Codes with hierarchical locality from covering maps of curves

TL;DR

This paper introduces a general method for constructing hierarchical locally recoverable codes using algebraic curves, providing flexible recovery options and new code families with good asymptotic properties.

Contribution

It presents a novel algebraic-curve-based construction of hierarchical LRC codes with multiple recoverability tiers and availability, including explicit parameter calculations and asymptotic families.

Findings

Constructed codes from various algebraic curves including rational, elliptic, Kummer, and Artin-Schreier.

Developed a general framework for codes with multiple recovery tiers and availability.

Produced asymptotically good families of hierarchical LRC codes from Garcia-Stichtenoth tower.

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.