Sequence reconstruction problem for deletion channels: A complete asymptotic solution

Transmit a codeword

, 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 (2001) [10], proposed the problem of determining $N(n,\ell ,t)+1$, the minimum number of distinct channel outputs required to uniquely reconstruct

. 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 ℓ and t. Specifically, we show that $N(n,\ell ,t)=\frac{\left(\begin{array}{c}2\ell \\ \ell \end{array}\right)}{(t-\ell )!}{n}^{t-\ell }-O\left(n^{t-\ell -1}\right)$. In the special instances: where $\ell =t$, we show that $N(n,\ell ,\ell )=\left(\begin{array}{c}2\ell \\ \ell \end{array}\right)$; and when $\ell =3$ and $t=4$, we show that $N(n,3,4)\le 20n-150$. We also provide a conjecture on the exact value of $N(n,\ell ,t)$ for all values of n, ℓ, and t.