Multilevel Channel Polarization for Arbitrary Discrete Memoryless Channels

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.

[1]  Aria Ghasemian Sahebi,et al.  Multilevel polarization of polar codes over arbitrary discrete memoryless channels , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[2]  Toshiyuki Tanaka,et al.  Channel polarization on q-ary discrete memoryless channels by arbitrary kernels , 2010, 2010 IEEE International Symposium on Information Theory.

[3]  Alexander Barg,et al.  Polar Codes for-Ary Channels , , 2013 .

[4]  Aria Ghasemian Sahebi,et al.  On the capacity of Abelian group codes over discrete memoryless channels , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[5]  Eren Sasoglu.  Polar Coding Theorems for Discrete Systems , 2011 .

[6]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[7]  Erdal Arikan,et al.  Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels , 2008, IEEE Transactions on Information Theory.

[8]  Giacomo Como,et al.  The Capacity of Finite Abelian Group Codes Over Symmetric Memoryless Channels , 2009, IEEE Transactions on Information Theory.

[9]  Emre Telatar,et al.  On the rate of channel polarization , 2008, 2009 IEEE International Symposium on Information Theory.

[10]  Emre Telatar,et al.  Polarization for arbitrary discrete memoryless channels , 2009, 2009 IEEE Information Theory Workshop.

[11]  N. Bloch.  Abstract Algebra With Applications , 1987 .

[12]  R. Ahlswede.  Group Codes do not Achieve Shannon's Channel Capacity for General Discrete Channels , 1971 .