Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels

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 second set of $N$ binary-input channels $\{W_N^{(i)}:1\le i\le N\}$ such that, as $N$ becomes large, the fraction of indices $i$ for which $I(W_N^{(i)})$ is near 1 approaches $I(W)$ and the fraction for which $I(W_N^{(i)})$ is near 0 approaches $1-I(W)$. The polarized channels $\{W_N^{(i)}\}$ are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC $W$ with $I(W)>0$ and any target rate $R < I(W)$, there exists a sequence of polar codes $\{{\mathscr C}_n;n\ge 1\}$ such that ${\mathscr C}_n$ has block length $N=2^n$, rate $\ge R$, and probability of block error under successive cancellation decoding bounded as $P_{e}(N,R) \le \bigoh(N^{-\frac14})$ independently of the code rate. This performance is achievable within encoding and decoding complexity both bounded by $O(N\log N)$. For the binary erasure channel, the construction complexity of a polar code of block-length $N$ is $O(N\log N)$. For general B-DMCs, the code construction complexity appears exponential in $N$; but, there exist approximate construction algorithms of complexity $O(N\log N)$.