On Optimal Locally Repairable Codes and Generalized Sector-Disk Codes

Cai, Han; Schwartz, Moshe

doi:10.1109/isit44484.2020.9174051

Cited by 11 publications

(27 citation statements)

References 33 publications

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“…Thus, we have τ = τ ′ . Finally, for Case 3, by (11) we have {τ } ⊆ {τ ′ , 2τ ′ mod a}. Obviously τ ≡ 2τ ′ (mod a) is impossible since 2|a and a 3.…”

Section: B Turning Non-cyclic Codes Into Cyclic Codesmentioning

confidence: 94%

“…Thus, an (n, r, h, δ, q)-MR code can recover from δ − 1 sector erasures in each stripe, and additional h erased sectors anywhere. We mention in passing that a more restricted type of codes, called sector-disk (SD) codes, are capable of recovering from δ − 1 disk erasures, and additional h erased sectors (see [11], [30]).…”

Section: B Maximally Recoverable Codesmentioning

confidence: 99%

“…Thus, we have ℓ = 1 and τ = τ ′ . For Case 2, by (11), we have {τ, 2τ mod a} ⊆ {τ ′ , 2τ ′ mod a, 3τ ′ mod a} .…”

Section: B Turning Non-cyclic Codes Into Cyclic Codesmentioning

confidence: 99%

See 2 more Smart Citations

A Bound on the Minimal Field Size of LRCs, and Cyclic MR Codes That Attain It

Han¹,

Schwartz²

2022

Preprint

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We prove a new lower bound on the field size of locally repairable codes (LRCs). Additionally, we construct maximally recoverable (MR) codes which are cyclic. While a known construction for MR codes has the same parameters, it produces noncyclic codes. Furthermore, we prove necessary and sufficient conditions that specify when the known non-cyclic MR codes may be permuted to become cyclic, thus proving our construction produces cyclic MR codes with new parameters. Furthermore, using our new bound on the field size, we show that the new cyclic MR codes have optimal field size in certain cases. Other known LRCs are also shown to have optimal field size in certain cases.

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“…Thus, we have τ = τ ′ . Finally, for Case 3, by (11) we have {τ } ⊆ {τ ′ , 2τ ′ mod a}. Obviously τ ≡ 2τ ′ (mod a) is impossible since 2|a and a 3.…”

Section: B Turning Non-cyclic Codes Into Cyclic Codesmentioning

confidence: 94%

Section: B Maximally Recoverable Codesmentioning

confidence: 99%

See 1 more Smart Citation

A Bound on the Minimal Field Size of LRCs, and Cyclic MR Codes That Attain It

Han¹,

Schwartz²

2022

Preprint

Self Cite

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“…When n = 6ℓ, we can take the tensor product I ℓ ⊗ D as the parity-check matrix. When n = 6ℓ − 1 or 6ℓ − 2, the structure of the parity-check matrix is similar to the matrices in ( 16) and (18).…”

Section: Optimal Quaternary (1 δ)-Lrcmentioning

confidence: 96%

“…There are a number of constructions of optimal code in this multiple-failure setting. In the literature, we can find constructions using cyclic or constacyclic codes [13]- [16], generalized RS codes [17], and Steiner system [18], and finite geometry [19], etc. Another source of code constructions is algebraic geometry.…”

Section: Introductionmentioning

confidence: 99%

Optimal Quaternary (r,delta)-Locally Repairable Codes Achieving the Singleton-type Bound

Shum¹,

Huang²

2021

Preprint

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Locally repairable codes enables fast repair of node failure in a distributed storage system. The code symbols in a codeword are stored in different storage nodes, such that a disk failure can be recovered by accessing a small fraction of the storage nodes. The number of storage nodes that are contacted during the repair of a failed node is a parameter called locality. We consider locally repairable codes that can be locally recovered in the presence of multiple node failures. The punctured code obtained by removing the code symbols in the complement of a repair group is called a local code. We aim at designing a code such that all local codes have a prescribed minimum distance, so that any node failure can be repaired locally, provided that the total number of node failures is less than the tolerance parameter.We consider linear locally repairable codes defined over a finite field of size four. This alphabet has characteristic 2, and hence is amenable to practical implementation. We classify all quaternary locally repairable codes that attain the Singleton-type upper bound for minimum distance. For each combination of achievable code parameters, an explicit code construction is given.

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A Family of Codes With Locality Containing Optimal Codes

et al. 2022

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Locally recoverable codes were introduced by Gopalan et al. in 2012, and in the same year Prakash et al. introduced the concept of codes with locality, which are a type of locally recoverable codes.In this work we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that elements of this family are "optimal codes", as defined by Prakash et al. INDEX TERMSLocally recoverable codes, affine cartesian codes.

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On Optimal Locally Repairable Codes and Generalized Sector-Disk Codes

Cited by 11 publications

References 33 publications

A Bound on the Minimal Field Size of LRCs, and Cyclic MR Codes That Attain It

A Bound on the Minimal Field Size of LRCs, and Cyclic MR Codes That Attain It

Optimal Quaternary (r,delta)-Locally Repairable Codes Achieving the Singleton-type Bound

A Family of Codes With Locality Containing Optimal Codes

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