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URN: urn:nbn:de:0030-drops-24133
URL: https://drops.dagstuhl.de/opus/volltexte/2010/2413/
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Alon, Noga ; Panigrahy, Rina ; Yekhanin, Sergey

Deterministic approximation algorithms for the nearest codeword problem

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09421.YekhaninSergey.Paper.2413.pdf (0.2 MB)

Abstract

The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v in F_2^n and a linear space L in F_2^n of dimension k NCP asks to find a point l in L that minimizes the (Hamming) distance from v. It is well-known that the nearest codeword problem is NP-hard. Therefore approximation algorithms are of interest. The best effcient approximation algorithms for the NCP to date are due to Berman and Karpinski. They are a deterministic algorithm that achieves an approximation ratio of O(k/c) for an arbitrary constant c; and a randomized algorithm that achieves an approximation ratio of O(k/ log n). In this paper we present new deterministic algorithms for approximating the NCP that improve substantially upon the earlier work, (almost) de-randomizing the randomized algorithm of Berman and Karpinski. We also initiate a study of the following Remote Point Problem (RPP). Given a linear space L in F_2^n of dimension k RPP asks to find a point v in F_2^n that is far from L. We say that an algorithm achieves a remoteness of r for the RPP if it always outputs a point v that is at least r-far from L. In this paper we present a deterministic polynomial time algorithm that achieves a remoteness of Omega(n log k / k) for all k < n/2. We motivate the remote point problem by relating it to both the nearest codeword problem and the matrix rigidity approach to circuit lower bounds in computational complexity theory.

BibTeX - Entry

@InProceedings{alon_et_al:DSP:2010:2413,
  author =	{Noga Alon and Rina Panigrahy and Sergey Yekhanin},
  title =	{Deterministic approximation algorithms for the nearest codeword problem},
  booktitle =	{Algebraic Methods in Computational Complexity},
  year =	{2010},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  number =	{09421},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2413},
  annote =	{Keywords: }
}

Collection: 09421 - Algebraic Methods in Computational Complexity
Issue Date: 2010
Date of publication: 19.01.2010

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