论文标题
在网格代码上
On Grid Codes
论文作者
论文摘要
相对于曼哈顿的距离，以$ n $维格的网格为$ r $ -sphere的大小生成功能，用于在网格点以最小和最大尺寸为中心的最小和最大尺寸。这使我们可以为这些网格中的代码提供版本的锤子和吉尔伯特·瓦尔沙莫夫的界限。研究了锤子，曼哈顿和李在$ g $中定义的距离之间的关系。提出了用于$ g $的循环子组的最小锤量距离的公式。此外，根据这些代码的最小锤锤和Lee距离的最小曼哈顿距离的几个下限。提供了说明主要结果的示例，包括几个SageMath实现。
Generating functions for the size of a $r$-sphere, with respect to the Manhattan distance in an $n$-dimensional grid, are used to provide explicit formulas for the minimum and maximum size of an $r$-ball centered at a point of the grid. This allows us to offer versions of the Hamming and Gilbert-Varshamov bounds for codes in these grids. Relations between the Hamming, Manhattan, and Lee distances defined in an abelian group $G$ are studied. A formula for the minimum Hamming distance of codes that are cyclic subgroups of $G$ is presented. Furthermore, several lower bounds for the minimum Manhattan distance of these codes based on their minimum Hamming and Lee distances are established. Examples illustrating the main results are presented, including several SageMath implementations.
