Abstract We construct Locally Recoverable Codes (LRCs) with availability 2 from a family of fibered surfaces. To obtain the locality and availability properties, and to estimate the minimum distance of the codes, we combine techniques coming from the theory of one-variable function fields and from the theory of fibrations on surfaces. When the locality parameter is $$r=3$$ r = 3 , we obtain a sharp bound on the minimum distance of the codes. In that case, we give a geometric interpretation of our codes in terms of doubly elliptic surfaces. In particular, this provides the first instance of an error correcting code constructed using a (doubly elliptic) K3 surface.

A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality r if, for every coordinate, its value at a codeword can be deduced from the value of (certain) r other coordinates of the codeword. These codes have found many recent applications, e.g., to distributed cloud storage. We will discuss the problem of constructing good locally recoverable codes and present some constructions using algebraic surfaces that improve previous constructions and sometimes provide codes that are optimal in a precise sense. The main conceptual contribution of this paper is to consider surfaces fibered over a curve in such a way that each recovery set is constructed from points in a single fiber. This allows us to use the geometry of the fiber to guarantee the local recoverability and use the global geometry of the surface to get a hold on the standard parameters of our codes. We look in detail at situations where the fibers are rational or elliptic curves and provide many examples applying our methods.