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

Hamed Hassani, Shrinivas Kudekar, Or Ordentlich, Yury Polyanskiy, Rudiger Urbanke

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

15 Scopus citations

Abstract

Consider a binary linear code of length N, minimum distance d-{\min}, transmission over the binary erasure channel with parameter 0 < \epsilon < 1 or the binary symmetric channel with parameter 0 < \epsilon < \frac{1}{2}, 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-{\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^{\frac{1}{2}-\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^{\frac{1}{2}}) 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.

Original languageAmerican English
Title of host publication2018 IEEE International Symposium on Information Theory, ISIT 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages311-315
Number of pages5
ISBN (Print)9781538647806
DOIs
StatePublished - 15 Aug 2018
Event2018 IEEE International Symposium on Information Theory, ISIT 2018 - Vail, United States
Duration: 17 Jun 201822 Jun 2018

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2018-June
ISSN (Print)2157-8095

Conference

Conference2018 IEEE International Symposium on Information Theory, ISIT 2018
Country/TerritoryUnited States
CityVail
Period17/06/1822/06/18

Bibliographical note

Publisher Copyright:
© 2018 IEEE.

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