Locally recoverable codes on algebraic curves

TL;DR

This paper introduces new locally recoverable codes on algebraic curves, extending Reed-Solomon codes, and addresses local recovery of erasures, multiple erasures, and availability using algebraic geometry techniques.

Contribution

It presents a novel construction of LRC codes on algebraic curves, including methods for multiple erasures and availability, expanding the scope of algebraic code design.

Findings

Constructed LRC codes on algebraic curves with various recovery properties

Developed codes with multiple disjoint recovery sets for each coordinate

Utilized automorphism groups of curves to enhance code availability

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 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.