## 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 language | American English |
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Title of host publication | 2018 IEEE International Symposium on Information Theory, ISIT 2018 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 311-315 |

Number of pages | 5 |

ISBN (Print) | 9781538647806 |

DOIs | |

State | Published - 15 Aug 2018 |

Event | 2018 IEEE International Symposium on Information Theory, ISIT 2018 - Vail, United States Duration: 17 Jun 2018 → 22 Jun 2018 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2018-June |

ISSN (Print) | 2157-8095 |

### Conference

Conference | 2018 IEEE International Symposium on Information Theory, ISIT 2018 |
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Country/Territory | United States |

City | Vail |

Period | 17/06/18 → 22/06/18 |

### Bibliographical note

Funding Information:The research of H. Hassani was supported by NSF-CRII award CCF-1755707. The work of O. Ordentlich was supported by ISF under Grant 1791/17. The research of Y. Polyanskiy was supported by the Center for Science of Information (CSoI), an NSF Science and Technology Center, under grant agreement CCF-09-39370, NSF CAREER award CCF-12-53205, and NSF grant CCF-17-17842. The work of R. Urbanke was supported by SNSF grant No. 200021–166106.

Publisher Copyright:

© 2018 IEEE.