On Algebraic-Geometric Constructions of Locally Recoverable Codes form a Family of Fibrated Surfaces

Locally Recoverable Codes (LRC) are used for storing information in such a way that the missing digits can be recovered easily. Adding availability makes the code more robust since digits can be recovered in more than one way. One line of work constructs LRC via, for example, cyclic codes, generalized Hamming codes and rank metric codes; while a more recent approach, due to Salgado and Vicino constructs LRC with availability via a family of fibered surfaces. In this work we give a detailed account of the construction of the latter, and we find that it in general it requires a bigger alphabet than other existing methods. However, the construction by Salgado Vicino allows for a proof of a sharp lower bound for the minimum distance in the case where the locality is 3 and also provides a geometric interpretation that gives an alternative, more conceptual recovery algorithm.