Lower bounds for linear locally decodable codes and private information retrieval

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 = 2^Ω(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 = Ω(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