Channel polarization: A method for constructing capacity-achieving codes

A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W. The symmetric capacity I(W) is the highest rate achievable subject to using the input letters of the channel equiprobably and equals the capacity C(W) if the channel has certain symmetry properties. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W, a different set of N binary-input channels such that the capacities of the latter set, except for a negligible fraction of them, are either near 1 or near 0. This second set of N channels are well-conditioned for channel coding: one need only send data at full rate through channels with capacity near 1 and at 0 rate through the others. The main coding theorem about polar coding states that, given any B-DMC W with I(W) > 0 and any fixed 0 < delta < I(W), there exist finite constants n₁ (W, delta) and c(W, delta) such that for all ngesn₁, there exist polar codes with block length N = 2ⁿ rate R > I(W)-delta, and probability of block decoding error P_e les cN^-1/4. The codes with this performance can be encoded and decoded within complexity O(N log N).