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. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, vol. 60, no. 8, 2014, pp. 4661–4676). In this paper we extend this construction to codes on algebraic curves. We give a general construction of LRC codes on curves and compute some examples, including asymptotically good families of codes derived from the Garcia-Stichtenoth towers. The local recovery procedure is performed by polynomial interpolation over r coordinates of the codevector. We also obtain a family of Hermitian codes with two disjoint recovering sets for every symbol of the codeword.
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