Query Complexity Lower Bounds for Reconstruction of Codes

13 years 3 months ago

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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 algorit...

Sourav Chakraborty, Eldar Fischer, Arie Matsliah

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