Some New Results on Sequence Reconstruction Problem for Deletion Channels
Xiang Wang, Weijun Fang, Han Li, Fang-Wei Fu
TL;DR
This paper studies the sequence reconstruction problem over binary deletion channels by analyzing $N(n,d,t)$, the maximum intersection of two deletion balls of radius $t$ for sequences at Levenshtein distance at least $d$. It first derives a concrete lower bound $N(n,3,t)\geq M(n,t)$ for all $n\geq 13$ and $t\geq 4$, where $M(n,t)$ is a linear combination of deletion-ball counts $D(\cdot,\cdot)$; notably, for $t=4$ this bound is tight, yielding $N(n,3,4)=20n-166$ for all $n\geq 13$, achieved by explicit constructions with $d_L(\mathbf{x},\mathbf{y})\ge 3$. The upper bound argument shows $|D_4(\mathbf{x})\cap D_4(\mathbf{y})|\le 20n-166$ for all $n\ge 13$ and $d_L(\mathbf{x},\mathbf{y})\ge 3$, verified for the base cases $n=13,14$ by computer and completed by an induction over structured subcases (a)–(f) aided by several technical lemmas. Together, these results establish the exact value $N(n,3,4)=20n-166$ for all $n\ge 13$ and outline a general conjecture for $t\ge 5$ that the same $M(n,t)$ governs $N(n,3,t)$ asymptotically, with small-$n$ instances also cataloged. The work advances understanding of deletion-channel reconstruction bounds and informs design of deletion-correcting constructs in binary sequences.
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 $n\geq 13$ 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 $n \geq 13$.