Corruption and Recovery-Efficient Locally Decodable Codes

A (q, i¾?, i¾?)-locally decodable code (LDC)C: {0,1}ni¾?{0,1}mis an encoding from n-bit strings to m-bit strings such that each bit x k can be recovered with probability at least $\frac{1}{2} + \epsilon$ from C(x) by a randomized algorithm that queries only qpositions of C(x), even if up to i¾?mpositions of C(x) are corrupted. If Cis a linear map, then the LDC is linear. We give improved constructions of LDCs in terms of the corruption parameter i¾?and recovery parameter i¾?. The key property of our LDCs is that they are non-linear, whereas all previous LDCs were linear. 1 For any i¾?, i¾?i¾? [i¾?(ni¾? 1/2), O(1)], we give a family of (2, i¾?, i¾?)-LDCs with length . For linear (2, i¾?, i¾?)-LDCs, Obata has shown that $m \geq \exp \left (\delta n \right )$. Thus, for small enough constants i¾?, i¾?, two-query non-linear LDCs are shorter than two-query linear LDCs. 1 We improve the dependence on i¾?and i¾?of all constant-query LDCs by providing general transformations to non-linear LDCs. Taking Yekhanin's linear (3, i¾?, 1/2 i¾? 6i¾?)-LDCs with $m = \exp \left (n^{1/t} \right )$ for any prime of the form 2ti¾? 1, we obtain non-linear (3, i¾?, i¾?)-LDCs with . Now consider a (q, i¾?, i¾?)-LDC Cwith a decoder that has nmatchings M 1 , ..., M n on the complete q-uniform hypergraph, whose vertices are identified with the positions of C(x). On input ki¾? [n] and received word y, the decoder chooses e= {a 1 , ..., a q } i¾? M k uniformly at random and outputs $\bigoplus_{j=1}^q y_{a_j}$. All known LDCs and ours have such a decoder, which we call a matching sum decoder. We show that if Cis a two-query LDC with such a decoder, then $m \geq \exp \left (\max(\delta, \epsilon)\delta n \right )$. Interestingly, our techniques used here can further improve the dependence on i¾?of Yekhanin's three-query LDCs. Namely, if i¾?i¾? 1/12 then Yekhanin's three-query LDCs become trivial (have recovery probability less than half), whereas we obtain three-query LDCs of length $\exp \left (n^{1/t} \right )$ for any prime of the form 2ti¾? 1 with non-trivial recovery probability for any i¾?< 1/6.