Binary Reconstruction Codes for Correcting One Deletion and One Substitution

In this paper, we investigate binary reconstruction codes capable of correcting one deletion and one substitution. We define the \emph{single-deletion single-substitution ball} function $\mathcal{B}$ as a mapping from a sequence to the set of sequences that can be derived from it by performing one deletion and one substitution. A binary \emph{$(n,N;\mathcal{B})$-reconstruction code} is defined as a collection of binary sequences of length $n$ such that the intersection size between the single-deletion single-substitution balls of any two distinct codewords is strictly less than $N$. This property ensures that each codeword can be uniquely reconstructed from $N$ distinct elements in its single-deletion single-substitution ball. Our main contribution is to demonstrate that when $N$ is set to $4n - 8$, $3n - 4$, $2n+9$, $n+21$, $31$, and $7$, the redundancy of binary $(n,N;\mathcal{B})$-reconstruction codes can be $0$, $1$, $2$, $\log\log n + 3$, $\log n + 1$, and $3\log n + 4$, respectively, where the logarithm is on base two.