Query-Efficient Locally Decodable Codes of Subexponential Length

A k-query locally decodable code (LDC) C: Σⁿ → Γ^N encodes each message x into a codeword C(x) such that each symbol of x can be probabilistically recovered by querying only k coordinates of C(x), even after a constant fraction of the coordinates has been corrupted. Yekhanin (in J ACM 55:1-16, 2008) constructed a 3-query LDC of subexponential length, N = exp(exp(O(log n/log log n))), under the assumption that there are infinitely many Mersenne primes. Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009) constructed a 3-query LDC of length, with no assumption, and a 2^r-query LDC of length, for every integer r ≥ 2. Itoh and Suzuki (in IEICE Trans Inform Syst E93-D 2:263-270, 2010) gave a composition method in Efremenko's framework and constructed a 3 · 2^r-2-query LDC of length N_r, for every integer r ≥ 4, which improved the query complexity of Efremenko's LDC of the same length by a factor of 3/4. The main ingredient of Efremenko's construction is the Grolmusz construction for super-polynomial size set-systems with restricted intersections, over ℤ_m, where m possesses a certain "good" algebraic property (related to the "algebraic niceness" property of Yekhanin in J ACM 55:1-16, 2008). Efremenko constructed a 3-query LDC based on m = 511 and left as an open problem to find other numbers that offer the same property for LDC constructions. In this paper, we develop the algebraic theory behind the constructions of Yekhanin (in J ACM 55:1-16, 2008) and Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009), in an attempt to understand the ℤ_m. We show that every integer m = pq = 2^t -1, where p, q, and t are prime, possesses the same good algebraic property as m = 511 that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki's composition method. More precisely, we construct a 3^⌈r/2⌉ -query LDC for every positive integer r < 104 and a ⌊(3/4)⁵¹· 2^r⌋-query LDC for every integer r ≥ 104, both of length N_r, improving the 2^r queries used by Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009) and 3 · 2^r-2 queries used by Itoh and Suzuki (in IEICE Trans Inform Syst E93-D 2:263-270, 2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.