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It is shown that polar codes achieve the symmetric capacity of discretememoryless channels with arbitrary input alphabet sizes. It is shown that ingeneral, channel polarization happens in several, rather than only two levelsso that the synthesized channels are either useless, perfect or "partiallyperfect". Any subset of the channel input alphabet which is closed underaddition, induces a coset partition of the alphabet through its shifts. For anysuch partition of the input alphabet, there exists a corresponding partiallyperfect channel whose outputs uniquely determine the coset to which the channelinput belongs. By a slight modification of the encoding and decoding rules, itis shown that perfect transmission of certain information symbols overpartially perfect channels is possible. Our result is general regarding boththe cardinality and the algebraic structure of the channel input alphabet; i.ewe show that for any channel input alphabet size and any Abelian groupstructure on the alphabet, polar codes are optimal. It is also shown through anexample that polar codes when considered as group/coset codes, do not achievethe capacity achievable using coset codes over arbitrary channels.

Multilevel Polarization of Polar Codes Over Arbitrary Discrete Memoryless Channels