Locally recoverable codes on algebraic curves

A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most $r$) other symbols of the codeword. In this paper we introduce a construction of LRC codes on algebraic curves, extending a recent construction of Reed-Solomon like codes with locality. We treat the following situations: local recovery of a single erasure, local recovery of multiple erasures, and codes with several disjoint recovery sets for every coordinate (the {\em availability problem}). For each of these three problems we describe a general construction of codes on curves and construct several families of LRC codes. We also describe a construction of codes with availability that relies on automorphism groups of curves.