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International Workshop on Approximation Algorithms for Combinatorial Optimization

International Workshop on Randomization and Approximation Techniques in Computer Science

APPROX 2008, RANDOM 2008: Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques pp 455–468Cite as

The Complexity of Local List Decoding

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5171))

Abstract

We study the complexity of locally list-decoding binary error correcting codes with good parameters (that are polynomially related to information theoretic bounds). We show that computing majority over Θ(1/ε) bits is essentially equivalent to locally list-decoding binary codes from relative distance 1/2 − ε with list size at most poly (1/ε). That is, a local-decoder for such a code can be used to construct a circuit of roughly the same size and depth that computes majority on Θ(1/ε) bits. On the other hand, there is an explicit locally list-decodable code with these parameters that has a very efficient (in terms of circuit size and depth) local-decoder that uses majority gates of fan-in Θ(1/ε).

Using known lower bounds for computing majority by constant depth circuits, our results imply that every constant-depth decoder for such a code must have size almost exponential in 1/ε (this extends even to sub-exponential list sizes). This shows that the list-decoding radius of the constant-depth local-list-decoders of Goldwasser et al. [STOC07] is essentially optimal.

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Authors and Affiliations

  1. Department of Mathematics and CSAIL, MIT,  ,  

    Dan Gutfreund

  2. CSAIL, MIT,  ,  

    Guy N. Rothblum

Authors
  1. Dan Gutfreund
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  2. Guy N. Rothblum
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Ashish Goel Klaus Jansen José D. P. Rolim Ronitt Rubinfeld

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Gutfreund, D., Rothblum, G.N. (2008). The Complexity of Local List Decoding. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_36

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  • DOI: https://doi.org/10.1007/978-3-540-85363-3_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85362-6

  • Online ISBN: 978-3-540-85363-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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