Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes

  • Kenji Obata
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)


This paper presents essentially optimal lower bounds on the size of linear codes
$$ C:\left\{ {0,1} \right\}^n \to \left\{ {0,1} \right\}^m $$
which have the property that, for constants δ, > 0, any bit of the message can be recovered with probability 1/2 + ∈ by an algorithm reading only 2 bits of a codeword corrupted in up to δm positions. Such codes are known to be applicable to, among other things, the construction and analysis of information-theoretically secure private information retrieval schemes. In this work, we show that m must be at least \( 2^{\Omega (\tfrac{\delta } {{1 - 2 \in }}n)} \). Our results extend work by Goldreich, Karloff, Schulman, and Trevisan [GKST02], which is based heavily on methods developed by Katz and Trevisan [KT00]. The key to our improved bounds is an analysis which bypasses an intermediate reduction used in both prior works. The resulting improvement in the efficiency of the overall analysis is sufficient to achieve a lower bound optimal within a constant factor in the exponent. A construction of a locally decodable linear code matching this bound is presented.


Linear Code Symmetric Distribution Maximum Match Recovery Parameter Private Information Retrieval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [CGKS98]
    B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan. Private information retrieval. Journal of the ACM, 45(6):965–982, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [GKST02]
    O. Goldreich, H. Karloff, L. Schulman, and Luca Trevisan. Lower bounds for linear locally decodable codes and private information retrieval. In Proc. of the 17th IEEE CCC, 2002.Google Scholar
  3. [KT00]
    J. Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In Proc. of the 32nd ACM STOC, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Kenji Obata
    • 1
  1. 1.Computer Science DivisionUniversity of CaliforniaBerkeley

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