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.2019.98
URN: urn:nbn:de:0030-drops-106745
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10674/
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Xing, Chaoping ; Yuan, Chen

Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph

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Abstract

Locally recoverable codes are a class of block codes with an additional property called locality. A locally recoverable code with locality r can recover a symbol by reading at most r other symbols. Recently, it was discovered by several authors that a q-ary optimal locally recoverable code, i.e., a locally recoverable code achieving the Singleton-type bound, can have length much bigger than q+1. In this paper, we present both the upper bound and the lower bound on the length of optimal locally recoverable codes. Our lower bound improves the best known result in [Yuan Luo et al., 2018] for all distance d >= 7. This result is built on the observation of the parity-check matrix equipped with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure produces an optimal locally recoverable code if it satisfies a certain expansion property for subsets of F_q. To our surprise, this expansion property is then shown to be equivalent to a well-studied problem in extremal graph theory. Our upper bound is derived by an refined analysis of the arguments of Theorem 3.3 in [Venkatesan Guruswami et al., 2018].

BibTeX - Entry

@InProceedings{xing_et_al:LIPIcs:2019:10674,
  author =	{Chaoping Xing and Chen Yuan},
  title =	{{Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{98:1--98:13},
  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/2019/10674},
  URN =		{urn:nbn:de:0030-drops-106745},
  doi =		{10.4230/LIPIcs.ICALP.2019.98},
  annote =	{Keywords: Locally Repairable Codes, Hypergraph}
}

Keywords: Locally Repairable Codes, Hypergraph
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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