Some New Results on Sequence Reconstruction Problem for Deletion Channels

TL;DR

This paper establishes a tight lower bound for the maximum intersection size of two metric balls in the sequence reconstruction problem for deletion channels, resolving an open question for specific parameters.

Contribution

It provides a new lower bound on N(n,3,t) for n ≥ max{13,t+8} and proves tightness for t=4, confirming a conjecture by Pham, Goyal, and Kiah.

Findings

Lower bound on N(n,3,t) for n ≥ max{13,t+8} and t ≥ 4

Proof that the bound is tight for t=4

Confirmed the exact value N(n,3,4)=20n-166 for all n ≥ 13

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 $n \geq 13$.