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Proceedings 17th IEEE Annual Conference on Computational Complexity (2002)
Montreal, Canada
May 21, 2002 to May 24, 2002
ISBN: 0-7695-1468-5
pp: 0175
Oded Goldreich , Weizmann Institute of Science
Howard Karloff , AT&T Labs
Luca Trevisan , University of California at Berkeley
We prove that if a linear error-correcting code C:\{0,1\}^n\to\{0,1\}^m is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then m = 2^{\Omega(n)}. We also present several extensions of this result.We show a reduction from the complexity of one-round, information-theoretic Private Information Retrieval Systems (with two servers) to Locally Decodable Codes, and conclude that if all the servers' answers are linear combinations of the database content, then t = \Omega(n/2^a), where t is the length of the user's query and a is the length of the servers' answers. Actually, 2^a can be replaced by O(a^k), where k is the number of bit locations in the answer that are actually inspected in the reconstruction.
Error Correcting Codes, Linear Codes, Private Information Retrieval

O. Goldreich, L. J. Schulman, H. Karloff and L. Trevisan, "Lower Bounds for Linear Locally Decodable Codes and Private Information Retrieval," Proceedings 17th IEEE Annual Conference on Computational Complexity(CCC), Montreal, Canada, 2002, pp. 0175.
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