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 $n\geq \max\{13,t+8\}$ 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：莱文斯坦于2001年首次提出序列重建问题。在组合学领域，序列重建问题等价于确定$N(n,d,t)$的值，该值表示半径为$t$的两个度量球交集的最大尺寸，前提是它们的中心距离至少为$d$且序列长度为$n$。本文中，我们给出了$N(n,3,t)$的一个下界，其中$n\geq \max\{13,t+8\}$且$t \geq 4$。对于$t=4$，我们证明了该下界是紧的。这解决了Pham、Goyal和Kiah提出的一个开放性问题，证实了对所有$n \geq 13$，有$N(n,3,4)=20n-166$。