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Locally Recoverable Codes with Availability t≥ 2 from Fiber Products of Curves [article]

Kathryn Haymaker, Beth Malmskog, Gretchen Matthews
2018 arXiv   pre-print
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 would like to acknowledge the hospitality of IPAM and the organizers of the 2016 Algebraic Geometry for Coding Theory and Cryptography Workshop: Everett Howe, Kristin Lauter, and Judy Walker  ... 
arXiv:1612.03841v4 fatcat:ansbd35nm5ah3lt24bieob3a7y

Optimal locally recoverable codes with hierarchy from nested F-adic expansions [article]

Austin Dukes and Giacomo Micheli and Vincenzo Pallozzi Lavorante
2022 arXiv   pre-print
In this paper we construct new optimal hierarchical locally recoverable codes.  ...  dimension of the final code), and to remove restrictions on the arithmetic properties of q compared with the size of the locality sets in the hierarchy.  ...  In this paper we construct new locally recoverable codes with hierarchy of locality sets.  ... 
arXiv:2207.10383v1 fatcat:fovfht2abzexpoc4uhqwvsrosm

Locally recoverable codes with availability t≥2 from fiber products of curves

Kathryn Haymaker, Beth Malmskog, Gretchen L. Matthews
2018 Advances in Mathematics of Communications  
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.  ...  Consider a locally recoverable code C of length n and availability t with symmetric recovery sets. Say that erasures occur.  ... 
doi:10.3934/amc.2018020 fatcat:xptauwuo4zf6xiy6r3nros2rfm

On One Generalization of LRC Codes with Availability [article]

Stanislav Kruglik, Marina Dudina, Valeriya Potapova, Alexey Frolov
2017 arXiv   pre-print
We investigate one possible generalization of locally recoverable codes (LRC) with all-symbol locality and availability when recovering sets can intersect in a small number of coordinates.  ...  In this paper we derive an upper bound for the rate of such codes and give explicit constructions of codes with such a property. These constructions utilize LRC codes developed by Wang et al.  ...  Barg for introducing this problem to him and for numerous fruitful discussions during his stay in University of Maryland.  ... 
arXiv:1705.11095v1 fatcat:gxy374datbchnblz4lnclbro5m

Bounding Multiple Unicasts through Index Coding and Locally Repairable Codes [article]

Karthikeyan Shanmugam, Alexandros G. Dimakis
2014 arXiv   pre-print
Specifically, we show that a natural extension of LRCs we call Generalized Locally Repairable Codes (GLCRs) are exactly dual to linear index codes.  ...  Our new upper bound is efficiently computable and relies on a strong approximation result for complementary index coding.  ...  and the side information packets available with them.  ... 
arXiv:1402.3895v1 fatcat:z5c35n6sjjhg7puu6ugor4zgcy

Codes for Distributed Storage

Vinayak Ramkumar, S. B. Balaji, Birenjith Sasidharan, Myna Vajha, M. Nikhil Krishnan, P. Vijay Kumar
2022 Foundations and Trends in Communications and Information Theory  
This concern led to the adoption by the storage industry of erasure-recovery codes such as Reed-Solomon (RS) codes and more generally, maximum distance separable codes, as these codes offer the lowest-possible  ...  The explosive growth in the amount of data generated on a daily basis brought up a second major concern, namely minimization of the overhead associated with such redundant storage.  ...  Acknowledgements References Full text available at: http://dx.doi.org/10.1561/0100000115  ... 
doi:10.1561/0100000115 fatcat:p2wafc3dffbmbbigq6dizuaile

Constructions of maximally recoverable local reconstruction codes via function fields

Venkatesan Guruswami, Lingfei Jin, Chaoping Xing
2020 IEEE Transactions on Information Theory  
Such a code is maximally recoverable if it can correct all patterns of a erasures per local group plus up to h additional erasures anywhere in the codeword.  ...  A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable  ...  Constructions of Maximally Recoverable Local Reconstruction Codes The first two bounds, and the bounds in (vii) and (viii) of Theorem 1 are derived from the rational function field F 2 (x).  ... 
doi:10.1109/tit.2020.2988459 fatcat:da2ltwoqffe7nmcn5fgxfqej5e

The CORE Storage Primitive: Cross-Object Redundancy for Efficient Data Repair & Access in Erasure Coded Storage [article]

