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 [4] 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. [24] as the generalized Cayley metric and the block permutation metric have the same magnitude. From the mathematical point of view, the block permutation metric is not natural as the last pair $(n,1)$ is not included in the characteristic set. In this paper, by including $(n,1)$ in the characteristic set, we introduce a new metric that is called cyclic block permutation metric. Under this new metric, we introduce a class of codes that are called cyclic block permutation codes. Based on some techniques from algebraic function fields originated in [21], we give an algebraic-geometric construction of cyclic block permutation codes with reasonably good parameters. By observing a trivial relation between cyclic block permutation metric and block permutation metric, we produce non-systematic codes in block permutation metric that improve all known results given in [24],[23]. More importantly, based on our non-systematic codes, we provide an explicit and systematic construction of block permutation codes which improves the systemic result shown in [24]. In the end, we demonstrate that our cyclic block permutation codes indeed have reasonably good parameters by showing that our construction beats the Gilbert-Varshamov bound.
翻译:由于各种应用,对不同应用应用来说,对不同的应用,有一个需要不同度量下的变异代码。通用的Cayley测量标准由Chee和Vu[4]采用,而这一测量标准包括了其他几个标准作为特例。然而,通用的Cayley测量标准一般不容易计算。因此,由Yang等人(24)采用区块变异标准,因为通用的Cayley测量标准和区块变异测量标准具有同等规模。从数学角度看,区块变异参数不是自然的,因为最后一对(n,1)美元没有列入特征集。在本文件中,将(n,1)美元纳入通用的Cayley测量标准包括了其他几个标准作为特例。但是,通用的Cayley测量标准并非容易计算。因此,由Yang等人(Yang等人)采用区块变异变量衡量标准。根据一些变异函数字段,我们从[21] 开始,我们给出了一种变异基因变异的测算值参数,作为最后一对立方块变异的构造构造构造构造的构造构造构造构造构造构造构造构造构造结构, 显示一个不甚高的校正关系,我们通过测量结果显示每个平的校正的校正的校正的校正的校正的校正的校正的校正结果。