On the Size of Error Ball in Single-Deletion Double-Insertion Channels

Recent experiments have demonstrated the feasibility of storing digital information in macromolecules such as DNA and proteins. While DNA storage systems benefit from extreme density and long durability, they suffer from various types of errors, including deletions, insertions, and substitutions. One way to address this problem is to store a sequence multiple times which can be modelled by sending a transmitted sequence over several identical channels that produce distinct outputs, allowing the receiver to collect the outputs and attempt to reconstruct the original sequence. The maximum number of possible outputs from the channels is referred to as the size of the error ball. The error ball encompasses all possible sequences that can arise from a limited number of errors applied to the original sequence. In this paper, we derive the size of the error ball generated by one deletion and two insertions. The method of derivation employs the inclusion-exclusion principle. We characterize the intersection size of any number of 2-insertion balls of subsequences belong to 1-deletion ball of a particular sequence. Furthermore, we generalize the notion of Type-B-confusable sequences for non-binary sequences and prove that the intersection size of two 1-insertion spheres of two sequences is one if and only if they are Type-B-confusable.