## Abstract

Let Z^{n} be iid Bernoulli (\delta) and U^{n} be uniform on the set of all binary vectors of weight \delta n (Hamming sphere). As is well known, the entropies of Z^{n} and U^{n} are within O(\log n). However, if X^{n} is another binary random variable independent of Z^{n} and U^{n}, we show that H(X^{n}+U^{n}) and H(X^{n}+Z^{n}) are within O(\sqrt{n}) and this estimate is tight. The bound is shown via coupling method. Tightness follows from the observation that the channels x^{n}\mapsto x^{n}+U^{n} and x^{n}\mapsto x^{n}+Z^{n} have similar capacities, but the former has zero dispersion. Finally, we show that despite the \sqrt{n} slack in general, the Mrs. Gerber Lemma for H(X^{n}+U^{n}) holds with only an O(\log n) correction compared to its brethren for H(X^{n}+Z^{n}).

Original language | 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 | 521-525 |

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

Publisher Copyright:© 2018 IEEE.