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Polar codes are constructed for arbitrary channels by imposing an arbitraryquasigroup structure on the input alphabet. Just as with "usual" polar codes,the block error probability under successive cancellation decoding is$o(2^{-N^{1/2-\epsilon}})$, where $N$ is the block length. Encoding anddecoding for these codes can be implemented with a complexity of $O(N\log N)$.It is shown that the same technique can be used to construct polar codes forarbitrary multiple access channels (MAC) by using an appropriate Abelian groupstructure. Although the symmetric sum capacity is achieved by this codingscheme, some points in the symmetric capacity region may not be achieved. Inthe case where the channel is a combination of linear channels, we provide anecessary and sufficient condition characterizing the channels whose symmetriccapacity region is preserved by the polarization process. We also provide asufficient condition for having a maximal loss in the dominant face.

Polar Codes for Arbitrary DMCs and Arbitrary MACs