Locally recoverable codes from automorphism groups of function fields of genus $g\ge 1$

Abstract

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 $\delta$ non overlapping subsets of cardinality $r_i$ that can be used to recover the missing coordinate we say that a linear code $\mathcal{C}$ with length $n$, dimension $k$, minimum distance $d$ has $(r_1,\dots ,r_{\delta })$-locality and denote it by $[n,k,d;r_1,r_2,\dots ,r_{\delta }].$ 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 $r_i+1$ of the automorphism group of a function field $\mathcal{F}|{\mathbb{F}}_q$ of genus $g\ge 1$, we propose a construction of $[n,k,d;r_1,r_2,\dots ,r_{\delta }]$-codes and apply the results to some well known families of function fields.