Sequence Reconstruction for the Single-Deletion Single-Substitution Channel

In this work, we study the sequence reconstruction problem for the single-deletion single-substitution channel, assuming that the transmitted sequence belongs to a $q$-ary code with minimum Hamming distance at least 2, where $q \geq 2$ is any fixed integer. Specifically, we prove that for any two $q$-ary sequences of length $n$ and with Hamming distance $d \geq 2$, the size of the intersection of their error balls is upper bounded by $2 q n-3 q-2-\delta_{q, 2}$, where $\delta_{i, j}$ is the Kronecker delta. We also prove the tightness of this bound by constructing two sequences whose error ball intersection size achieves this bound.