Universal Bounds on the Scaling Behavior of Polar Codes

TL;DR

This paper establishes universal bounds on the rate-block length trade-off for polar codes, showing that the rate approaches capacity at a specific polynomial rate characterized by a universal parameter.

Contribution

It introduces a universal parameter  that bounds the rate of polar codes relative to capacity, providing lower bounds and conjecturing its exact value for certain channels.

Findings

Existence of a universal parameter  governing rate decay

Lower bound  .553 on the parameter 

Conjecture that  equals 3.627 for the binary erasure channel

Abstract

We consider the problem of determining the trade-off between the rate and the block-length of polar codes for a given block error probability when we use the successive cancellation decoder. We take the sum of the Bhattacharyya parameters as a proxy for the block error probability, and show that there exists a universal parameter $\mu$ such that for any binary memoryless symmetric channel $W$ with capacity $I(W)$, reliable communication requires rates that satisfy $R< I(W)-\alpha N^{-\frac{1}{\mu}}$, where $\alpha$ is a positive constant and $N$ is the block-length. We provide lower bounds on $\mu$, namely $\mu \geq 3.553$, and we conjecture that indeed $\mu=3.627$, the parameter for the binary erasure channel.