Optimal $(r,δ)$-LRCs from zero-dimensional affine variety codes and their subfield-subcodes

We introduce zero-dimensional affine variety codes (ZAVCs) 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 ZAVCs admit subfield-subcodes with the same parameters of the optimal codes but over smaller supporting fields. This fact allows us to determine infinitely many sets of new $(r,\delta)$-optimal LRCs and their parameters.