New locally decodable codes and private information retrieval schemes

A q query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new constructions of three query LDCs of vastly shorter length than that of previous constructions. Specifically, given any Mersenne prime p = 2 t − 1, we design three query LDCs of length N = exp � n 1/t �, for every n. Based on the largest known Mersenne prime, this translates to a length of less than exp n 10−7� compared to exp � n1/2 � in the previous constructions. It has often been conjectured that there are infinitely many Mersenne�primes. Under this conjecture, our constructions yield three query locally decodable codes of length N = exp n