Lower bounds for linear locally decodable codes and private information retrieval

@article{Goldreich2002LowerBF,
  title={Lower bounds for linear locally decodable codes and private information retrieval},
  author={Oded Goldreich and Howard J. Karloff and Leonard J. Schulman and Luca Trevisan},
  journal={computational complexity},
  year={2002},
  volume={15},
  pages={263-296}
}
We prove that if a linear error-correcting code C:{0, 1} n →{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 = 2Ω (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… CONTINUE READING

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