Multilinear algebra for minimum storage regenerating codes: a generalization of the product-matrix construction

Duursma, Iwan; Wang, Hsin Po

doi:10.1007/s00200-021-00526-3

Cited by 6 publications

(16 citation statements)

References 34 publications

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“…Using the method in the proof of Lemma 4, we can show that both Q (4) z and Q (5) z are invertible. Recall the definitions of y (2) and y (3) in Case 3. The first 2 z−1 rows of M z y = 0 give us…”

Section: Optimal Repair Bandwidth For Single Node Failurementioning

confidence: 99%

“…Yet it is limited to the low code rate regime. After that, explicit constructions of high-rate MSR codes were given in [8], [11], [12], [13], [7], [6], [9], [4], [2], [10]. Among these MSR code constructions, the most relevant one to this paper is the (n = k + 2, k) MSR code construction with subpacketization 2 k/3 proposed in [11].…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, Wang et al [11] proposed a framework of constructing MDS array codes with subpacketization r k/(r+1) that can optimally repair all the systematic nodes from d = n − 1 helper nodes. For the special case of r = 2, they further gave explicit constructions of such codes over any finite field whose size is larger than 2 3 k. This construction has the smallest subpacketization among all the existing MSR code constructions with parameters d = k + 1 and n = k + 2.…”

Section: Introductionmentioning

confidence: 99%

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Constructing MSR codes with subpacketization $2^{n/3}$ for $k+1$ helper nodes

Wang,

Li,

et al. 2022

Preprint

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Wang et al. (IEEE Transactions on Information Theory, vol. 62, no. 8, 2016) proposed an explicit construction of an (n = k + 2, k) Minimum Storage Regenerating (MSR) code with 2 parity nodes and subpacketization 2 k/3 . The number of helper nodes for this code is d = k + 1 = n − 1, and this code has the smallest subpacketization among all the existing explicit constructions of MSR codes with the same n, k and d. In this paper, we present a new construction of MSR codes for a wider range of parameters. More precisely, we still fix d = k + 1, but we allow the code length n to be any integer satisfying n ≥ k + 2. The field size of our code is linear in n, and the subpacketization of our code is 2 n/3 . This value is slightly larger than the subpacketization of the construction by Wang et al. because their code construction only guarantees optimal repair for all the systematic nodes while our code construction guarantees optimal repair for all nodes.

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Section: Optimal Repair Bandwidth For Single Node Failurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constructing MSR codes with subpacketization $2^{n/3}$ for $k+1$ helper nodes

Wang,

Li,

et al. 2022

Preprint

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“…The MSR case is the most widely studied in the literature. Several general constructions of MSR codes have been proposed in recent years, among them product matrix codes [17] and their generalization in [5], diagonal matrix codes [26], and others. In this work we study MSR codes as well as codes for the interior points of the trade-off curve.…”

Section: Introductionmentioning

confidence: 99%

“…In doing so, we shift the perspective, viewing them as evaluation codes, i.e., codes whose encoding can be phrased as evaluation of a linear functional written in a convenient algebraic form. We rewrite the IP repair procedure of product-matrix codes from [15], which enables us to extend it to a much more general class of codes introduced recently by Duursma and Wang [5]. This in turn prepares the way for the analysis of intermediate-point codes, and we begin with implementing IP repair for the Moulin codes of Duursma et al [4] which also fall under the evaluation category.…”

Section: Introductionmentioning

confidence: 99%

Node Repair on Connected Graphs

Patra

Barg

2022

IEEE Trans. Inform. Theory

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We continue our study of regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this problem, the failed node downloads the information stored at a subset of vertices of the graph for the purpose of recovering the lost data. This information is moved across the network, and the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. This problem was formulated in our recent work (IEEE IT Transactions, May 2022) where we showed that processing of the information at the intermediate nodes can yield savings in repair bandwidth over the direct forwarding of the data.While the previous paper was limited to the MSR case, here we extend our study to the case of general regenerating codes. We derive a lower bound on the repair bandwidth and formulate repair procedures with intermediate processing for several families of regenerating codes, with an emphasis on the recent constructions from multilinear algebra. We also consider the task of data retrieval for codes on graphs, deriving a lower bound on the communication bandwidth and showing that it can be attained at the MBR point of the storagebandwidth tradeoff curve.

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Regenerating codes on graphs

Patra

Barg

2021

2021 IEEE International Symposium on Information Theory (ISIT)

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We consider regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this problem, the failed node downloads the information stored at a subset of vertices of the graph for the purpose of recovering the lost data. Compared to the standard setting, regenerating codes on graphs address two additional features. The repair information is moved across the network, and the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. Accordingly, the helpers far away from the failed node may be expected to contribute less data for repair than the nodes in the neighborhood of that node. We analyze regenerating codes with nonuniform download for repair on graphs. Moreover, in the process of repair, the information moved from the helpers to the failed node may be combined through intermediate processing, reducing the repair bandwidth. We derive lower bounds for communication complexity of node repair on graphs, including repair schemes with nonuniform download and intermediate processing, and construct codes that attain these bounds.Additionally, some of the nodes may act as adversaries, introducing errors into the data moved in the network. For repair on graphs in the presence of adversarial nodes, we construct codes that support node repair and error correction in systematic nodes.

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Multilinear algebra for minimum storage regenerating codes: a generalization of the product-matrix construction

Cited by 6 publications

References 34 publications

Constructing MSR codes with subpacketization $2^{n/3}$ for $k+1$ helper nodes

Constructing MSR codes with subpacketization $2^{n/3}$ for $k+1$ helper nodes

Node Repair on Connected Graphs

Regenerating codes on graphs

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