Codes With Hierarchical Locality From Covering Maps of Curves

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 r...

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Bibliographic Details
Published in:IEEE transactions on information theory 2019-10, Vol.65 (10), p.6056-6071
Main Authors: Ballentine, Sean, Barg, Alexander, Vladut, Serge
Format: Article
Language:English
Subjects:
Algebraic Geometry
Automorphisms
Codes
codes with availability
elliptic curves
fiber products
Information processing
Linear codes
Locally recoverable codes
Maintenance engineering
Mathematical analysis
Mathematics
Poles and towers
Quotients
Recoverability
Reed-Solomon codes
Subgroups
Upper bound
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Summary: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.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2919830