Levenshtein first introduced the sequence reconstruction problem in $2001$. In the realm of combinatorics, the sequence reconstruction problem is equivalent to determining the value of $N(n,d,t)$, which represents the maximum size of the intersection of two metric balls of radius $t$, given that the distance between their centers is at least $d$ and the sequence length is $n$. In this paper, We present a lower bound on $N(n,3,t)$ for $n\geq 13$ and $t \geq 4$. For $t=4$, we prove that this lower bound is tight. This settles an open question posed by Pham, Goyal, and Kiah, confirming that $N(n,3,4)=20n-166$ for all $n \geq 13$.

翻译：Levenshtein 于 2001 年首次提出了序列重构问题。在组合学领域，序列重构问题等价于确定 $N(n,d,t)$ 的值，该值表示在序列长度为 $n$、且两个中心点之间的距离至少为 $d$ 的条件下，两个半径为 $t$ 的度量球交集的最大尺寸。在本文中，我们针对 $n\geq 13$ 和 $t \geq 4$ 给出了 $N(n,3,t)$ 的一个下界。对于 $t=4$，我们证明该下界是紧的。这解决了 Pham、Goyal 和 Kiah 提出的一个公开问题，确认了对于所有 $n \geq 13$，$N(n,3,4)=20n-166$ 成立。