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

Abstract

AbstractWe study monomial-Cartesian codes (MCCs) which can be regarded as $$(r,\delta )$$ ( r , δ ) -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 , δ ) -optimal LRCs for that distance, which are in fact $$(r,\delta )$$ ( r , δ ) -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 , δ ) -optimal LRCs and their parameters.