On the Rate of Channel Polarization

TL;DR

This paper proves that polar codes achieve exponentially decreasing error probabilities for any rate below the channel capacity, with the decay rate approaching a square root exponent as block length increases.

Contribution

It establishes a bound on the block decoding error probability for polar codes, showing it decreases exponentially with block length for rates below capacity.

Findings

Error probability decreases as 2^{-N^β} for β<1/2

Polar codes achieve reliable communication below channel capacity

Error decay rate approaches a square root exponent as block length grows

Abstract

It is shown that for any binary-input discrete memoryless channel $W$ with symmetric capacity $I(W)$ and any rate $R <I(W)$, the probability of block decoding error for polar coding under successive cancellation decoding satisfies $P_e \le 2^{-N^\beta}$ for any $\beta<\frac12$ when the block-length $N$ is large enough.