  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.
