Optimal (r,δ)-LRCs from zero-dimensional affine variety codes and their subfield-subcodes
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We introduce zero-dimensional affine variety codes (ZAVCs) which can be regarded as (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,δ)-optimal LRCs for that distance, which are in fact (r,δ)-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,δ)-optimal LRCs and their parameters.
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