On t-designs and bounds relating query complexity to error resilience in locally correctable codes

Lalitha, V and Prakash, N and Kamath, Govinda M and V ijay Kumar, P (2012) On t-designs and bounds relating query complexity to error resilience in locally correctable codes. In: 2012 National Conference on Communications (NCC), 3-5 Feb. 2012, Kharagpur, India.

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Abstract

An n-length block code C is said to be r-query locally correctable, if for any codeword x ∈ C, one can probabilistically recover any one of the n coordinates of the codeword x by querying at most r coordinates of a possibly corrupted version of x. It is known that linear codes whose duals contain 2-designs are locally correctable. In this article, we consider linear codes whose duals contain t-designs for larger t. It is shown here that for such codes, for a given number of queries r, under linear decoding, one can, in general, handle a larger number of corrupted bits. We exhibit to our knowledge, for the first time, a finite length code, whose dual contains 4-designs, which can tolerate a fraction of up to 0.567/r corrupted symbols as against a maximum of 0.5/r in prior constructions. We also present an upper bound that shows that 0.567 is the best possible for this code length and query complexity over this symbol alphabet thereby establishing optimality of this code in this respect. A second result in the article is a finite-length bound which relates the number of queries r and the fraction of errors that can be tolerated, for a locally correctable code that employs a randomized algorithm in which each instance of the algorithm involves t-error correction.

Item Type:	Conference Paper
Publisher:	IEEE
Additional Information:	Copyright of this article belongs to IEEE.
Department/Centre:	Division of Electrical Sciences > Electrical Communication Engineering
Date Deposited:	27 Jan 2014 06:31
Last Modified:	27 Jan 2014 06:32
URI:	http://eprints.iisc.ac.in/id/eprint/48286

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