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New bounds of permutation codes under Hamming metric and Kendall’s \(\tau \)-metric

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Abstract

Permutation codes are widely studied objects due to their numerous applications in various areas, such as power line communications, block ciphers, and the rank modulation scheme for flash memories. Several kinds of metrics are considered for permutation codes according to their specific applications. This paper concerns some improvements on the bounds of permutation codes under Hamming metric and Kendall’s \(\tau \)-metric respectively, using mainly a graph coloring approach. Specifically, under Hamming metric, we improve the Gilbert–Varshamov bound asymptotically by a factor n, when the minimum Hamming distance d is fixed and the code length n goes to infinity. Under Kendall’s \(\tau \)-metric, we narrow the gap between the known lower bounds and upper bounds. Besides, we also obtain some sporadic results under Kendall’s \(\tau \)-metric for small parameters.

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Acknowledgements

The authors express their gratitude to the anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of this paper. Research supported by the National Natural Science Foundation of China under Grant Nos. 11431003 and 61571310.

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Xin Wang, Yiwei Zhang & Gennian Ge

  2. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang, China

    Xin Wang

  3. Department of Mathematics, Tongji University, Shanghai, 200092, China

    Yiting Yang

  4. Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 100048, China

    Gennian Ge

Authors
  1. Xin Wang
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  2. Yiwei Zhang
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  3. Yiting Yang
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  4. Gennian Ge
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Correspondence to Gennian Ge.

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Communicated by C. Ding.

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Wang, X., Zhang, Y., Yang, Y. et al. New bounds of permutation codes under Hamming metric and Kendall’s \(\tau \)-metric. Des. Codes Cryptogr. 85, 533–545 (2017). https://doi.org/10.1007/s10623-016-0321-5

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  • DOI: https://doi.org/10.1007/s10623-016-0321-5

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