Bounds on the Size of Locally Recoverable Codes

Cadambe, Viveck R.; Mazumdar, Arya

doi:10.1109/tit.2015.2477406

Cited by 158 publications

(158 citation statements)

References 26 publications

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“…As a corollary of Theorem 1, we can force the parameter λ to only be a multiple of κ. This gives a bound resembling the original bound in [24].…”

Section: Mathematical Preliminaries and Locality Revisitedsupporting

confidence: 64%

“…To remedy this, we proposed a new definition of locality called dimension-locality that restricts the dimension of the local codes instead of the size of the repair sets. We derived a new alphabet-dependent bound for LRCs of the same form as the bound in [24], which intrinsically uses the Griesmer bound. We showed that the bound is at least as good as the extension of the bound [24] for LRCs with locality (r, δ) and the Singleton-type bound.…”

Section: Discussionmentioning

confidence: 99%

“…We derived a new alphabet-dependent bound for LRCs of the same form as the bound in [24], which intrinsically uses the Griesmer bound. We showed that the bound is at least as good as the extension of the bound [24] for LRCs with locality (r, δ) and the Singleton-type bound. We derived an asymptotic upper bound on the rate-distance tradeoff of LRCs and proved that the bound is the tightest known upper bound for large relative minimum distances.…”

Section: Discussionmentioning

confidence: 99%

“…To do this, we distinguish two cases. If there exists a repair set R such that H(R) ≥ b, then we can Finally, now that we have the existence of this set, by the same argument as in [24] with Lemma 1, we can approximate the entropy of the shortened part by k opt with n minus the bound on the size and the minimum distance d. Then, the best upper bound on k is the minimum bound obtained over all λ.…”

Section: Acknowledgmentmentioning

confidence: 99%

“…The pioneering work done in [24] improves on the bound (1) by including a dependence on the alphabet size in the bound, that is, for any (n, k, d, r)-LRC over the alphabet Q with |Q| = q, we have k ≤ min 1≤t≤n/(r+1)…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Grezet

Freij-Hollanti

Westerbäck

et al. 2019

IEEE Trans. Inform. Theory

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Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality (r, δ), implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.

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“…As a corollary of Theorem 1, we can force the parameter λ to only be a multiple of κ. This gives a bound resembling the original bound in [24].…”

Section: Mathematical Preliminaries and Locality Revisitedsupporting

confidence: 64%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Acknowledgmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Grezet

Freij-Hollanti

Westerbäck

et al. 2019

IEEE Trans. Inform. Theory

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Locally Recoverable Codes from Algebraic Curves and Surfaces

Barg

Haymaker

Howe

et al. 2017

Association for Women in Mathematics Series

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Constructions and Bounds for Batch Codes with Small Parameters

Thomas

Skachek

2017

Coding Theory and Applications

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Linear batch codes and codes for private information retrieval (PIR) with a query size t and a restricted size r of the reconstruction sets are studied. New bounds on the parameters of such codes are derived for small values of t or r by providing corresponding constructions. By building on the ideas of Cadambe and Mazumdar, a new bound in a recursive form is derived for batch codes and PIR codes.

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Bounds on the Size of Locally Recoverable Codes

Cited by 158 publications

References 26 publications

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Locally Recoverable Codes from Algebraic Curves and Surfaces

Constructions and Bounds for Batch Codes with Small Parameters

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