Unique Insertion Error Patterns in Levenshtein's Reconstruction Problem

Levenshtein's sequence reconstruction model plays an essential role in information retrieval of advanced memory systems, such as the DNA-based storage systems. In the model, a word $\mathbf{x}\in\mathbb{Z}_q^n$ is transmitted through $N$ noisy channels, and the goal is to recover it. Errors occurring in the channels usually involve substitutions, insertions and deletions. Our focus is on insertions. One of the main questions in this context is determining the minimum number of channels $N$ required to recover the transmitted word $\mathbf{x}$. The original formulation of the reconstruction problem requires that all the output words from the channels are distinct. However, different insertion errors may lead to the same output words. In this paper, we investigate two reconstruction models where the channels are allowed to produce identical output words even though different insertion errors occur in the channels. These two models, called \textit{the multiset model} and \textit{non-multiset model}, generalize the Levenshtein's model. We denote the minimum number of channels required to \textit{unambiguously} recover the transmitted word $\mathbf{x}\in\mathbb{Z}_q^n$ by $N_q^m(n,t)+1$ in the multiset model and $N_q^{nm}(n,t)+1$ in the non-multiset model, where $t$ is the exact number of insertions occurring in a channel. We determine $N_q^m(n,1)$ and $N_q^{nm}(n,1)$ for all $n$ and $q$, and show the somewhat surprising fact that $N_q^m(n,1)=N_q^{nm}(n,1)$. We also provide a full characterization of the words attaining this value and give a general lower bound on $N_q^m(n,t)$ for $t\ge1$ and a recursive upper bound. For $t=1$, we construct codes $C'\subseteq\mathbb{Z}_q^{n+2}$ from codes $C\subseteq\mathbb{Z}_q^n$ such that the number of channels required to determine the transmitted word $\mathbf{x}\in C'$ is small. This construction is shown to be optimal for certain parameters.