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Computer Science > Information Theory

arXiv:1807.05473 (cs)
[Submitted on 15 Jul 2018 (v1), last revised 2 Apr 2019 (this version, v3)]

Title:Codes with hierarchical locality from covering maps of curves

Authors:Sean Ballentine, Alexander Barg, Serge Vladuts
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Abstract: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.
Comments: V2: We added a construction of codes with two-level hierarchical locality and availability in each of the levels. \\ V3: The section on locality and availability has been rewritten; a new section was added with two general families of codes with two-level locality and availability in each of the levels, Sec. VIII.E
Subjects: Information Theory (cs.IT); Algebraic Geometry (math.AG)
Cite as: arXiv:1807.05473 [cs.IT]
  (or arXiv:1807.05473v3 [cs.IT] for this version)
  https://doi.org/10.48550/arXiv.1807.05473

Submission history

From: Alexander Barg [view email]
[v1] Sun, 15 Jul 2018 01:14:58 UTC (27 KB)
[v2] Thu, 16 Aug 2018 02:07:46 UTC (29 KB)
[v3] Tue, 2 Apr 2019 14:02:01 UTC (39 KB)
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