Binary Locally Repairable Codes - Sequential Repair for Multiple Erasures

Song, Wentu; Yuen, Chau

doi:10.1109/glocom.2016.7841631

Cited by 19 publications

(43 citation statements)

References 20 publications

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“…There are also several optimal code constructions in the literature [4], [5], [7], [13]- [15], [19]- [21] that achieve the equality in (2). The rate upper bound for (r, δ)-LRCs has been shown in [15], [22] to be k n ≤ r r + δ + 1 , which can also be expressed as an upper bound on the dimension as k ≤ n r + δ + 1 · r.…”

Section: (R δ)-Localitymentioning

confidence: 99%

Locally Repairable Codes With Unequal Local Erasure Correction

Kim

Lee

2018

IEEE Trans. Inform. Theory

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When a node in a distributed storage system fails, it needs to be promptly repaired to maintain system integrity.While typical erasure codes can provide a significant storage advantage over replication, they suffer from poor repair efficiency. Locally repairable codes (LRCs) tackle this issue by reducing the number of nodes participating in the repair process (locality), at the cost of reduced minimum distance. In this paper, we study the tradeoff between locality and minimum distance of LRCs with local codes that have arbitrary distance requirements. Unlike existing methods where both the locality and the local distance requirements imposed on every node are identical, we allow the requirements to vary arbitrarily from node to node. Such a property can be an advantage for distributed storage systems with non-homogeneous characteristics. We present Singleton-type distance upper bounds and also provide an optimal code construction with respect to these bounds. In addition, the feasible rate region is characterized by dimension upper bounds that do not depend on the distance. DRAFT 2 of nodes that are accessed during repair, for given code parameters such as length, dimension, and minimum distance. The tradeoff between locality and other parameters has been studied extensively since the discovery of the Singleton-type bound in [3].A natural extension to the conventional locality is the (r, δ)-locality [4], [5], where more flexible repair options are provided by generalizing the constraint on the minimum distance of local codes to at least δ instead of 2 (single parity checks). Such flexibility is beneficial to modern large-scale DSSs where multiple node failures have become more common. For example, in conventional optimal LRCs [3], [6], [7] with locality r, if another node included in the local repair group of a failed node simultaneously fails, repair from r nodes is no longer valid, and a large number of nodes have to be accessed to perform ordinary erasure correction. On the other hand, (r, δ)-LRCs can still perform repair from r nodes even if δ − 1 nodes in a local repair group simultaneously fail.Recently, there has been interest in the case where locality is specified differently for different nodes [8]- [11].Such situations may occur, for example, when the underlying storage network is not homogeneous. It would also be beneficial in the scenarios where hot data symbols require faster repair or reduced download latency [8]. In [8],[10], relevant Singleton-type bounds have been found and some optimal code constructions are also given, which show the tightness of the bounds. B. Contributions and OrganizationIn this paper, we study the tradeoff between locality and minimum distance for (r, δ)-LRCs, where the locality parameter r and the local distance parameter δ are not necessarily the same for each node. Our main contribution is different from previous work on unequal locality [8], [10] in two ways. First, we extend the results on conventional r-locality to (r, δ)-locality. Specifically, our new Singleton-typ...

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Section: (R δ)-Localitymentioning

confidence: 99%

Locally Repairable Codes With Unequal Local Erasure Correction

Kim

Lee

2018

IEEE Trans. Inform. Theory

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“…The sequential approach to recovery from multiple erasures was introduced by Prakash et al [12] and has been further investigated in [31], [32], [33], [34], [35], [30], [1], [2] as discussed below. a) Two Erasures: Seq-LRC with t = 2 are considered in [12] (see also [32]) where a tight upper bound on the rate and a matching construction achieving the upper bound on rate for t = 2 is provided. A lower bound on block length and a construction achieving the lower bound on block length for t = 2 is provided in [32].…”

Section: Background On Seq-lrcmentioning

confidence: 99%

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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“…Rate bounds on LRCs from any of these classes yield upper bounds on the rate of an LRCs with (r,t)-availability. A discussion on the hierarchy of these classes can be found in [36]. a) LRCs with strong local codes, wherein every symbol is protected by an (r + t, r,t + 1) local code, were considered in [20,17,15,37].…”

Section: Relationship To Previous Workmentioning

confidence: 99%

“…Our result shows the uniqueness of such a construction for rate optimal codes with exact covering. For the sequential recovery of t = 3 erasures with locality r under functional repair model, [36] presents a lower bound on the length of a code. Under suitable divisibility assumptions, this bound for r = 2 translates to the rate bound of 4/9.…”

Section: Relationship To Previous Workmentioning

confidence: 99%

Rate optimal binary linear locally repairable codes with small availability

Kadhe

Calderbank

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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A locally repairable code with availability has the property that every code symbol can be recovered from multiple, disjoint subsets of other symbols of small size. In particular, a code symbol is said to have (r,t)-availability if it can be recovered from t disjoint subsets, each of size at most r. A code with availability is said to be rate-optimal, if its rate is maximum among the class of codes with given locality, availability, and alphabet size.This paper focuses on rate-optimal binary, linear codes with small availability, and makes four contributions. First, it establishes tight upper bounds on the rate of binary linear codes with (r, 2) and (2, 3) availability. Second, it establishes a uniqueness result for binary rate-optimal codes, showing that for certain classes of binary linear codes with (r, 2) and (2, 3)-availability, any rate optimal code must be a direct sum of shorter rate optimal codes. Third, it presents novel upper bounds on the rates of binary linear codes with (2,t) and (r, 3)-availability. In particular, the main contribution here is a new method for bounding the number of cosets of the dual of a code with availability, using its covering properties. Finally, it presents a class of locally repairable linear codes associated with convex polyhedra, focusing on the codes associated with the Platonic solids. It demonstrates that these codes are locally repairable with t = 2, and that the codes associated with (geometric) dual polyhedra are (coding theoretic) duals of each other.

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Binary Locally Repairable Codes - Sequential Repair for Multiple Erasures

Cited by 19 publications

References 20 publications

Locally Repairable Codes With Unequal Local Erasure Correction

Locally Repairable Codes With Unequal Local Erasure Correction

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

Rate optimal binary linear locally repairable codes with small availability

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