Query Complexity Lower Bounds for Reconstruction of Codes

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 an oracle access to a corrupted encoding w ∈ {0, 1} n of some message x ∈ {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 xi. The reconstructor can (and indeed must) be randomized, but all its randomness is specified in advance by a single random seed, such that with high probability all k values xi for 1 ≤ i ≤ k are reconstructed correctly. Using the reconstructor as a filter allows to evaluate certain classes of algorithms on x efficiently. For instance, in case of a parallel algorithm, one can initialize several copies of the reconstructor with the same random seed, and they can autonomously handle decoding requests while producing outputs that are consistent with the original message x. Another example is that of adaptive querying algorithms, that need to know the value of some xi before deciding which index should be decoded next. The second problem that we address is codeword reconstruction, which is similarly defined, but instead of reconstructing the message our goal is to reconstruct the codeword itself, given an 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 Correctible Codes (SCC) respectively. The main contribution of this paper is a proof that in terms of query complexity, these are close to be the best possible constructions, even when we disregard the length of the encoding.

Query complexity lower bounds for reconstruction of codes

Abstract: 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 ∈ {0, 1} n of some message x ∈ {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 xi. 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 x1,..., xk are reconstructed correctly (notice that swapping the order of “for most random seeds ” and “for all x1,..., xk ” makes 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 an 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 Correctible Codes (SCC) respectively. The main contribution of this paper is a proof that in terms of query complexity, these are close to be the best possible constructions, even when we disregard the length of the encoding