Locally recoverable codes from rational maps

Locally recoverable codes are error-correcting codes allowing local recovery of lost encoded data in a codeword. We give a method to construct locally recoverable codes from rational maps between affine spaces, whose fibres are used as recovery sets. The recovery of erasures is carried out by Lagrangian interpolation in general and simply by one addition in some good cases. We first state the general construction of these codes and study its main properties. Next we apply it to several types of codes, including algebraic geometry codes, Reed-Muller, and other related codes. The existence of several recovering sets for the same coordinate and the possibility of recover more than one erasure at the same time are also treated.