Locally Recoverable Codes with Availability $t\geq 2$ from Fiber Products of Curves

TL;DR

This paper extends the construction of locally recoverable codes using fiber products of algebraic curves, creating new families with multiple recovery sets and analyzing their error correction capabilities.

Contribution

It generalizes existing code constructions to include arbitrarily many recovery sets via fiber products, introducing new codes from maximal curves like GK, Suzuki, and Hermitian curves.

Findings

Constructed new locally recoverable codes with multiple recovery sets

Developed codes from generalized GK and Suzuki curves

Analyzed the relationship between local and global error correction

Abstract

We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vl\u{a}du\c{t} to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. 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 Korchm\'{a}ros (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 consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures.