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In this paper, we study codes with locality that can recover from twoerasures via a sequence of two local, parity-check computations. By a localparity-check computation, we mean recovery via a single parity-check equationassociated to small Hamming weight. Earlier approaches considered recovery inparallel; the sequential approach allows us to potentially construct codes withimproved minimum distance. These codes, which we refer to as locally2-reconstructible codes, are a natural generalization along one direction, ofcodes with all-symbol locality introduced by Gopalan \textit{et al}, in whichrecovery from a single erasure is considered. By studying the GeneralizedHamming Weights of the dual code, we derive upper bounds on the minimumdistance of locally 2-reconstructible codes and provide constructions for afamily of codes based on Tur\'an graphs, that are optimal with respect to thisbound. The minimum distance bound derived here is universal in the sense thatno code which permits all-symbol local recovery from $2$ erasures can havelarger minimum distance regardless of approach adopted. Our approach also leadsto a new bound on the minimum distance of codes with all-symbol locality forthe single-erasure case.

Codes with Locality for Two Erasures