Constructions of optimal and almost optimal locally repairable codes

Ernvall, Toni; Westerbäck, Thomas; Hollanti, Camilla

doi:10.1109/vitae.2014.6934442

Cited by 14 publications

(12 citation statements)

References 13 publications

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“…In the last section, we saw that the p-c matrix of a rate-optimal seq-LRC can be assumed without loss of generality, to have the staircase form appearing in equations (11) and (12). It will be shown in the present section, just as was done in the case of the examples presented in Section III, that this form of p-c matrix leads to a graphical representation of the code.…”

Section: A Graphical Representation For the Rate-optimal Seq-lrcmentioning

confidence: 92%

“…Also contained in [10] is a construction of codes with all-symbol locality having field size of O(n) whose minimum distance differs by at most 1 from the bound given in (1) when r k, n = 1 (mod r + 1). In [11], the authors show that codes with all-symbol locality whose minimum distance differs by atmost 1 from the bound given in (1) can be constructed for any n, k, r. However the construction provided in [11] has field size exponential in n.…”

Section: A Background On Single-erasure Lrcmentioning

confidence: 99%

“…In the case t even, we recall from equation (11), that the p-c matrix of a rate-optimal code can be put into the form:…”

Section: A T Even Casementioning

confidence: 99%

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A Tight Rate Bound and Matching Construction for Locally Recoverable Codes With Sequential Recovery From Any Number of Multiple Erasures

Balaji¹,

Kini

Kumar

2020

IEEE Trans. Inform. Theory

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An [n, k] code C is said to be locally recoverable in the presence of a single erasure, and with locality parameter r, if each of the n code symbols of C can be recovered by accessing at most r other code symbols. An [n, k] code is said to be a locally recoverable code with sequential recovery from t erasures, if for any set of s ≤ t erasures, there is an s-step sequential recovery process, in which at each step, a single erased symbol is recovered by accessing at most r other code symbols. This is equivalent to the requirement that for any set of s ≤ t erasures, the dual code contain a codeword whose support contains the co-ordinate of precisely one of the s erased symbols.In this paper, a tight upper bound on the rate of such a code, for any value of number of erasures t and any value r ≥ 3, of the locality parameter is derived. This bound proves an earlier conjecture due to Song, Cai and Yuen. While the bound is valid irrespective of the field over which the code is defined, a matching construction of binary codes that are rate-optimal is also provided, again for any value of t and any value r ≥ 3.

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Section: A Graphical Representation For the Rate-optimal Seq-lrcmentioning

confidence: 92%

Section: A Background On Single-erasure Lrcmentioning

confidence: 99%

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A Tight Rate Bound and Matching Construction for Locally Recoverable Codes With Sequential Recovery From Any Number of Multiple Erasures

Balaji¹,

Kini

Kumar

2020

IEEE Trans. Inform. Theory

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“…5) It is shown in [103] that one can construct codes with AS locality and field size of order n whose minimum distance is within 1 of the bound in (18) provided r k, n = 1 (mod r + 1). In [104], it is shown that this can be achieved for any parameter set if one permits the field size to be exponential in n.…”

Section: ) the Pyramid-code Construction In Vii-b1 Provides A Generamentioning

confidence: 99%

Erasure coding for distributed storage: an overview

Balaji

Krishnan

Vajha

et al. 2018

Sci. China Inf. Sci.

127

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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“…In this last construction the field size has to be at least the length of the code. In addition to the constructions mentioned above there are several other constructions of LRCs, see e.g., [4], [7]- [9], [12], [19].…”

Section: Introductionmentioning

confidence: 99%

Cyclic linear binary locally repairable codes

Huang

Yaakobi

Uchikawa

et al. 2015

2015 IEEE Information Theory Workshop (ITW)

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Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size q, i.e., the size of the code alphabet. In particular, for the binary case, only a few specific results are known by Goparaju and Calderbank. Recently, however, an upper bound on the dimension k of LRCs was presented by Cadambe and Mazumdar. The bound takes into account the length n, minimum distance d, locality r, and field size q, and it is applicable to both non-linear and linear codes.In this work, we first develop an improved version of the bound mentioned above for linear codes. We then focus on cyclic linear binary codes. By leveraging the cyclic structure, we notice that the locality of such a code is determined by the minimum distance of its dual code. Using this result, we investigate the locality of a variety of well known cyclic linear binary codes, e.g., Hamming codes and Simplex codes, and also prove their optimality with our improved bound for linear codes. We also discuss the locality of codes which are obtained by applying the operations of Extend, Shorten, Expurgate, Augment, and Lengthen to cyclic linear binary codes. Several families of such modified codes are considered and their optimality is addressed. Finally, we investigate the locality of Reed-Muller codes. Even though they are not cyclic, it is shown that some of the locality results for cyclic codes still apply.

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Constructions of optimal and almost optimal locally repairable codes

Cited by 14 publications

References 13 publications

A Tight Rate Bound and Matching Construction for Locally Recoverable Codes With Sequential Recovery From Any Number of Multiple Erasures

A Tight Rate Bound and Matching Construction for Locally Recoverable Codes With Sequential Recovery From Any Number of Multiple Erasures

Erasure coding for distributed storage: an overview

Cyclic linear binary locally repairable codes

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