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Zigzag Codes: MDS Array Codes With Optimal Rebuilding

Tamo, Itzhak and Wang, Zhiying and Bruck, Jehoshua (2013) Zigzag Codes: MDS Array Codes With Optimal Rebuilding. IEEE Transactions on Information Theory, 59 (3). pp. 1597-1616. ISSN 0018-9448. https://resolver.caltech.edu/CaltechAUTHORS:20130321-102330661

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Abstract

Maximum distance separable (MDS) array codes are widely used in storage systems to protect data against erasures. We address the rebuilding ratio problem, namely, in the case of erasures, what is the fraction of the remaining information that needs to be accessed in order to rebuild exactly the lost information? It is clear that when the number of erasures equals the maximum number of erasures that an MDS code can correct, then the rebuilding ratio is 1 (access all the remaining information). However, the interesting and more practical case is when the number of erasures is smaller than the erasure correcting capability of the code. For example, consider an MDS code that can correct two erasures: What is the smallest amount of information that one needs to access in order to correct a single erasure? Previous work showed that the rebuilding ratio is bounded between 1/2 and 3/4; however, the exact value was left as an open problem. In this paper, we solve this open problem and prove that for the case of a single erasure with a two-erasure correcting code, the rebuilding ratio is 1/2. In general, we construct a new family of r-erasure correcting MDS array codes that has optimal rebuilding ratio of 1/(r) in the case of a single erasure. Our array codes have efficient encoding and decoding algorithms (for the cases r=2 and r=3, they use a finite field of size 3 and 4, respectively) and an optimal update property.

Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1109/TIT.2012.2227110DOIArticle
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6352912PublisherArticle
http://arxiv.org/abs/1112.0371arXivDiscussion Paper
ORCID:
AuthorORCID
Bruck, Jehoshua0000-0001-8474-0812
Additional Information:© 2012 IEEE. Manuscript received October 22, 2011; revised June 30, 2012; accepted September 23, 2012. Date of publication November 16, 2012; date of current version February 12, 2013. This work was supported in part by NSF Grant ECCS-0801795 and in part by BSF Grant 2010075. This paper was presented in part at the 2011 IEEE International Symposium on Information Theory and in part at the 2011 Allerton Conference on Communication, Control, and Computing, Monticello, IL.
Funders:
Funding AgencyGrant Number
NSFECCS-0801795
Binational Science Foundation (USA-Israel)2010075
Subject Keywords:Distributed storage; network coding; optimal rebuilding; RAID
Other Numbering System:
Other Numbering System NameOther Numbering System ID
INSPEC Accession Number13307118
Issue or Number:3
Record Number:CaltechAUTHORS:20130321-102330661
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20130321-102330661
Official Citation:Tamo, I.; Zhiying Wang; Bruck, J., "Zigzag Codes: MDS Array Codes With Optimal Rebuilding," Information Theory, IEEE Transactions on , vol.59, no.3, pp.1597,1616, March 2013 doi: 10.1109/TIT.2012.2227110
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:37585
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:04 Apr 2013 23:28