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.