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On the Hardness of the Decoding and the Minimum Distance Problems for Rank Codes.

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Názov: On the Hardness of the Decoding and the Minimum Distance Problems for Rank Codes.
Autori: Gaborit, Philippe, Zemor, Gilles
Zdroj: IEEE Transactions on Information Theory; Dec2016, Vol. 62 Issue 12, p7245-7252, 8p
Predmety: HAMMING codes, ERROR-correcting codes, PERFECT codes, LINEAR codes, ALGEBRAIC codes
Abstrakt: We give a randomized reduction for the Rank Syndrome Decoding problem and Rank Minimum Distance problem for rank codes over extension fields. Our results are based on embedding linear codes in the Hamming space into linear codes over an extension field equipped with the rank metric. We prove that if any of the previous problems for the rank metric is in ZPP = RP $\cap $ coRP, then we would have NP = ZPP. We also give complexity results for the respective rank metric approximation problems. [ABSTRACT FROM PUBLISHER]
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  Data: On the Hardness of the Decoding and the Minimum Distance Problems for Rank Codes.
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  Data: <searchLink fieldCode="AR" term="%22Gaborit%2C+Philippe%22">Gaborit, Philippe</searchLink><br /><searchLink fieldCode="AR" term="%22Zemor%2C+Gilles%22">Zemor, Gilles</searchLink>
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  Data: IEEE Transactions on Information Theory; Dec2016, Vol. 62 Issue 12, p7245-7252, 8p
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  Data: <searchLink fieldCode="DE" term="%22HAMMING+codes%22">HAMMING codes</searchLink><br /><searchLink fieldCode="DE" term="%22ERROR-correcting+codes%22">ERROR-correcting codes</searchLink><br /><searchLink fieldCode="DE" term="%22PERFECT+codes%22">PERFECT codes</searchLink><br /><searchLink fieldCode="DE" term="%22LINEAR+codes%22">LINEAR codes</searchLink><br /><searchLink fieldCode="DE" term="%22ALGEBRAIC+codes%22">ALGEBRAIC codes</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We give a randomized reduction for the Rank Syndrome Decoding problem and Rank Minimum Distance problem for rank codes over extension fields. Our results are based on embedding linear codes in the Hamming space into linear codes over an extension field equipped with the rank metric. We prove that if any of the previous problems for the rank metric is in ZPP = RP $\cap $ coRP, then we would have NP = ZPP. We also give complexity results for the respective rank metric approximation problems. [ABSTRACT FROM PUBLISHER]
– Name: Abstract
  Label:
  Group: Ab
  Data: <i>Copyright of IEEE Transactions on Information Theory is the property of IEEE and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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