Zigzag Codes: MDS Array Codes With Optimal Rebuilding

• Published in: IEEE transactions on information theory Vol. 59; no. 3; pp. 1597 - 1616
• Main Authors: Tamo, I., Zhiying Wang, Bruck, J.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Arrays; Bandwidth; Codes; Coding, codes; Distributed storage; Encoding; Exact sciences and technology; Frequency modulation; Information theory; Information, signal and communications theory; Maintenance engineering; Mathematical problems; network coding; optimal rebuilding; RAID; Signal and communications theory; Switching and signalling; Systematics; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Vectors; Distributed memory; Finite field; Packet switching; optimal rebuilding; Storage system; Network coding; Updating; Decoding; RAID; Algorithm; Distributed storage
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2012.2227110

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.