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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Abstract
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 |
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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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