Codes on Curves with Hierarchical Locality

Ballentine, Sean; Barg, Alexander

doi:10.1109/isit.2018.8437791

Cited by 5 publications

(4 citation statements)

References 16 publications

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“…(this is similar to deg ψ pxq defined before Proposition IV.1). Therefore, h 1 ii " s i , i " 1, 2 and, if the fibers of these maps are transversal, then also h 1 12 " h 1 21 " 1. Let V be the space of functions given by V " spantf i x j1 1 x j2 2 y k1 1 y k2 2 | i " 1, .…”

Section: H-lrc Codes With Availabilitymentioning

confidence: 97%

“…The repair groups of this LRC code are of size r 2 `1. Assume that there is a polynomial y P rxs of degree r 2 `1 that is constant on these repair groups 1 . Further, we choose a polynomial f P rxs of degree ν that is constant on each of the subsets D α .…”

Section: A a Family Of Optimal Rs-like H-lrc Codesmentioning

confidence: 99%

“…A preliminary version of this work was presented at the 2018 IEEE International Symposium on Information Theory [1]. At the same time, most of the material in Sections VI-VIII was not included in [1] and appears here for the first time.…”

Section: Introductionmentioning

confidence: 99%

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Codes With Hierarchical Locality From Covering Maps of Curves

Ballentine¹,

Barg²,

Vlăduţ³

2019

IEEE Trans. Inform. Theory

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bstract Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes with 2-level hierarchical locality from maps between algebraic curves and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin-Schreier curves. We further address the question of H-LRC codes with availability, and suggest a general construction of such codes from fiber products of curves. Detailed calculations of parameters for H-LRC codes with availability are performed for Reed-Solomon-and Hermitian-like code families. Finally, we construct asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.:

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Section: H-lrc Codes With Availabilitymentioning

confidence: 97%

Section: A a Family Of Optimal Rs-like H-lrc Codesmentioning

confidence: 99%

See 1 more Smart Citation

Codes With Hierarchical Locality From Covering Maps of Curves

Ballentine¹,

Barg²,

Vlăduţ³

2019

IEEE Trans. Inform. Theory

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“…Optimal constructions are provided in [158], [159]. We note that in the context of a practical distributed-storage system, the authors in [160] had previously suggested the topology of hierarchical codes and compared hierarchical codes with Reed-Solomon codes in terms of repair-efficiency using real data.…”

Section: E Codes With Hierarchical Localitymentioning

confidence: 99%

Erasure coding for distributed storage: an overview

Balaji

Krishnan

Vajha

et al. 2018

Sci. China Inf. Sci.

131

102

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In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted. Coding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade. I. INTRODUCTIONThis survey article deals with the use of erasure coding for the reliable and efficient storage of large amounts of data in settings such as that of a data center. The amount of data stored in a single data center can run into tens or hundreds of petabytes. Reliability of data storage is ensured in part by introducing redundancy in some form, ranging from simple replication to the use of more sophisticated erasure-coding schemes such as Reed-Solomon codes. Minimizing the storage overhead that comes with ensuring reliability is a key consideration in the choice of erasure-coding scheme. More recently a second problem has surfaced, namely, that of node repair.In [1], [2] the authors study the Facebook warehouse cluster and analyze the frequency of node failures as well as the resultant network traffic relating to node repair. It was observed in [1] that a median of 50 nodes are unavailable per day and that a median of 180TB of cross-rack traffic is generated as a result of node unavailability. It was also reported that 98.08% of the cases have exactly one block missing in a stripe. The erasure code that was deployed in this instance was an [n = 14, k = 10] Reed Solomon (RS) code. Here n denotes the block length of the code and k the dimension. The conventional repair of an [n, k] RS code is inefficient in that the repair of a single node, calls for contacting k other (helper) nodes and downloading k times the amount of data stored in the failed node, which is clearly inefficient. Thus there is significant practical interest in the design of erasure-coding techniques that offer both low overhead and which can also be repaired efficiently.Coding theorists have responded to this need by coming up with two new classes of codes, namely ReGenerating (RG) and Locally Recoverable (LR) codes. The focus in a RG code is on minimizing the amount of data download needed to repair a failed node, termed the repair bandwidth while LR codes seek to minimize the number of helper nodes contacted for node repair, termed the repair degree. In a different direction, coding theorists have also re-examined the problem of node repair in RS codes and have come up with new and more efficient ...

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