Almost Optimal Scaling of Reed-Muller Codes on BEC and BSC Channels

TL;DR

This paper demonstrates that Reed-Muller codes can achieve nearly optimal error probability transition widths on BEC and BSC channels, even without linear minimum distance, under certain weight distribution conditions.

Contribution

It proves that Reed-Muller codes exhibit almost optimal transition scaling, improving previous bounds and introducing a new estimate on the EXIT function derivative.

Findings

Transition width can be as small as Θ(1/N^{1/2-κ}) for Reed-Muller codes.

The result applies even when the minimum distance is not linear.

Introduces a new estimate on the derivative of the EXIT function.

Abstract

Consider a binary linear code of length $N$, minimum distance $d_{\text{min}}$, transmission over the binary erasure channel with parameter $0 < \epsilon < 1$ or the binary symmetric channel with parameter $0 < \epsilon < \frac12$, and block-MAP decoding. It was shown by Tillich and Zemor that in this case the error probability of the block-MAP decoder transitions "quickly" from $\delta$ to $1-\delta$ for any $\delta>0$ if the minimum distance is large. In particular the width of the transition is of order $O(1/\sqrt{d_{\text{min}}})$. We strengthen this result by showing that under suitable conditions on the weight distribution of the code, the transition width can be as small as $\Theta(1/N^{\frac12-\kappa})$, for any $\kappa>0$, even if the minimum distance of the code is not linear. This condition applies e.g., to Reed-Mueller codes. Since $\Theta(1/N^{\frac12})$ is the smallest transition possible for any code, we speak of "almost" optimal scaling. We emphasize that the width of the transition says nothing about the location of the transition. Therefore this result has no bearing on whether a code is capacity-achieving or not. As a second contribution, we present a new estimate on the derivative of the EXIT function, the proof of which is based on the Blowing-Up Lemma.