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A new metric on symmetric groups and applications to block permutation codes

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Abstract

Permutation codes have received a great attention due to various applications. For different applications, one needs permutation codes under different metrics. The generalized Cayley metric was introduced by Chee and Vu (in: 2014 IEEE international symposium on information theory, Honolulu, June 29–July 4, 2014, pp 2959–2963, 2014) and this metric includes several other metrics as special cases. However, the generalized Cayley metric is not easily computable in general. Therefore the block permutation metric was introduced by Yang et al. (IEEE Trans Inf Theory 65(8):4746–4763, 2019) as the generalized Cayley metric and the block permutation metric have the same magnitude. In this paper, by introducing a novel metric closely related to the block permutation metric, we build a bridge between some advanced algebraic methods and codes in the block permutation metric. More specifically, based on some techniques from algebraic function fields originated in Xing (IEEE Trans Inf Theory 48(11):2995–2997, 2002), we give an algebraic-geometric construction of codes in the novel metric with reasonably good parameters. By observing a trivial relation between the novel metric and block permutation metric, we then produce non-systematic codes in block permutation metric that improve all known results given in Xu et al. (Des Codes Cryptogr 87(11):2625–2637, 2019) and Yang et al. (2019). More importantly, based on our non-systematic codes, we provide an explicit and systematic construction of codes in the block permutation metric which improves the systematic result shown in Yang et al. (2019). In the end, we demonstrate that our codes in the novel metric itself have reasonably good parameters by showing that our construction beats the corresponding Gilbert–Varshamov bound.

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Notes

  1. For a permutation \(\sigma \in {\mathcal {S}}_n\), denote \(\sigma _{(k)}\) by the permutation in \({\mathcal {S}}_k\) obtained from \(\sigma \) after deleting all the elements of \(\{k+1,k+2,\ldots ,n\}\) in \(\sigma \). Recall that a permutation code \({\mathcal {C}}\subset {\mathcal {S}}_n\) is called (n, k) systematic if for every \(\alpha \in {\mathcal {S}}_k\) there exists exactly one codeword \(\sigma \) of \({\mathcal {C}}\) such that \(\sigma _{(k)}=\alpha .\) Otherwise, we call the permutation codes non-systematic (See [2, Section II]).

  2. Bertrand–Chebyshev theorem states that for any positive integer n, there exist at least one prime number p such that \(n\le p\le 2n\).

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Acknowledgements

The author thanks professor Chaoping Xing, for patiently and carefully instructing him to rewrite this paper in a more reasonable way. Since this work was done by the author when he was an undergraduate student in Sichuan University, the author also thanks professor Qifan Zhang, for his guidance during the last several years. Meanwhile, the author thanks Zixiang Xu for generously introducing this topic to him as well as discussing and giving useful suggestions about Sect. 5, Siyi Yang for some useful advice in explicitly constructing systematic block permutation codes. Last but not least, the author thanks professor Zeyu Guo for kindly helping him revise this paper and two anonymous reviewers for giving their useful advice.

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  1. Zihan Zhang

    Present address: Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, 43210, USA

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  1. School of Mathematics, Sichuan University, Chengdu, 610065, China

    Zihan Zhang

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Zhang, Z. A new metric on symmetric groups and applications to block permutation codes. Des. Codes Cryptogr. 91, 2255–2271 (2023). https://doi.org/10.1007/s10623-023-01197-6

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