Some results on optimal locally repairable codes

Hao, Jie; Xia, Shu-Tao; Chen, Bin

doi:10.1109/isit.2016.7541337

Cited by 59 publications

(53 citation statements)

References 11 publications

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“…For δ 1 = δ 2 = 2, the condition r 1 ⌈(∆−1)/(δ 1 − 1)⌉ + (∆ − 1) < n 1 is naturally satisfied, and the bound (6) reduces to the bound (2) in [13]. For δ 1 > 2, the condition r 1 ⌈(∆ − 1)/(δ 1 − 1)⌉ + (∆ − 1) < n 1 , i.e., r 1 + δ 1 − 1 | n 1 .…”

Section: Remarkmentioning

confidence: 99%

Locally repairable codes with multiple (r<inf>i</inf>, δ<inf>i</inf>)-localities

Chen

Xia

Hao

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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In distributed storage systems, locally repairable codes (LRCs) are introduced to realize low disk I/O and repair cost. In order to tolerate multiple node failures, the LRCs with (r, δ)-locality are further proposed. Since hot data is not uncommon in a distributed storage system, both Zeh et al. and Kadhe et al. focus on the LRCs with multiple localities or unequal localities (ML-LRCs) recently, which said that the localities among the code symbols can be different. ML-LRCs are attractive and useful in reducing repair cost for hot data. In this paper, we generalize the ML-LRCs to the (r, δ)-locality case of multiple node failures, and define an LRC with multiple (ri, δi) i∈[s] localities (s ≥ 2), where r1 ≤ r2 ≤ · · · ≤ rs and δ1 ≥ δ2 ≥ · · · ≥ δs ≥ 2. Such codes ensure that some hot data could be repaired more quickly and have better failure-tolerance in certain cases because of relatively smaller ri and larger δi. Then, we derive a Singleton-like upper bound on the minimum distance for the proposed LRCs by employing the regeneratingset technique. Finally, we obtain a class of explicit and structured constructions of optimal ML-LRCs, and further extend them to the cases of multiple (ri, δ) i∈[s] localities.

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Section: Remarkmentioning

confidence: 99%

Locally repairable codes with multiple (r<inf>i</inf>, δ<inf>i</inf>)-localities

Chen

Xia

Hao

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…These constructions are for the case d min = 6. In [124], the authors present a characterization of binary LR codes that achieve the Singleton bound (18). In [125], the authors present constructions of binary codes meeting the Singleton bound.…”

Section: ) Construction Of Binary Codesmentioning

confidence: 99%

“…In [125], the authors present constructions of binary codes meeting the Singleton bound. These codes are a subclass of the codes characterized in [124] for the case d min ≤ 4.…”

Section: ) Construction Of Binary Codesmentioning

confidence: 99%

Erasure coding for distributed storage: an overview

Balaji

Krishnan

Vajha

et al. 2018

Sci. China Inf. Sci.

125

102

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In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted. Coding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade. I. INTRODUCTIONThis survey article deals with the use of erasure coding for the reliable and efficient storage of large amounts of data in settings such as that of a data center. The amount of data stored in a single data center can run into tens or hundreds of petabytes. Reliability of data storage is ensured in part by introducing redundancy in some form, ranging from simple replication to the use of more sophisticated erasure-coding schemes such as Reed-Solomon codes. Minimizing the storage overhead that comes with ensuring reliability is a key consideration in the choice of erasure-coding scheme. More recently a second problem has surfaced, namely, that of node repair.In [1], [2] the authors study the Facebook warehouse cluster and analyze the frequency of node failures as well as the resultant network traffic relating to node repair. It was observed in [1] that a median of 50 nodes are unavailable per day and that a median of 180TB of cross-rack traffic is generated as a result of node unavailability. It was also reported that 98.08% of the cases have exactly one block missing in a stripe. The erasure code that was deployed in this instance was an [n = 14, k = 10] Reed Solomon (RS) code. Here n denotes the block length of the code and k the dimension. The conventional repair of an [n, k] RS code is inefficient in that the repair of a single node, calls for contacting k other (helper) nodes and downloading k times the amount of data stored in the failed node, which is clearly inefficient. Thus there is significant practical interest in the design of erasure-coding techniques that offer both low overhead and which can also be repaired efficiently.Coding theorists have responded to this need by coming up with two new classes of codes, namely ReGenerating (RG) and Locally Recoverable (LR) codes. The focus in a RG code is on minimizing the amount of data download needed to repair a failed node, termed the repair bandwidth while LR codes seek to minimize the number of helper nodes contacted for node repair, termed the repair degree. In a different direction, coding theorists have also re-examined the problem of node repair in RS codes and have come up with new and more efficient ...

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“…Without loss of generality, we choose the first 4 subspaces of them. Thus we obtain an optimal [12,4,6; 2] 2 LRC with parity check matrix Now we give an almost complete construction for optimal binary LRCs with r = 3. We modify the parity check matrix H of r = 2 by adding a column and a row in each H (i) .…”

Section: So We Havementioning

confidence: 99%

“…In the binary case, the bound (1) is not tight in almost all cases. In fact, Hao et al [12] prove that there are only 4 classes of d-optimal binary LRCs meeting the Singleton-like bound in the sense of equivalence of linear codes. A new bound taking field size into consideration is derived in [4], which is called Cadambe-Mazumdar (C-M) bound,…”

Section: Introductionmentioning

confidence: 99%

Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups

Ma¹,

Ge²

2019

SIAM J. Discrete Math.

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In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However, most known constructions are over large fields with sizes close to the code length, which lead to the systems computationally expensive. Due to this, binary LRCs are of interest in practice.In this paper, we focus on binary linear LRCs with disjoint repair groups. We first derive an explicit bound for the dimension k of such codes, which can be served as a generalization of the bounds given in [11], [36], [37]. We also give several new constructions of binary LRCs with minimum distance d = 6 based on weakly independent sets and partial spreads, which are optimal with respect to our newly obtained bound. In particular, for locality r ∈ {2, 3} and minimum distance d = 6, we obtain the desired optimal binary linear LRCs with disjoint repair groups for almost all parameters.which is also called as Singleton-like bound since it degenerates to classical Singleton bound when r = k. Later, in [7], [19], the bound (1) is generalized to vector codes and nonlinear codes. Although it certainly holds for all LRCs, it is not tight in many cases. The tightness of the bound (1) is studied in [28], [34].We say an LRC is d-optimal if it satisfies bound (1) with equality for given n, k and r. For the case (r + 1)|n, d-optimal LRCs are constructed explicitly in [32] and [25] by using Reed-Solomon codes and Gabidulin codes respectively. However, both constructions are built over a finite field whose size is an exponential function of the code length n. In [30], for the same case (r + 1)|n, the authors construct a d-optimal code over a finite field of size sightly greater than n by using "good" polynomials. This construction can be extended to the case (r + 1) ∤ n with the minimum distance d ≥ n − k − ⌈ k r ⌉ + 1 which

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Some results on optimal locally repairable codes

Cited by 59 publications

References 11 publications

Locally repairable codes with multiple (r<inf>i</inf>, δ<inf>i</inf>)-localities

Locally repairable codes with multiple (r<inf>i</inf>, δ<inf>i</inf>)-localities

Erasure coding for distributed storage: an overview

Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups

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