Corruption and recovery-efficient locally decodable codes

Abstract

A (q, δ, ε)-locally decodable code (LDC) C: {0,1} n →{0,1} m is an encoding from n-bit strings to m-bit strings such that each bit x k can be recovered with probability at least 1/2 + ε 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 ε. The key property of our LDCs is that they are non-linear, whereas all previous LDCs were linear. 1 For any δ, ε ∈[Ω(n-1/2), O(1)], we give a family of (2, δ, ε)-LDCs with length m = poly (δ-1, ε-1)exp (max (δ, ε)δn). For linear (2, δ, ε)-LDCs, Obata has shown that m > exp (δn). Thus, for small enough constants δ, ε, two-query non-linear LDCs are shorter than two-query linear LDCs. We improve the dependence on δ and ε of all constant-query LDCs by providing general transformations to non-linear LDCs. Taking Yekhanin's linear (3, δ, 1/2 - 6δ)-LDCs with m = exp (n 1/t) for any prime of the form 2t - 1, we obtain non-linear (3, δ, ε)-LDCs with m = poly (δ-1, ε-1)exp ((max(δ, ε)δn)1/t). Now consider a (q, δ, ε)-LDC C with a decoder that has n matchings M 1, ..., M n on the complete q-uniform hypergraph, whose vertices are identified with the positions of C(x). On input k ∈ [n] and received word y, the decoder chooses e = {a1, ..., a q}∈[M] k uniformly at random and outputs j=1qyaj. All known LDCs and ours have such a decoder, which we call a matching sum decoder. We show that if C is a two-query LDC with such a decoder, then m ≥ exp(max (δ, ε)δ n). Interestingly, our techniques used here can further improve the dependence on δ of Yekhanin's three-query LDCs. Namely, if δ ≥ 1/12 then Yekhanin's three-query LDCs become trivial (have recovery probability less than half), whereas we obtain three-query LDCs of length exp (n1/t)for any prime of the form 2t - 1 with non-trivial recovery probability for any δ < 1/6. © 2008 Springer-Verlag Berlin Heidelberg.