  We consider the problem of correcting insertion and deletion errors in the
$d$-dimensional space. This problem is well understood for vectors
(one-dimensional space) and was recently studied for arrays (two-dimensional
space). For vectors and arrays, the problem is motivated by several practical
applications such as DNA-based storage and racetrack memories. From a
theoretical perspective, it is interesting to know whether the same properties
of insertion/deletion correcting codes generalize to the $d$-dimensional space.
In this work, we show that the equivalence between insertion and deletion
correcting codes generalizes to the $d$-dimensional space. As a particular
result, we show the following missing equivalence for arrays: a code that can
correct $t_\mathrm{r}$ and $t_\mathrm{c}$ row/column deletions can correct any
combination of
$t_\mathrm{r}^{\mathrm{ins}}+t_\mathrm{r}^{\mathrm{del}}=t_\mathrm{r}$ and
$t_\mathrm{c}^{\mathrm{ins}}+t_\mathrm{c}^{\mathrm{del}}=t_\mathrm{c}$
row/column insertions and deletions. The fundamental limit on the redundancy
and a construction of insertion/deletion correcting codes in the
$d$-dimensional space remain open for future work.
