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DOI: 10.4230/LIPIcs.ITCS.2020.85
URN: urn:nbn:de:0030-drops-117704
URL: https://drops.dagstuhl.de/opus/volltexte/2020/11770/
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Bhattacharyya, Arnab ; Chandran, L. Sunil ; Ghoshal, Suprovat

Combinatorial Lower Bounds for 3-Query LDCs

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Abstract

A code is called a q-query locally decodable code (LDC) if there is a randomized decoding algorithm that, given an index i and a received word w close to an encoding of a message x, outputs x_i by querying only at most q coordinates of w. Understanding the tradeoffs between the dimension, length and query complexity of LDCs is a fascinating and unresolved research challenge. In particular, for 3-query binary LDC’s of dimension k and length n, the best known bounds are: 2^{k^o(1)} ≥ n ≥ Ω ̃(k²). In this work, we take a second look at binary 3-query LDCs. We investigate a class of 3-uniform hypergraphs that are equivalent to strong binary 3-query LDCs. We prove an upper bound on the number of edges in these hypergraphs, reproducing the known lower bound of Ω ̃(k²) for the length of strong 3-query LDCs. In contrast to previous work, our techniques are purely combinatorial and do not rely on a direct reduction to 2-query LDCs, opening up a potentially different approach to analyzing 3-query LDCs.

BibTeX - Entry

@InProceedings{bhattacharyya_et_al:LIPIcs:2020:11770,
  author =	{Arnab Bhattacharyya and L. Sunil Chandran and Suprovat Ghoshal},
  title =	{{Combinatorial Lower Bounds for 3-Query LDCs}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{85:1--85:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Thomas Vidick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11770},
  URN =		{urn:nbn:de:0030-drops-117704},
  doi =		{10.4230/LIPIcs.ITCS.2020.85},
  annote =	{Keywords: Coding theory, Graph theory, Hypergraphs}
}

Keywords: Coding theory, Graph theory, Hypergraphs
Seminar: 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Issue Date: 2020
Date of publication: 10.01.2020


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