Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels

TLDR

The symmetric capacity is the maximum achievable rate when channel inputs are used with equal probability. The authors propose channel polarization as a method to construct code sequences that achieve the symmetric capacity of any binary‑input discrete memoryless channel. Channel polarization synthesizes N copies of a B‑DMC into N binary‑input channels whose capacities polarize toward 0 or 1, allowing data to be sent only through the near‑capacity channels. The paper proves that for any B‑DMC with positive symmetric capacity and any target rate below it, there exists a sequence of polar codes with block length 2ⁿ, rate at least R, and block‑error probability bounded by O(N⁻¹⁄⁴), achievable with O(N log N) encoding and decoding complexity.

Abstract

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.