Corruption and Recovery-Efficient Locally Decodable Codes

A (q, δ, )-locally decodable code (LDC) C : {0, 1} → {0, 1} is an encoding from n-bit strings to m-bit strings such that each bit xk can be recovered with probability at least 1 2 + from C(x) by a randomized algorithm that queries only q positions of C(x), even if up to δm positions of C(x) are corrupted. If C is a linear map, then the LDC is linear. We give improved constructions of LDCs in terms of the corruption parameter δ and recovery parameter . The key property of our LDCs is that they are non-linear, whereas all previous LDCs were linear. 1. For any δ, ∈ [Ω(n−1/2), O(1)], we give a family of (2, δ, )-LDCs with length m = poly(δ−1, −1) exp (max(δ, )δn). For linear (2, δ, )LDCs, Obata has shown that m ≥ exp (δn). Thus, for small enough constants δ, , two-query non-linear LDCs are shorter than two-query