We investigate the problem of local reconstruction , as defined by Saks and Seshadhri (2008), in the context of error correcting codes.

The first problem we address is that of message reconstruction , where given oracle access to a corrupted encoding $w \in \{0,1\}^n$ of some message $x \in \{0,1\}^k$ our goal is to probabilistically recover $x$ (or some portion of it). This should be done by a procedure (reconstructor) that given an index $i$ as input, probes $w$ at few locations and outputs the value of $x_i$. The reconstructor can (and indeed must) be randomized, with all its randomness specified in advance by a single random seed, and the main requirement is that for most random seeds, all values $x_1,\ldots,x_k$ are reconstructed correctly. (Notice that swapping the order of “for most random seeds” and “for all $x_1,\ldots,x_k$” would make the definition equivalent to standard Local Decoding ).

A message reconstructor can serve as a “filter” that allows evaluating certain classes of algorithms on $x$ safely and efficiently. For instance, to run a parallel algorithm, one can initialize several copies of the reconstructor with the same random seed, and then they can autonomously handle decoding requests while producing outputs that are consistent with the original message $x$. Another motivation for studying message reconstruction arises from the theory of Locally Decodable Codes.