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.

翻译：利用$n$维网格中曼哈顿距离下$r$-球面大小的生成函数，本文给出了以网格中某点为中心的$r$-球最小与最大规模的显式公式。基于此，我们为这类网格中的编码提出了汉明界与吉尔伯特-瓦尔沙莫夫界的对应形式。研究了在阿贝尔群$G$上定义的汉明距离、曼哈顿距离与李距离之间的关系，给出了作为$G$的循环子群的编码的最小汉明距离计算公式。此外，基于这些编码的最小汉明距离与最小李距离，建立了其最小曼哈顿距离的若干下界。文中给出了阐释主要结果的示例，包括若干SageMath实现。