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We generalize the construction of locally recoverable codes on algebraiccurves given by Barg, Tamo and Vl\u{a}du\c{t} to those with arbitrarily manyrecovery sets by exploiting the structure of fiber products of curves.Employing maximal curves, we create several new families of locally recoverablecodes with multiple recovery sets, including codes with two recovery sets fromthe generalized Giulietti and Korchm\'{a}ros (GK) curves and the Suzuki curves,and new locally recoverable codes with many recovery sets based on theHermitian curve, using a fiber product construction of van der Geer and van derVlugt. In addition, we consider the relationship between local error recoveryand global error correction as well as the availability required to locallyrecover any pattern of a fixed number of erasures.

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