Query-Efficient Locally Decodable Codes of Subexponential Length

A k-query locally decodable code (LDC) C : Σn → ΓN 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 have been corrupted. Yekhanin (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 (2009) constructed a 3-query LDC of length N2 = exp(exp(O( √ log n log log n))) with no assumption, and a 2r-query LDC of length Nr = exp(exp(O( r √ log n(log log n)r−1))), for every integer r ≥ 2. Itoh and Suzuki (2010) gave a composition method in Efremenko’s framework and constructed a 3 · 2r−2-query LDC of length Nr, 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 superpolynomial size set-systems with restricted intersections, over Zm, where m possesses a certain “good” algebraic property (related to the “algebraic niceness” property of Yekhanin (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 (2008) and Efremenko (2009), in an attempt to understand the “algebraic niceness” phenomenon in Zm. We show that every integer m = pq = 2t − 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 · 2r ⌋ query LDC for every integer r ≥ 104, both of length Nr, improving the 2r queries used by Efremenko (2009) and 3 · 2r−2 queries used by Itoh and Suzuki (2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.