Corruption and Recovery-Efficient Locally Decodable Codes

• Authors: David Woodruff
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

A (q, 茂戮驴, 茂戮驴)-locally decodable code (LDC)C: {0,1}n茂戮驴{0,1}mis an encoding from n-bit strings to m-bit strings such that each bit xkcan 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 茂戮驴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 茂戮驴and recovery parameter 茂戮驴. The key property of our LDCs is that they are non-linear, whereas all previous LDCs were linear.1For any 茂戮驴, 茂戮驴茂戮驴 [茂戮驴(n茂戮驴 1/2), O(1)], we give a family of (2, 茂戮驴, 茂戮驴)-LDCs with length . For linear (2, 茂戮驴, 茂戮驴)-LDCs, Obata has shown that $m \geq \exp \left (\delta n \right )$. Thus, for small enough constants 茂戮驴, 茂戮驴, two-query non-linear LDCs are shorter than two-query linear LDCs.1We improve the dependence on 茂戮驴and 茂戮驴of all constant-query LDCs by providing general transformations to non-linear LDCs. Taking Yekhanin's linear (3, 茂戮驴, 1/2 茂戮驴 6茂戮驴)-LDCs with $m = \exp \left (n^{1/t} \right )$ for any prime of the form 2t茂戮驴 1, we obtain non-linear (3, 茂戮驴, 茂戮驴)-LDCs with .Now consider a (q, 茂戮驴, 茂戮驴)-LDC Cwith a decoder that has nmatchings M1, ..., Mnon the complete q-uniform hypergraph, whose vertices are identified with the positions of C(x). On input k茂戮驴 [n] and received word y, the decoder chooses e= {a1, ..., aq} 茂戮驴 Mkuniformly 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 茂戮驴of Yekhanin's three-query LDCs. Namely, if 茂戮驴茂戮驴 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 2t茂戮驴 1 with non-trivial recovery probability for any 茂戮驴