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MDS Array Codes with Optimal Rebuilding
Information Theory · Submitted 19 Mar 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In general, we construct a new family of $r$-erasure correcting MDS array codes that has optimal rebuilding ratio of $\frac{1}{r}$ in the case of a single erasure.
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The abstract, sentence by sentence
MDS array codes are widely used in storage systems to protect data against erasures. We address the \emph{rebuilding ratio} problem, namely, in the case of erasures, what is the the fraction of the remaining information that needs to be accessed in order to rebuild \emph{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 2-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 $\frac{1}{r}$ in the case of a single erasure. Our array codes have efficient encoding and decoding algorithms (for the case $r=2$ they use a finite field of size 3) and an optimal update property.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | MDS array codes are widely used in storage systems to protect data against erasures. | background 1.00 | 0.04 | 0.01 | asserted 1.00 |
| s1 | We address the \emph{rebuilding ratio} problem, namely, in the case of erasures, what is the the fraction of the remaining information that needs to be accessed in order to rebuild \emph{exactly} the lost information? | background 0.81 | 0.22 | 0.04 | unclear 0.57 |
| s2 | 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). | background 1.00 | 0.11 | 0.02 | asserted 1.00 |
| s3 | However, the interesting (and more practical) case is when the number of erasures is smaller than the erasure correcting capability of the code. | background 0.92 | 0.07 | 0.02 | asserted 0.97 |
| s4 | 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? | background 0.84 | 0.05 | 0.02 | unclear 0.84 |
| s5 | 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. | gap 0.98 | 0.03 | 0.03 | asserted 1.00 |
| s6 | In this paper, we solve this open problem and prove that for the case of a single erasure with a 2-erasure correcting code, the rebuilding ratio is 1/2. | result 0.83 | 0.97 | 0.59 | asserted 1.00 |
| s7 | In general, we construct a new family of $r$-erasure correcting MDS array codes that has optimal rebuilding ratio of $\frac{1}{r}$ in the case of a single erasure. | contribution 0.79 | 0.96 | 0.74 | asserted 1.00 |
| s8 | Our array codes have efficient encoding and decoding algorithms (for the case $r=2$ they use a finite field of size 3) and an optimal update property. | contribution 0.76 | 0.92 | 0.08 | asserted 1.00 |
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Original abstract
MDS array codes are widely used in storage systems to protect data against erasures. We address the \emph{rebuilding ratio} problem, namely, in the case of erasures, what is the the fraction of the remaining information that needs to be accessed in order to rebuild \emph{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 2-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 $\frac{1}{r}$ in the case of a single erasure. Our array codes have efficient encoding and decoding algorithms (for the case $r=2$ they use a finite field of size 3) and an optimal update property.
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5 arXiv papers cite this in 9 sentences
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Compares with it 1 paper
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We show that these codes are equivalent to codes that involve permutation matrices in the manner of [ref] and [ref] under a similarity transformation.
Repair Optimal Erasure Codes through Hadamard Designs · Introduction · 0.86
Cites it as background 2 papers
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There are several metrics that can be optimized during repair: the total information read from existing disks during repair [ref] , the total information communicated in the network (repair bandwidth [ref] ), or the total number of disks required for each repair [ref] .
Repair Optimal Erasure Codes through Hadamard Designs · Introduction · 0.93
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There are several metrics that can be optimized during repair: the total information read from existing disks during repair [ref] , the total information communicated in the network [ref] (called repair bandwidth [ref] ), or the total number of disks required for each repair [ref] .
Simple Regenerating Codes: Network Coding for Cloud Storage · Introduction · 0.85
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