Locally recoverable codes on algebraic curves

TL;DR

This paper extends the construction of locally recoverable codes to algebraic curves, providing a general framework and examples including asymptotically good families and Hermitian codes with multiple recovery sets.

Contribution

It introduces a new general method for constructing LRC codes on algebraic curves, expanding beyond previous Reed-Solomon based approaches.

Findings

Constructed LRC codes on algebraic curves including Garcia-Stichtenoth towers.

Provided explicit examples of codes with good asymptotic properties.

Developed codes with multiple disjoint recovery sets, enhancing local repair options.

Abstract

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. 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.