Combinatorial lower bounds for 3-query LDCs

Bhattacharyya, A and Sunil Chandran, L and Ghoshal, S (2020) Combinatorial lower bounds for 3-query LDCs. In: Leibniz International Proceedings in Informatics, LIPIcs, 12 - 14 January 2020, Seattle.

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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 xi 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: 2ko(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.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing.
Keywords:	Computer programming; Decoding; Equivalence classes, 3-uniform hypergraphs; Coding Theory; Decoding algorithm; Direct Reduction; Hyper graph; Locally-decodable codes; Query complexity; Research challenges, Graph theory
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	09 Feb 2023 05:36
Last Modified:	09 Feb 2023 05:36
URI:	https://eprints.iisc.ac.in/id/eprint/80054

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