License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.92
URN: urn:nbn:de:0030-drops-106687
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10668/
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Guruswami, Venkatesan ; Riazanov, Andrii

Beating Fredman-Komlós for Perfect k-Hashing

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Abstract

We say a subset C subseteq {1,2,...,k}^n is a k-hash code (also called k-separated) if for every subset of k codewords from C, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as (log_2 |C|)/n, of a k-hash code is a classical problem. It arises in two equivalent contexts: (i) the smallest size possible for a perfect hash family that maps a universe of N elements into {1,2,...,k}, and (ii) the zero-error capacity for decoding with lists of size less than k for a certain combinatorial channel.
A general upper bound of k!/k^{k-1} on the rate of a k-hash code (in the limit of large n) was obtained by Fredman and Komlós in 1984 for any k >= 4. While better bounds have been obtained for k=4, their original bound has remained the best known for each k >= 5. In this work, we present a method to obtain the first improvement to the Fredman-Komlós bound for every k >= 5, and we apply this method to give explicit numerical bounds for k=5, 6.

BibTeX - Entry

@InProceedings{guruswami_et_al:LIPIcs:2019:10668,
  author =	{Venkatesan Guruswami and Andrii Riazanov},
  title =	{{Beating Fredman-Koml{\'o}s for Perfect k-Hashing}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{92:1--92:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10668},
  URN =		{urn:nbn:de:0030-drops-106687},
  doi =		{10.4230/LIPIcs.ICALP.2019.92},
  annote =	{Keywords: Coding theory, perfect hashing, hash family, graph entropy, zero-error information theory}
}

Keywords: Coding theory, perfect hashing, hash family, graph entropy, zero-error information theory
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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