New Constructions of Binary and Ternary Locally Repairable Codes Using Cyclic Codes

Kim, Chanki; No, Jong-Seon

doi:10.1109/lcomm.2017.2776141

Cited by 25 publications

(12 citation statements)

References 10 publications

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“…A BLRC that can satisfy the explicit bound given in Equation (4) is also proposed in [60] as follows:…”

Section: Construction (Cc8) [54]mentioning

confidence: 99%

“…Construction (CC9) [60]: For (r + 1)|n, let v = n r+1 and u = r + 1, where gcd(u, v) = 1 and u, v ≥ 2. Let g(x) be a generator polynomial of the cyclic BLRC and β be the uth root of unity.…”

Section: Construction (Cc8) [54]mentioning

confidence: 99%

“…Another example of BLRC from the code modification methods is presented in [60] using the shortened expurgated Hamming code.…”

Section: Blrcs From Code Modificationmentioning

confidence: 99%

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Overview of Binary Locally Repairable Codes for Distributed Storage Systems

Kim

2019

Electronics

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This paper summarizes the details of recently proposed binary locally repairable codes (BLRCs) and their features. The construction of codes over a small alphabet size of symbols is of particular interest for efficient hardware implementation. Therefore, BLRCs are highly noteworthy because no multiplication is required during the encoding, decoding, and repair processes. We explain the various construction approaches of BLRCs such as cyclic code based, bipartite graph based, anticode based, partial spread based, and generalized Hamming code based techniques. We also describe code generation methods based on modifications for linear codes such as extending, shorting, expurgating, and augmenting. Finally, we summarize and compare the parameters of the discussed constructions.

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“…A BLRC that can satisfy the explicit bound given in Equation (4) is also proposed in [60] as follows:…”

Section: Construction (Cc8) [54]mentioning

confidence: 99%

Section: Construction (Cc8) [54]mentioning

confidence: 99%

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Overview of Binary Locally Repairable Codes for Distributed Storage Systems

Kim

2019

Electronics

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“…In the last decade, many constructions of optimal LRCs have been proposed, for example see [2][3][4][5][6][7][8][9][10][11][12][14][15][16][17][18][19][20][21][22] and references therein. A breakthrough construction of optimal LRCs was proposed by Tamo and Barg in [18] via a generalization of the classical construction of Reed-Solomon codes.…”

Section: Introductionmentioning

confidence: 99%

“…Below we give an example to illustrate Corollary 4.Let n = 18 and α be an 18-th primitive root of unity in F 19 . Let A = {α, α 2 , α 3 , α 4 , α 5 , α 9 }, and this is the case of t = 1, b = 1, m = 5 and ℓ = 8 in Corollary 4.4.Magma verifies that C ⊥A has the parameters[18,6,10], where C ⊥ A is the dual of the cyclic code with the complete defining set A.Let d ⊥ A denote the minimum distance of C ⊥ A . It is clear that δ = 2, 3 or 4 satisfies 0 ≤ δ − 2 < n − m − d ⊥ A .…”

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confidence: 99%

New Constructions of Optimal Cyclic (r,δ) Locally Repairable Codes from Their Zeros

Qiu¹,

Zheng²,

Fu³

2020

Preprint

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An (r, δ)-locally repairable code ((r, δ)-LRC for short) was introduced by Prakash et al. [14] for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of r-LRCs produced by Gopalan et al. [5]. An (r, δ)-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. [2] generalized the construction of cyclic r-LRCs proposed by Tamo et al. [19,20] and constructed several classes of optimal (r, δ)-LRCs of length n for n | (q − 1) or n | (q + 1), respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of [2,3], this paper first characterizes (r, δ)-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic (r, δ)-LRCs of length n for n | (q − 1) or n | (q + 1), respectively from the product of two sets of zeros. Our constructions include all optimal cyclic (r, δ)-LRCs proposed in [2, 3], and our method seems more convenient to obtain optimal cyclic (r, δ)-LRCs with flexible parameters. Moreover, many optimal cyclic (r, δ)-LRCs of length n for n | (q − 1) or n | (q + 1), respectively such that (r + δ − 1) ∤ n can be obtained from our method.

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