Towards 3-query locally decodable codes of subexponential length

A q-query Locally Decodable Code (LDC) encodes an n-bitmessage x as an n-bit codeword C(x), such that one canprobabilistically recover any bit xi of the message by queryingonly q bits of the codeword C(x), even after some constantfraction of codeword bits has been corrupted.We give new constructions of three query LDCs of vastly shorterlength than that of previous constructions. Specifically, givenany Mersenne prime p = 2t - 1, we design three query LDCs of length N=(n1/t), for every n. Based on thelargest known Mersenne prime, this translates to a length of less than exp(n10-7), compared to exp(n1/2) in the previous constructions. It hasoften been conjectured that there are infinitely many Mersenneprimes. Under this conjecture, our constructions yield three querylocally decodable codes of length N=exp(nO(1/(log log n))) forinfinitely many n. We also obtain analogous improvements for Private InformationRetrieval (PIR) schemes. We give 3-server PIR schemes withcommunication complexity of O(n10-7) to accessan n-bit database, compared to the previous best scheme withcomplexity O(n1/5.25). Assuming again that there areinfinitely many Mersenne primes, we get 3-server PIR schemes ofcommunication complexity nO(1/(log log n))for infinitely many n. Previous families of LDCs and PIR schemes were based on theproperties of low-degree multivariate polynomials over finitefields. Our constructions are completely different and areobtained by constructing a large number of vectors in a smalldimensional vector space whose inner products are restricted tolie in an algebraically nice set.

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