Repairing Reed-solomon codes

Guruswami, Venkatesan; Wootters, Mary

doi:10.1145/2897518.2897525

Cited by 93 publications

(240 citation statements)

References 18 publications

(12 reference statements)

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“…We then say that C is MRD if equality holds in (10). In another direction, a Singleton bound on the relative minimum rank distance of a pair of F q m -linear codes C 2 C 1 ⊆ F n q m was first given in [16,Prop.…”

Section: Mrd Codes and Coset Coding Schemes With Optimal Communicatiomentioning

confidence: 99%

Universal secure rank-metric coding schemes with optimal communication overheads

Martínez-Peñas

2018

Cryptogr. Commun.

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We study the problem of reducing the communication overhead from a noisy wire-tap channel or storage system where data is encoded as a matrix, when more columns (or their linear combinations) are available. We present its applications to reducing communication overheads in universal secure linear network coding and secure distributed storage with crisscross errors and erasures and in the presence of a wire-tapper. Our main contribution is a method to transform coding schemes based on linear rank-metric codes, with certain properties, to schemes with lower communication overheads. By applying this method to pairs of Gabidulin codes, we obtain coding schemes with optimal information rate with respect to their security and rank error correction capability, and with universally optimal communication overheads, when n ≤ m, being n and m the number of columns and number of rows, respectively. Moreover, our method can be applied to other families of maximum rank distance codes when n > m. The downside of the method is generally expanding the packet length, but some practical instances come at no cost.

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Section: Mrd Codes and Coset Coding Schemes With Optimal Communicatiomentioning

confidence: 99%

Universal secure rank-metric coding schemes with optimal communication overheads

Martínez-Peñas

2018

Cryptogr. Commun.

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“…The problem of devising exact repair mechanism with small repair bandwidth for known MDS codes has been studied in [10,24,30,33,35]. In particular, [10,24,35] consider the exact repair problem for the well-known Reed-Solomon codes. In [10], Guruswami and Wootters present a general framework to design linear schemes to repair RS codes.…”

Section: Background and Related Workmentioning

confidence: 99%

“…In particular, [10,24,35] consider the exact repair problem for the well-known Reed-Solomon codes. In [10], Guruswami and Wootters present a general framework to design linear schemes to repair RS codes. They further characterize optimal repair bandwidth for RS codes in certain regimes of system parameters.…”

Section: Background and Related Workmentioning

confidence: 99%

“…They further characterize optimal repair bandwidth for RS codes in certain regimes of system parameters. In [35], utilizing the framework from [10], Ye and Barg show that it is possible to construct RS codes with asymptotically optimal repair bandwidth (as the code length n tends to ∞). The repair scheme in [35] works with a sub-packetization level which is exponential in n. In particular, the repair scheme treats each code symbol of the underlying RS codes as a vector of length (n − k) n over a base field.…”

Section: Background and Related Workmentioning

confidence: 99%

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MDS Code Constructions with Small Sub-packetization and Near-optimal Repair Bandwidth

Rawat

Tamo

Guruswami

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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An (n, M ) vector code C ⊆ F n is a collection of M codewords where n elements (from the field F) in each of the codewords are referred to as code blocks. Assuming that F ∼ = B , the code blocks are treated as -length vectors over the base field B. Equivalently, the code is said to have the sub-packetization level . This paper addresses the problem of constructing MDS vector codes which enable exact reconstruction of each code block by downloading small amount of information from the remaining code blocks. The repair bandwidth of a code measures the information flow from the remaining code blocks during the reconstruction of a single code block. This problem naturally arises in the context of distributed storage systems as the node repair problem [4]. Assuming that M = |B| k , the repair bandwidth of an MDS vector code is lower bounded by (n − 1)/(n − k) · symbols (over the base field B) which is also referred to as the cut-set bound [4]. For all values of n and k, the MDS vector codes that attain the cut-set bound with the sub-packetization level = (n − k) n/(n−k) are known in the literature [23,36].This paper presents a construction for MDS vector codes which simultaneously ensures both small repair bandwidth and small sub-packetization level. The obtained codes have the smallest possible sub-packetization level = O(n − k) for an MDS vector code and the repair bandwidth which is at most twice the cut-set bound. The paper then generalizes this code construction so that the repair bandwidth of the obtained codes approach the cut-set bound at the cost of increased sub-packetization level. The constructions presented in this paper give MDS vector codes which are linear over the base field B.

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“…The same approach can be used to prove Lemma 5. [4,2,3] binary array code with optimal repair bandwidth for a single systematic node failure, described by (x 1 , x 2 ,…”

Section: Primer: Code Construction Formentioning

confidence: 99%

Minimum storage regenerating codes for all parameters

Goparaju

Vardy

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Abstract-Regenerating codes for distributed storage have attracted much research interest in the past decade. Such codes trade the bandwidth needed to repair a failed node with the overall amount of data stored in the network. Minimum storage regenerating (MSR) codes are an important class of optimal regenerating codes that minimize (first) the amount of data stored per node and (then) the repair bandwidth. Specifically, an [n, k, d]-(α) MSR code C over F q stores a file F consisting of αk symbols over F q among n nodes, each storing α symbols, in such a way that:• the file F can be recovered by downloading the content of any k of the n nodes; and • the content of any failed node can be reconstructed by accessing any d of the remaining n − 1 nodes and downloading α/(d−k+1) symbols from each of these nodes. In practice, the file F is typically available in uncoded form on some k of the n nodes, known as systematic nodes, and the defining node-repair condition above can be relaxed to requiring the optimal repair bandwidth for systematic nodes only. Such codes are called systematic-repair MSR codes.Unfortunately, finite-α constructions of [n, k, d] MSR codes are known only for certain special cases: either low rate, namely k/n 0.5, or high repair connectivity, namely d = n − 1. Our main result in this paper is a finite-α construction of systematic-repair [n, k, d] MSR codes for all possible values of parameters n, k, d. We also introduce a generalized construction for [n, k] MSR codes to achieve the optimal repair bandwidth for all values of d simultaneously.

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Repairing Reed-solomon codes

Cited by 93 publications

References 18 publications

Universal secure rank-metric coding schemes with optimal communication overheads

Universal secure rank-metric coding schemes with optimal communication overheads

MDS Code Constructions with Small Sub-packetization and Near-optimal Repair Bandwidth

Minimum storage regenerating codes for all parameters

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