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

TL;DR

This paper proves that the equivalence between insertion and deletion correcting codes extends to $d$-dimensional arrays, generalizing known results from vectors and 2D arrays, with implications for error correction in complex data structures.

Contribution

It establishes the equivalence of insertion and deletion correcting codes in $d$-dimensional space, filling a theoretical gap for arrays beyond two dimensions.

Findings

Proves the generalization of insertion/deletion code equivalence to $d$-dimensions.

Shows that correcting row/column deletions implies correction of insertions and deletions.

Identifies open problems regarding redundancy bounds and code constructions.

Abstract

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.