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 $n \geq 13$.

The sequence reconstruction problem was proposed by Levenshtein in 2001. In this model, a sequence from a code is transmitted over several channels, and the decoder receives the distinct outputs from each channel. The main problem is to determine the minimum number of channels required to reconstruct the transmitted sequence. In the combinatorial context, the sequence reconstruction problem is equivalent to finding the value ofNq(n,d,t), defined as the size of the largest intersection of two metric balls of radiust, where the distance between their centers is at leastdand the sequences areq-ary sequences of lengthn. Levenshtein first discussed this problem in the uncoded sequence setting and determined the value ofNq(n, 1, t)for anyn≥t. Moreover, Gabrys and Yaakobi studied this problem in the context of binary one-deletion-correcting codes and determined the value ofN2(n, 2, t)fort≥ 2. In this paper we study this problem for 3-ary sequences of lengthnover the deletion channel, where the transmitted sequence belongs to a one-deletion-correcting code and there aretdeletions in every channel. Specifically, we determineN3(n, 2, t)fort≥ 2.