On the Sequence Reconstruction Problem for the Single-Deletion Two-Substitution Channel

Abstract: The Levenshtein sequence reconstruction problem studies the reconstruction of a transmitted sequence from multiple erroneous copies of it. A fundamental question in this field is to determine the minimum number of erroneous copies required to guarantee correct reconstruction of the original sequence. This problem is equivalent to determining the maximum possible intersection size of two error balls associated with the underlying channel. Existing research on the sequence reconstruction problem has largely focused on channels with a single type of error, such as insertions, deletions, or substitutions alone. However, relatively little is known for channels that involve a mixture of error types, for instance, channels allowing both deletions and substitutions. In this work, we study the sequence reconstruction problem for the single-deletion two-substitution channel, which allows one deletion and at most two substitutions applied to the transmitted sequence. Specifically, we prove that if two $q$-ary length-$n$ sequences have the Hamming distance $d\geq 2$, where $q\geq 2$ is any fixed integer, then the intersection size of their error balls under the single-deletion two-substitution channel is upper bounded by $(q^2-1)n^2-(3q^2+5q-5)n+O_q(1)$, where $O_q(1)$ is a constant independent from $n$ but dependent on $q$. Moreover, we show that this upper bound is tight up to an additive constant.