The techniques developed in this paper, particularly the use of Turán density and the analytic framework involving measurable subsets, can be adapted to analyze covering codes for other edit distance metrics, such as deletions and transpositions. For instance, the concept of covering codes can be generalized to account for sequences that allow for deletions by defining a new type of covering code that considers the subsequences formed by removing elements from a sequence.
In the case of transpositions, one could define a covering code that allows for the rearrangement of elements within a sequence. This would involve extending the definitions of covering codes to include permutations of sequences, thereby creating a new class of codes that can cover sequences under the transposition metric.
The analytic methods employed in this paper, such as the use of measurable sets and the application of Bonferroni inequalities, can be similarly applied to derive bounds for these new types of codes. By establishing relationships between the densities of these codes and their corresponding Turán densities, researchers can explore the optimal densities for covering codes under various edit distance metrics, leading to a deeper understanding of their combinatorial properties.