Sequence Reconstruction Problem for Deletion Channels: A Complete Asymptotic Solution

TL;DR

This paper provides an asymptotic solution for the minimum number of distinct outputs needed to uniquely reconstruct codewords transmitted over deletion channels, extending prior results to all parameter ranges.

Contribution

It offers the first asymptotically exact formula for the reconstruction number for all deletion parameters, generalizing previous partial results.

Findings

Derived an asymptotic formula for N(n,ℓ,t) involving binomial coefficients and polynomial terms.

Established the exact value of N(n,ℓ,ℓ) as a binomial coefficient.

Proposed a conjecture for the precise value of N(n,ℓ,t) for all parameters.

Abstract

Transmit a codeword $x$, that belongs to an $(\ell-1)$-deletion-correcting code of length $n$, over a $t$-deletion channel for some $1\le \ell\le t<n$. Levenshtein, in 2001, proposed the problem of determining $N(n,\ell,t)+1$, the minimum number of distinct channel outputs required to uniquely reconstruct $x$. Prior to this work, $N(n,\ell,t)$ is known only when $\ell\in\{1,2\}$. Here, we provide an asymptotically exact solution for all values of $\ell$ and $t$. Specifically, we show that $N(n,\ell,t)=\binom{2\ell}{\ell}/(t-\ell)! n^{t-\ell} - O(n^{t-\ell-1})$ and in the special instance where $\ell=t$, we show that $N(n,\ell,\ell)=\binom{2\ell}{\ell}$. We also provide a conjecture on the exact value of $N(n,\ell,t)$ for all values of $n$, $\ell$, and $t$.