English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 inthis case the error probability of the block-MAP decoder transitions "quickly"from $\delta$ to $1-\delta$ for any $\delta>0$ if the minimum distance islarge. In particular the width of the transition is of order$O(1/\sqrt{d_{\text{min}}})$. We strengthen this result by showing that undersuitable conditions on the weight distribution of the code, the transitionwidth 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 appliese.g., to Reed-Mueller codes. Since $\Theta(1/N^{\frac12})$ is the smallesttransition possible for any code, we speak of "almost" optimal scaling. Weemphasize that the width of the transition says nothing about the location ofthe transition. Therefore this result has no bearing on whether a code iscapacity-achieving or not. As a second contribution, we present a new estimateon the derivative of the EXIT function, the proof of which is based on theBlowing-Up Lemma.

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