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.

翻译：莱文斯坦序列重构问题研究如何从多个包含错误的副本中重构出原始传输序列。该领域的一个基本问题是确定保证正确重构原始序列所需的最小错误副本数量。该问题等价于确定与底层信道相关的两个错误球的最大可能交集大小。现有关于序列重构问题的研究主要集中于仅包含单一类型错误的信道，例如仅包含插入、删除或替换操作的信道。然而，对于涉及多种错误类型混合的信道（例如同时允许删除和替换操作的信道），相关研究相对较少。本文研究了单删除双替换信道下的序列重构问题，该信道允许对传输序列施加一次删除操作和至多两次替换操作。具体而言，我们证明了若两个$q$元长度为$n$的序列之间的汉明距离$d\geq 2$（其中$q\geq 2$为任意固定整数），则它们在单删除双替换信道下的错误球交集大小上界为$(q^2-1)n^2-(3q^2+5q-5)n+O_q(1)$，其中$O_q(1)$是与$n$无关但依赖于$q$的常数。此外，我们证明了该上界在加法常数意义下是紧致的。