Kyumars Sheykh Esmaili, Lluis Pamies-Juarez, Anwitaman Datta
2013 arXiv   pre-print
In this paper, we take an engineering alternative, exploring the use of simple and mature techniques -juxtaposing a standard erasure code with RAID-4 like parity.  ...  Consequently, novel codes optimized to cope with distributed storage system nuances are vigorously being researched.  ...  the form G g = [I k , H], and (ii) a systematic (k + 1, k )-code with local generator matrix G l of the form G l = [I k , 1 k ], for k = k/2.  ... 
arXiv:1302.5192v4 fatcat:rwljkazcc5azfi7khqpsy7vc5i

Capacity bounds for distributed storage [article]

Michael Luby
2018 arXiv   pre-print
To maintain recoverability of the source data, a repairer continually reads data over a network from nodes at a rate R, and generates and writes data to nodes based on the read data.  ...  Storage nodes fail randomly over time and are replaced with nodes of equal capacity initialized to zeroes, and thus bits are erased at some rate E.  ...  from the nodes to generate and store a fragment at the new node.  ... 
arXiv:1610.03541v5 fatcat:qxwcyex2qfbuvbvymiyeeelepa

Bounds and Constructions of Codes with All-Symbol Locality and Availability [article]

Stanislav Kruglik, Alexey Frolov
2017 arXiv   pre-print
We investigate the distance properties of linear locally recoverable codes (LRC codes) with all-symbol locality and availability.  ...  To reduce the gap in between upper and lower bounds we do not restrict the alphabet size and propose explicit constructions of codes with locality and availability via rank-metric codes.  ...  New upper and lower bounds on the minimum distance of LRC codes with availability are derived.  ... 
arXiv:1702.01314v1 fatcat:gsccxznyuzgqjkz72qxwdvwa5u

Repair rate lower bounds for distributed storage [article]

Michael Luby
2020 arXiv   pre-print
To maintain recoverability of the source data, a repairer continually reads data over a network from nodes at some average rate rrate, and generates and writes data to nodes based on the read data.  ...  This inequality provides a fundamental lower bound on the average rate that any repairer needs to read data from the system in order to maintain recoverability of the source data.  ...  In the real system, only a small portion of the global memory state may be available in the local memory of a node when local-computation repair is used.  ... 
arXiv:2002.07904v1 fatcat:nfqnb3yqz5eyjjlywlvudqpgsm

Irregular Recovery and Unequal Locality for Locally Recoverable Codes with Availability [article]

Sourbh Bhadane, Andrew Thangaraj
2017 arXiv   pre-print
A code is said to be a Locally Recoverable Code (LRC) with availability if every coordinate can be recovered from multiple disjoint sets of other coordinates called recovering sets.  ...  for information locality with availability.  ...  CONCLUSION We derived new upper bounds on minimum distance for codes with unequal all-symbol locality and availability.  ... 
arXiv:1705.05005v1 fatcat:yq2mqk6apvfgddjaas7uwgn4wa

Symbolic Debugging of Optimized Code

John Hennessy
1982 ACM Transactions on Programming Languages and Systems  
The effects of local and global optimizations on the variables of a program are categorized, and models for representing the effect of optimizations are given.  ...  The long-standing conflict between code optimization and symbolic debugging is examined.  ...  A MODEL OF LOCAL OPTIMIZATION AND CODE GENERATION Describing algorithms that locate noncurrent variables requires a model for code generation.  ... 
doi:10.1145/357172.357173 fatcat:p7eu5tg57nedtja4mcal7bpy3e

Fractional repetition codes with flexible repair from combinatorial designs [article]

Oktay Olmez, Aditya Ramamoorthy
2016 arXiv   pre-print
Furthermore, we present the Kronecker product technique for generating new codes from existing ones and elaborate on their properties.  ...  FR codes with locality are those where the repair degree is smaller than the number of nodes contacted for reconstructing the stored file.  ...  (lñ, lθ,α,ρ)β tθ +α tñ + 1 Codes meet the minimum distance bound for locally recoverable codes with exact and uncoded repair (cf.  ... 
arXiv:1408.5780v2 fatcat:7ddowqa5ubgrdgr5jonsliolsq

Locally recoverable codes from planar graphs

Kathryn Haymaker, Justin O'Pella
2019 Journal of Algebra Combinatorics Discrete Structures and Applications  
The resulting edge codes are locally recoverable with availability two.  ...  We prove that the minimum distance of planar graph LRCs is equal to the girth of the graph, and we also establish a new bound on the rate of planar graph edge codes.  ...  The rate R of a locally recoverable code with availability two and recovery set sizes at most r satisfies: R ≤ r r + 2 . (1) In [4] , Kadhe and Calderbank introduced codes generated by convex polyhedra  ... 
doi:10.13069/jacodesmath.645021 fatcat:j6k42ueqtngn3pxujok2lcmwxu
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