Let $\mathcal{B}(\cdot)$ be an error ball function. A set of $q$-ary sequences of length $n$ is referred to as an \emph{$(n,q,N;\mathcal{B})$-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 is either the \emph{$t$-deletion ball} $\mathcal{D}_t(\cdot)$ or the \emph{$t$-insertion ball} $\mathcal{I}_t(\cdot)$. Firstly, we establish a fundamental connection between reconstruction codes for deletions and insertions. 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 $\rho(n,q,N;\mathcal{D}_t)\leq \rho(n,q,N;\mathcal{I}_t)$. Then, we identify 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 $\rho(n,q,\tfrac{2(q-1)^{t-1}}{q^{t-1}(t-1)!}n^{t-1}+O(n^{t-2});\mathcal{D}_t)=O(1)$, which disproves a conjecture posed in \cite{Chrisnata-22-IT}. For insertions, we show that $\rho(n,q,\tfrac{(q-1)^{t-1}}{(t-1)!}n^{t-1}+O(n^{t-2});\mathcal{I}_t)=\log\log n + O(1)$, which extends a key result from \cite{Ye-23-IT}. Finally, we construct $(n,q,N;\mathcal{B})$-reconstruction codes, where $\mathcal{B}\in \{\mathcal{D}_2,\mathcal{I}_2\}$, for $N \in \{2,3, 4, 5\}$ and establish 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 in \cite{Sun-23-IT}.

翻译：令 $\mathcal{B}(\cdot)$ 表示一个误差球函数。一个长度为 $n$ 的 $q$ 元序列集合被称为 \emph{$(n,q,N;\mathcal{B})$-重构码}，如果该集合中的每个序列 $\boldsymbol{x}$ 都可以从其误差球 $\mathcal{B}(\boldsymbol{x})$ 中的任意 $N$ 个不同元素唯一地重构出来。该领域的主要目标是确定或建立 $(n,q,N;\mathcal{B})$-重构码的最小冗余度（记为 $\rho(n,q,N;\mathcal{B})$）的界限。在本文中，我们研究误差球为 \emph{$t$-删除球} $\mathcal{D}_t(\cdot)$ 或 \emph{$t$-插入球} $\mathcal{I}_t(\cdot)$ 的重构码。首先，我们建立了删除与插入重构码之间的一个基本联系。对于任意正整数 $n,t,q,N$，任何 $(n,q,N;\mathcal{I}_t)$-重构码同时也是 $(n,q,N;\mathcal{D}_t)$-重构码。这导致了不等式 $\rho(n,q,N;\mathcal{D}_t)\leq \rho(n,q,N;\mathcal{I}_t)$。接着，当 $N=O(n^{t-1})$ 且 $t\geq 2$ 时，我们指出了删除与插入重构码之间的一个重要差异。对于删除，我们证明了 $\rho(n,q,\tfrac{2(q-1)^{t-1}}{q^{t-1}(t-1)!}n^{t-1}+O(n^{t-2});\mathcal{D}_t)=O(1)$，这推翻了 \cite{Chrisnata-22-IT} 中提出的一个猜想。对于插入，我们证明了 $\rho(n,q,\tfrac{(q-1)^{t-1}}{(t-1)!}n^{t-1}+O(n^{t-2});\mathcal{I}_t)=\log\log n + O(1)$，这扩展了 \cite{Ye-23-IT} 中的一个关键结果。最后，我们针对 $\mathcal{B}\in \{\mathcal{D}_2,\mathcal{I}_2\}$ 以及 $N \in \{2,3, 4, 5\}$ 构造了 $(n,q,N;\mathcal{B})$-重构码，并分别建立了最小冗余度 $\rho(n,q,N;\mathcal{B})$ 的上界：$3\log n+O(\log\log n)$、$3\log n+O(1)$、$2\log n+O(\log\log n)$ 和 $\log n+O(\log\log n)$。这推广了 \cite{Sun-23-IT} 中先前建立的结果。