Locally Recoverable Codes from Automorphism Group of Function Fields of Genus g ≥ 1

A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have δ non-overlapping subsets of cardinality ri that can be used to recover the missing coordinate we say that a linear code C with length n, dimension k, minimum distance d has (r1,⋯, rδ) -locality and denote by [n, k, d; r1, r2, ⋯, rδ]. In this paper we provide a new upper bound for the minimum distance of these codes. Working with a finite number of subgroups of cardinality ri+1 of the automorphism group of a function field F| Fq of genus g ≥ 1 we propose a construction of [n, k, d; r1, r2, ⋯, rδ] -codes and apply the results to some well known families of function fields.