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.

由于各种应用，排列码受到了极大的关注。对于不同的应用，需要不同度量下的排列码。广义 Cayley 度量由 Chee 和 Vu 引入（在：2014 年 IEEE 国际信息论研讨会，火奴鲁鲁，2014 年 6 月 29 日至 7 月 4 日，第 2959-2963 页，2014 年）并且该度量包括其他几个度量作为特例。然而，广义的 Cayley 度量通常不容易计算。因此，Yang 等人引入了块置换度量。(IEEE Trans Inf Theory 65(8):4746–4763, 2019) 因为广义 Cayley 度量和块排列度量具有相同的量级。在本文中，通过引入与块置换度量密切相关的新度量，我们在块置换度量中的一些高级代数方法和代码之间建立了桥梁。更具体地说，基于源自 Xing 的代数函数域的一些技术 (IEEE Trans Inf Theory 48(11):2995–2997, 2002)，我们给出了具有相当好的参数的新度量中代码的代数几何构造。通过观察新度量和块置换度量之间的微不足道的关系，我们然后在块置换度量中生成非系统代码，以改进 Xu 等人给出的所有已知结果。(Des Codes Cryptogr 87(11):2625–2637, 2019) 和 Yang 等人。（2019）。更重要的是，基于我们的非系统代码，我们在块置换度量中提供了明确和系统的代码构造，从而改进了 Yang 等人所示的系统结果。（2019）。到底，