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A Tight Rate Bound and Matching Construction for Locally Recoverable Codes with Sequential Recovery from Any Number of Multiple Erasures

Balaji, S B and Kini, G R and Kumar, P V (2020) A Tight Rate Bound and Matching Construction for Locally Recoverable Codes with Sequential Recovery from Any Number of Multiple Erasures. In: IEEE Transactions on Information Theory, 66 (2). pp. 1023-1052.

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Official URL: https://dx.doi.org/10.1109/TIT.2019.2958970


This paper considers the natural extension of locally recoverable codes (LRC) to the case of t > 1 erased symbols. While several approaches have been proposed for the handling of multiple erasures, in the approach considered here, the t erased symbols are recovered in succession, each time contacting at most r other symbols for assistance. Under the local-recovery constraint, this sequential approach is the most general and hence offers the maximum possible code rate. We characterize the rate of an LRC with sequential recovery for any r \geq 3 and any t, by first deriving an upper bound on the code rate and then constructing a binary code achieving this optimal rate. The upper bound derived here proves an earlier conjecture. Our approach permits us to deduce the structure of the parity-check matrix of a rate-optimal LRC with sequential recovery. The derived structure of parity-check matrix leads to a graphical description of the code used in code construction. A subclass of binary codes that are both rate and block-length optimal, are shown to correspond to certain regular graphs known as Moore graphs, that have the smallest number of vertices for a given girth. A connection with Tornado codes is also made.

Item Type: Journal Article
Publication: IEEE Transactions on Information Theory
Publisher: Institute of Electrical and Electronics Engineers Inc.
Additional Information: Copyright of this article belongs to IEEE
Keywords: Binary codes; Graph theory; Matrix algebra; Recovery, Code construction; Distributed storage; Graphical description; Local recoveries; locally repairable codes; Natural extension; Parity check matrices; Sequential approach, Optimal systems
Department/Centre: Division of Electrical Sciences > Electrical Communication Engineering
Date Deposited: 25 Feb 2020 06:41
Last Modified: 25 Feb 2020 06:41
URI: http://eprints.iisc.ac.in/id/eprint/64512

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