect channels is possible. It is also shown through an example that polar codes do not achieve the capacity of coset codes over arbitrary channels.