Almost Optimal Scaling of Reed-Muller Codes on BEC and BSC Channels
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1801.09481.pdf
Description
Submitted version
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157.58 KB
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Adobe PDF
Checksum (MD5)
0f76e2d587647f9de0666bfd6402faf3
Author(s)
Ordentlich, Or
Polyanskiy, Yury
Date Issued
June 2018
Journal
2018 IEEE International Symposium on Information Theory (ISIT)
Publisher
IEEE
Citation
Hassani, Hamed, Shrinivas Kudekar, Or Ordentlich, Yury Polyanskiy and Rüdiger Urbanke. "Almost Optimal Scaling of Reed-Muller Codes on BEC and BSC Channels." In 2018 IEEE International Symposium on Information Theory (ISIT), Vail, CO, USA, 17-22 June 2018.
Version
Original manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1109/isit.2018.8437453
