On Levenshtein’s Reconstruction Problem for Channels with Unique Insertion Error Patterns

Levenshtein’s sequence reconstruction model plays an essential role in information retrieval in DNA-based storage systems. In this model, a word ${\text{x}} \in {\mathbb{Z}}_q^n$ is transmitted through N noisy channels, and the goal is to recover the original word exactly, or with a small uncertainty ${\mathcal{L}}$, using the outputs from these channels. Errors occurring in the channels usually involve substitutions, insertions or deletions. In this paper, we focus on insertion errors, which we represent using (so-called) insertion vectors. One of the main questions in this context is determining the minimum number of channels N required to recover the word either unambiguously or within a given precision ${\mathcal{L}}$. The original formulation of Levenshtein’s reconstruction problem requires that all the outputs from the channels are distinct. However, different channels may produce the same output word even when different errors occur. In this paper, we investigate two generalized reconstruction models where the channels are allowed to produce the same output word as long as, in each channel, different errors occur (that is, the errors correspond to different insertion vectors). Our objective is to determine the number of channels N required to uniquely recover the transmitted word x under these conditions. We present several results in this direction, some of which are optimal.