On the Correlation Between Polarized BECs

TL;DR

This paper demonstrates that in polar codes for BECs, the correlations between synthetic channels' erasure events diminish extremely rapidly with increasing code length, enabling very tight error probability bounds.

Contribution

It establishes that pairwise correlations between synthetic BEC channels decay faster than any exponential in the block length, improving error analysis for polar codes.

Findings

Correlation coefficients vanish faster than exponential in n

Union bound on block error probability is very tight

Asymptotic independence of synthetic channels

Abstract

We consider the $2^n$ channels synthesized by the $n$-fold application of Ar\i{}kan's polar transform to a binary erasure channel (BEC). The synthetic channels are BECs themselves, and we show that, asymptotically for almost all these channels, the pairwise correlations between their erasure events are extremely small: the correlation coefficients vanish faster than any exponential in $n$. Such a fast decay of correlations allows us to conclude that the union bound on the block error probability of polar codes is very tight.