Revised: August 25, 2014

Published: December 29, 2014

**Keywords:**property testing, locally decodable codes, locally correctable codes, reconstruction

**Categories:**complexity theory, query complexity, property testing, codes, locally testable codes, short

**ACM Classification:**F.2.2, G.1.6

**AMS Classification:**68W25

**Abstract:**
[Plain Text Version]

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.

The second problem that we address is *codeword reconstruction*, which is similarly defined, but instead of reconstructing the message the goal is to reconstruct the codeword itself, given
oracle access to its corrupted version.

Error correcting codes that admit message and codeword reconstruction can be obtained from Locally Decodable Codes (LDC) and Self Correctable Codes (SCC) respectively. The main contribution of this paper is a proof that in terms of query complexity, these constructions are close to best possible in many settings, even when we disregard the length of the encoding.

A conference version of this paper appeared in the Proceedings of the Second Symposium on Innovations in Computer Science (ICS 2011).