  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 is the highest
rate achievable subject to using the input letters of the channel with equal
probability. 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 by encoders and decoders with complexity $O(N\log N)$ for each.
