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Snake-in-the-box codes under the \(\ell _{\infty }\)-metric for rank modulation

  • Xiang WangEmail author
  • Fang-Wei Fu
Article
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Abstract

In the rank modulation scheme, Gray codes are very useful in the realization of flash memories. For a Gray code in this scheme, two adjacent codewords are obtained by using some “push-to-the-top” operations. Moreover, snake-in-the-box codes under the \(\ell _{\infty }\)-metric (\(\ell _{\infty }\)-snakes) are Gray codes, which can be capable of detecting one \(\ell _{\infty }\)-error. In this paper, we give two constructions of \(\ell _{\infty }\)-snakes. On the one hand, inspired by Yehezkeally and Schwartz’s construction, we present a new construction of the \(\ell _{\infty }\)-snake. The length of this \(\ell _{\infty }\)-snake is longer than the length of the \(\ell _{\infty }\)-snake constructed by Yehezkeally and Schwartz. On the other hand, we also give another construction of \(\ell _{\infty }\)-snakes by using \({\mathcal {K}}\)-snakes and obtain the longer \(\ell _{\infty }\)-snakes than the previously known ones.

Keywords

Flash memory Rank modulation Gray codes Snake-in-the-box codes \({\mathcal {K}}\)-snakes \(\ell _{\infty }\)-snakes 

Mathematics Subject Classification

68P30 94A15 

Notes

Acknowledgements

This work was supported by the 973 Program of China (Grant No. 2013CB834204) and the National Natural Science Foundation of China (Grant Nos. 61571243, U1836111).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Computer Network Emergency Response Technical TeamBeijingChina
  2. 2.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina

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