Reconstruction Codes for Deletions and Insertions: Connection, Distinction, and Construction

Abstract

Let $\mathcal {B}(\cdot)$ be an error ball function. A set of $q$ -ary sequences of length $n$ is referred to as an $(n,q,N;\mathcal {B})$ -reconstruction code if each sequence $\boldsymbol {x}$ within this set can be uniquely reconstructed from any $N$ distinct elements within its error ball $\mathcal {B}(\boldsymbol {x})$ . The main objective in this area is to determine or establish bounds for the minimum redundancy of $(n,q,N;\mathcal {B})$ -reconstruction codes, denoted by $\rho (n,q,N;\mathcal {B})$ . In this paper, we investigate reconstruction codes where the error ball corresponds to either the $t$ -deletion ball $\mathcal {D}_{t}(\cdot)$ or the $t$ -insertion ball $\mathcal {I}_{t}(\cdot)$ . Our primary technical contributions include: 1) establishing a fundamental connection between reconstruction codes for deletions and insertions. Specifically, for any positive integers $n,t,q,N$ , any $(n,q,N;\mathcal {I}_{t})$ -reconstruction code is also an $(n,q,N;\mathcal {D}_{t})$ -reconstruction code. This leads to the inequality $\rho (n,q,N;\mathcal {D}_{t})\leq \rho (n,q,N;\mathcal {I}_{t})$ ; 2) identifying a significant distinction between reconstruction codes for deletions and insertions when $N=O(n^{t-1})$ and $t\geq 2$ . For deletions, we prove that there exists some constant $K$ such that $\rho \left ({ n,q,\tfrac {2(q-1)^{t-1}}{q^{t-1}(t-1)!}n^{t-1}+K n^{t-2};\mathcal {D}_{t}}\right)=O(1)$ , which disproves a conjecture posed by Chrisnata et al. (2022). In contrast, for insertions, we show that there exists some constant $K'$ such that $\rho \left ({ n,q,\tfrac {(q-1)^{t-1}}{(t-1)!}n^{t-1}+K' n^{t-2};\mathcal {I}_{t}}\right)=\log \log n + O(1)$ , which extends a key result by Ye et al. (2023); and 3) constructing $(n,q,N;\mathcal {B})$ -reconstruction codes, where $\mathcal {B}\in \{\mathcal {D}_{2},\mathcal {I}_{2}\}$ , for $N \in \{2,3, 4, 5\}$ and establishing respective upper bounds of $3\log n+O(\log \log n)$ , $3\log n+O(1)$ , $2\log n+O(\log \log n)$ and $\log n+O(\log \log n)$ on the minimum redundancy $\rho (n,q,N;\mathcal {B})$ . This generalizes results previously established by Sun et al. (2023).