Skip to Main Content
It is shown that polar codes, with their original (u,u+v) kernel, 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.” Any subset of the channel input alphabet which is closed under addition induces a coset partition of the alphabet through its shifts. For any such partition of the input alphabet, there exists a corresponding partially perfect channel whose outputs uniquely determine the coset to which the channel input belongs. By a slight modification of the encoding and decoding rules, it is shown that perfect transmission of certain information symbols over partially perfect channels is possible. Our result is general regarding both the cardinality and the algebraic structure of the channel input alphabet; i.e., we show that for any channel input alphabet size and any Abelian group structure on the alphabet, polar codes are optimal. Due to the modifications, we make to the encoding rule of polar codes, the constructed codes fall into a larger class of structured codes called nested group codes.