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arXiv · 2601.06503

Some New Results on Sequence Reconstruction Problem for Deletion Channels

Xiang Wang·Weijun Fang·Han Li·Fang-Wei Fu
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Abstract

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 $n \geq 13$.

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Xiang Wang, Weijun Fang, Han Li, Fang-Wei Fu. 2026-01-10. Some New Results on Sequence Reconstruction Problem for Deletion Channels. https://arxiv.org/abs/2601.06503

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