Equivalence of Insertion/Deletion Correcting Codes for d-dimensional Arrays

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
r
and t
c
row/column deletions can correct any combination of $t_{\text{r}}^{{\text{ins}}} + t_{\text{r}}^{{\text{del}}} = {t_{\text{r}}}{\text{ and }}t_{\text{c}}^{{\text{ins}}} + t_{\text{c}}^{{\text{del}}} = {t_{\text{c}}}$ 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.