Cyclic LRC codes with hierarchy and availability

Chen, Zitan; Barg, Alexander

doi:10.1109/isit44484.2020.9174243

Cited by 7 publications

(6 citation statements)

References 19 publications

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“…After focusing on shortened and punctured codes, let us look again at the entire code. If we permute the columns of the parity-check matrix of C using ℓ, we arrive at the following parity-check matrix for C ℓ , to (14),…”

Section: B Turning Non-cyclic Codes Into Cyclic Codesmentioning

confidence: 99%

“…For example, LRCs with (r, δ)locality [31] allow an erased symbol to be recovered by reading r other symbols, even if the repair set suffered δ − 1 more erasures. Other examples include: locality which guarantees disjoint multiple repairable sets (also named as availability) [6], [9], [35], [40], locality which has a hierarchical structure [14], [36], and unequal localities [22], [26], [44].…”

Section: Introductionmentioning

confidence: 99%

“…Among the known optimal LRCs, some of them also achieve order-optimal field size [2], [12], [25], [43] when δ = 2, and [10] when δ 2. Otherwise, constructions of optimal cyclic LRCs were introduced in [13], [14], [32], [33], [39]. When considering pre-determined recoverable erasure patterns beyond the minimum Hamming distance, codes that can recover from all information-theoretically recoverable erasure patterns are called maximally recoverable (MR) codes [17], also known as partial MDS codes [3].…”

Section: Introductionmentioning

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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Section: B Turning Non-cyclic Codes Into Cyclic Codesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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 now present the natural extension of H-LRCs to multiple levels of hierarchy. This generalization was mentioned in [15] and precisely defined in [5,10]. Adapted to the notation above, we have the following definition.…”

Section: Codes With H-level Hierarchy From Reed-muller and Generalize...mentioning

confidence: 99%

Locally recoverable codes with availability <i>t</i>≥2 from fiber products of curves

Haymaker¹,

Malmskog²,

Matthews³

2018

Advances in Mathematics of Communications

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We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vlȃduţ [4] to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets, including codes with two recovery sets from the generalized Giulietti and Korchmáros (GK) curves and the Suzuki curves, and new locally recoverable codes with many recovery sets based on the Hermitian curve, using a fiber product construction of van der Geer and van der Vlugt. In addition, we consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures.

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“…To allow different requirements for local recovery, hierarchical and unequal locality were introduced in [34] and [23], [42], respectively. Over the past decade, many bounds and constructions for LRCs have been introduced, e.g., [3], [6], [7], [9], [17], [20], [25], [32], [37], [40], [41] for (r, δ)-locality [4], [5], [22], [33], [38] for multiple repair sets, [10], [27], [34], [43] for hierarchical locality, and [23], [42] for unequal locality.…”

Section: Introductionmentioning

confidence: 99%