Corruption and Recovery-Efficient Locally Decodable Codes View Full Text


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Chapter Info

DATE

2008-01-01

AUTHORS

David Woodruff

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 xk can be recovered with probability at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{1}{2} + \epsilon$\end{document} 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.For any δ, ε ∈ [Ω(n− 1/2), O(1)], we give a family of (2, δ, ε)-LDCs with length . For linear (2, δ, ε)-LDCs, Obata has shown that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m \geq \exp \left (\delta n \right )$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m = \exp \left (n^{1/t} \right )$\end{document} for any prime of the form 2t − 1, we obtain non-linear (3, δ, ε)-LDCs with .Now consider a (q, δ, ε)-LDC C with a decoder that has n matchings M1, ..., Mn 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, ..., aq} ∈ Mk uniformly at random and outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bigoplus_{j=1}^q y_{a_j}$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m \geq \exp \left (\max(\delta, \epsilon)\delta n \right )$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\exp \left (n^{1/t} \right )$\end{document} for any prime of the form 2t − 1 with non-trivial recovery probability for any δ< 1/6. More... »

PAGES

584-595

Book

TITLE

Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques

ISBN

978-3-540-85362-6
978-3-540-85363-3

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-540-85363-3_46

DOI

http://dx.doi.org/10.1007/978-3-540-85363-3_46

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1016904024


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