Consider a systematic linear code where some (local) parity symbols depend on few prescribed symbols, while other (heavy) parity symbols may depend on all data symbols. Local parities allow to quickly recover any single symbol when it is erased, while heavy parities provide tolerance to a large number of simultaneous erasures. A code as above is maximally-recoverable, if it corrects all erasure patterns which are information theoretically recoverable given the code topology. In this paper we present explicit families of maximally-recoverable codes with locality. We also initiate the study of the trade-off between maximal recoverability and alphabet size.Definition 3. Let C be a data-local (k, r, h)-code. We say that C is maximally-recoverable if for any set E ⊆ [n], where E is obtained by picking one coordinate from each of k r local groups, puncturing C in coordinates in E yields a maximum distance separable [k + h, k] code.A [k+h, k] MDS code obviously corrects all patterns of h erasures. Therefore a maximally-recoverable data-local (k, r, h)-code corrects all erasure patterns E ⊆ [n] that involve erasing one coordinate per local group, and h additional coordinates. We now argue that any erasure pattern that is not dominated by a pattern above has to be uncorrectable.Lemma 4. Let C be an arbitrary data-local (k, r, h)-code. Let E ⊆ [n] be an erasure pattern. Suppose E affects t local groups and |E| > t + h; then E is not correctable.
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