Bounds and Constructions of Locally Repairable Codes: Parity-Check Matrix Approach

Hao, Jie; Xia, Shu-Tao; Shum, Kenneth W.; Chen, Bin; Fu, Fang-Wei; Yang, Yixian

doi:10.1109/tit.2020.3021707

Cited by 42 publications

(43 citation statements)

References 36 publications

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“…There exists two equivalent descriptions of an LRC, i.e, the generator-matrix approach [1] and the parity-check matrix approach [8]. The latter says that a code symbol has locality r if and only if there exists a dual codeword (or parity-check equation) whose Hamming weight is at most r + 1 and the corresponding non-zero components cover the coordinate of the symbol.…”

Section: Parity-check Matrix Frameworkmentioning

confidence: 99%

“…Reference [22] proposes a more general definition of LRCs in the matroid language, some generalized upper bounds and constructions are also obtained. The parity-check matrix approach [8] is frequently used in characterizing the structure of optimal LRCs and deriving the bounds on the parameters recently. References [9] and [12] carefully analyze the structure of parity-check matrix of optimal LRCs, and completely determine all binary and ternary optimal LRCs.…”

Section: Introductionmentioning

confidence: 99%

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Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and $K-1$

Xia

Chen

2019

IEEE Access

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A locally repairable code (LRC) is a [n, k, d] linear code with length n, dimension k, minimum distance d and locality r, which means that every code symbol can be repaired by at most r other symbols. LRCs have become an important candidate in distributed storage systems due to their relatively low I/O cost. An LRC is said to be optimal if its minimum distance meets one of the Singleton-like bounds. This paper considers the optimal constructions of LRCs with locality r = 1 and r = k − 1, which involves three types: r-local LRCs, (r, δ)-LRCs and LRCs with t-availability. Specifically, we first prove that the existence of an optimal LRC with locality r = 1 is equivalent to that of an MDS code with certain parameters. Thus we can completely characterize the three types of optimal LRCs with r = 1 based on some known constructions of MDS codes. Near MDS codes is a special class of sub-optimal linear code whose minimum distance d = n − k and the i-th generalized Hamming weight achieves the generalized Singleton bound for 2 ≤ i ≤ k. For r = k − 1, we have established the connections between optimal r-local LRCs/ LRCs with t-availability and near MDS codes. Such connections can help to construct optimal LRCs with r = k − 1 from some known classes of near MDS codes.INDEX TERMS Distributed storage systems, erasure codes, maximum distance separable (MDS) codes, near MDS codes, locally repairable codes.Note that when δ = 2, an (r, δ)-LRC could reduce to an r-local LRC.

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Section: Parity-check Matrix Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and $K-1$

Xia

Chen

2019

IEEE Access

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“…A family of optimal (n, k, r) LRCs against the bound (I.1) was constructed by Tamo and Barg [11]. Hao et al [5] overviewed (n, k, r) LRCs via parity check matrix and gave different possible classes of optimal binary (n, k, r) LRCs.…”

Section: Introductionmentioning

confidence: 99%

A Class of $(n, k, r, t)$ IS-LRCs Via Parity Check Matrix

Mukhopadhyay¹,

Bhowmick²,

Hansda³

et al. 2021

Preprint

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A code is called (n, k, r, t) information symbol locally repairable code (IS-LRC) if each information coordinate can be achieved by at least t disjoint repair sets containing at most r other coordinates. This letter considers a class of (n, k, r, t) IS-LRCs, where each repair set contains exactly one parity coordinate. We explore the systematic code in terms of the standard parity check matrix. First, we propose some structural features of the parity check matrix by showing a connection with the membership matrix. After that, we place parity check matrix based proof of several bounds associated with the code. In addition, we provide two constructions of optimal parameters of (n, k, r, t) IS-LRCs with the help of two Cayley tables of a finite field. Finally, we present a generalized result on optimal q-ary (n, k, r, t) IS-LRCs related to MDS codes.

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“…We take a slightly different perspective and enumerate the code parameters of all optimal LRCs over a fixed finite field. For field size two and three, binary and ternary locally repairable codes attaining the Singleton-like bound are already characterized in [25]- [28]. In this paper, we classify optimal LRCs over GF (4) attaining the Singleton-like bound for LRCs that can locally correct multiple node erasures.…”

Section: Introductionmentioning

confidence: 99%

Optimal Quaternary (r,delta)-Locally Repairable Codes Achieving the Singleton-type Bound

Shum¹,

Huang²

2021

Preprint

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Locally repairable codes enables fast repair of node failure in a distributed storage system. The code symbols in a codeword are stored in different storage nodes, such that a disk failure can be recovered by accessing a small fraction of the storage nodes. The number of storage nodes that are contacted during the repair of a failed node is a parameter called locality. We consider locally repairable codes that can be locally recovered in the presence of multiple node failures. The punctured code obtained by removing the code symbols in the complement of a repair group is called a local code. We aim at designing a code such that all local codes have a prescribed minimum distance, so that any node failure can be repaired locally, provided that the total number of node failures is less than the tolerance parameter.We consider linear locally repairable codes defined over a finite field of size four. This alphabet has characteristic 2, and hence is amenable to practical implementation. We classify all quaternary locally repairable codes that attain the Singleton-type upper bound for minimum distance. For each combination of achievable code parameters, an explicit code construction is given.

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Bounds and Constructions of Locally Repairable Codes: Parity-Check Matrix Approach

Cited by 42 publications

References 36 publications

Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and $K-1$

Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and $K-1$

A Class of $(n, k, r, t)$ IS-LRCs Via Parity Check Matrix

Optimal Quaternary (r,delta)-Locally Repairable Codes Achieving the Singleton-type Bound

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