Bounds on the maximal minimum distance of linear locally repairable codes

Pöllänen, Antti; Westerbäck, Thomas; Freij-Hollanti, Ragnar; Hollanti, Camilla

doi:10.1109/isit.2016.7541566

Cited by 3 publications

(3 citation statements)

References 21 publications

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“…where (a) is due to the fact thatq j < r j + δ j − 1, and (b) follows from (20) and (22) (p j − p j +φ j ).…”

Section: Upper Bounds Based On Unequal Localitymentioning

confidence: 99%

“…where (a) is obtained by removing some non-negative subtrahends, and (b) comes from (20) and (22) with a derivation similar to Lemma 7.…”

Section: Upper Bounds Based On Unequal Localitymentioning

confidence: 99%

“…− p j + φj ),where (a) is due to the fact that qj < r j + δ j − 1, and (b) follows from(20) and(22) with a derivation similar to Lemma 7. As an intermediate result, we get ⌊A⌋ ≥ jL j=1 (p j − p j + φj ).…”

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confidence: 99%

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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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“…where (a) is due to the fact thatq j < r j + δ j − 1, and (b) follows from (20) and (22) (p j − p j +φ j ).…”

Section: Upper Bounds Based On Unequal Localitymentioning

confidence: 99%

“…where (a) is obtained by removing some non-negative subtrahends, and (b) comes from (20) and (22) with a derivation similar to Lemma 7.…”

Section: Upper Bounds Based On Unequal Localitymentioning

confidence: 99%