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A Locally Recoverable Code is a code such that the value of any singlecoordinate of a codeword can be recovered from the values of a small subset ofother 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 linearcode $\mathcal{C}$ with length $n$, dimension $k$, minimum distance $d$ has$(r_1,\ldots, r_\delta)$-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 distanceof 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 $g \geq 1$, we propose a construction of $[n, k, d; r_1,r_2,\dots, r_\delta]$-codes and apply the results to some well known familiesof function fields.

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