Optimal $(r,\delta)$-LRCs from monomial-Cartesian codes and their subfield-subcodes

We study monomial-Cartesian codes (MCCs) which can be regarded as $(r,\delta)$-locally recoverable codes (LRCs). These codes come with a natural bound for their minimum distance and we determine those giving rise to $(r,\delta)$-optimal LRCs for that distance, which are in fact $(r,\delta)$-optimal. A large subfamily of MCCs admits subfield-subcodes with the same parameters of certain optimal MCCs but over smaller supporting fields. This fact allows us to determine infinitely many sets of new $(r,\delta)$-optimal LRCs and their parameters.
Comment: This is a revised version of the manuscript "Optimal $(r,\delta)$-LRCs from zero-dimensional affine variety codes and their subfield-subcodes". We have modified the title and the new one is "Optimal $(r,\delta)$-LRCs from monomial-Cartesian codes and their subfield-subcodes". This new version contains rather changes, the main ones appear in Section 4