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

We study monomial-Cartesian codes (MCCs) which can be regarded as \((r,\delta )\) $(r,\delta )$ -locally recoverable codes (LRCs). These codes come with a natural bound for their minimum distance and we determine those giving rise to \((r,\delta )\) $(r,\delta )$ -optimal LRCs for that distance, which are in fact \((r,\delta )\) $(r,\delta )$ -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 \((r,\delta )\) $(r,\delta )$ -optimal LRCs and their parameters.