On the locality of codeword symbols in non-linear codes

Forbes, Michael A.; Yekhanin, Sergey

doi:10.1016/j.disc.2014.01.016

Cited by 62 publications

(51 citation statements)

References 4 publications

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“…We will show that ∂(x) is the same polynomial as δ(x) defined in (7). Each f i (x) is a linear combination of powers of g, therefore it is also constant on the set A j , i.e., for any β ∈ A j and any coefficient polynomial…”

Section: Recovery Of the Erased Symbolmentioning

confidence: 99%

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A family of optimal locally recoverable codes

Tamo

Barg

2014

2014 IEEE International Symposium on Information Theory

194

554

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Abstract-A code over a finite alphabet is called locally recoverable (LRC) if every symbol in the encoding is a function of a small number (at most r) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to a Reed-Solomon code if the locality parameter r is set to be equal to the code dimension. The size of the code alphabet for most parameters is only slightly greater than the code length. The recovery procedure is performed by polynomial interpolation over r points. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data ("hot data").

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Section: Recovery Of the Erased Symbolmentioning

confidence: 99%

“…Codes with information locality property were also studied in [9], [7]. A natural question to ask is as follows: given an (n, k, r) LRC code C, what is the best possible minimum distance d(C)?…”

Section: Introductionmentioning

confidence: 99%

A family of optimal locally recoverable codes

Tamo

Barg

2014

2014 IEEE International Symposium on Information Theory

194

554

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“…Due to applications to distributed storage systems, locally repairable codes have recently attracted great attention of researchers [5,4,12,13,7,8,3,11,14,15,1]. A local repairable code is nothing but a block code with an additional parameter called locality.…”

Section: Introductionmentioning

confidence: 99%

Optimal Locally Repairable Codes of Distance 3 and 4 via Cyclic Codes

Luo

Xing

Yuan

2019

IEEE Trans. Inform. Theory

107

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Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work of [14], several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in [14] are upper bounded by the code alphabet size q.Recently, it was proved through extension of construction in [14] that length of q-ary optimal locally repairable codes can be q + 1 in [7]. Surprisingly, [2] presented a few examples of q-ary optimal locally repairable codes of small distance and locality with code length achieving roughly q 2 . Very recently, it was further shown in [8] that there exist q-ary optimal locally repairable codes with length bigger than q + 1 and distance propositional to n. Thus, it becomes an interesting and challenging problem to construct new families of q-ary optimal locally repairable codes of length bigger than q + 1.In this paper, we construct a class of optimal locally repairable codes of distance 3 and 4 with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen.

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“…Similarly when n mod (r + 1) ≥ k + k r mod (r + 1), a tighter December 7, 2018 DRAFT upper bound on minimum distance of LRC and LRC achieving the upper bound can be found in [15]. In [16] and [17], the authors study codes with locality in the setting of nonlinear codes. In [17], authors show that same upper bound as in (1) continues to hold for non-linear codes.…”

Section: A Background On Single-erasure 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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On the locality of codeword symbols in non-linear codes

Cited by 62 publications

References 4 publications

A family of optimal locally recoverable codes

A family of optimal locally recoverable codes

Optimal Locally Repairable Codes of Distance 3 and 4 via Cyclic Codes

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

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