Abstract : 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.