Sequence Reconstruction over the Deletion Channel
Fengxing Zhu
TL;DR
The paper addresses sequence reconstruction over the binary $t$-deletion channel by focusing on the maximal overlap of deletion balls, quantified as $N(n,\ell, t)$. It derives exact closed-form expressions: for $\ell=3$, $N(n,3,t)=D(n-3,t-1)+3D(n-4,t-2)$, and for all $\ell\ge4$, $N(n,\ell,t)=\sum_{i=1}^{\ell-2}D(n-2i,t-i)+2D(n-2(\ell-1),t-(\ell-1))$, with $D(n,t)$ satisfying $D(n,t)=D(n-1,t)+D(n-2,t-1)$ and $D(n,t)=\sum_{i=0}^t \binom{n-t}{i}$. The results are established via constructive lower bounds and induction-based upper bounds, jointly yielding exact intersection sizes. Consequently, the minimum number of distinct channel outputs needed to list-reconstruct the transmitted sequence is $N_{\ell}(n,t)+1$. These findings provide a precise, computable criterion for list-reconstruction performance on deletion channels, with potential impact on redundancy-efficient storage and DNA data storage systems.
Abstract
In this paper, we consider the Levenshtein's sequence reconstruction problem in the case where the transmitted codeword is chosen from $\{0,1\}^n$ and the channel can delete up to $t$ symbols from the transmitted codeword. We determine the minimum number of channel outputs (assuming that they are distinct) required to reconstruct a list of size $\ell-1$ of candidate sequences, one of which corresponds to the original transmitted sequence. More specifically, we determine the maximum possible size of the intersection of $\ell \geq 3$ deletion balls of radius $t$ centered at $x_1, x_2, \dots, x_{\ell}$, where $x_i \in \{0,1\}^n$ for all $i \in \{1,2,\dots,\ell\}$ and $x_i \neq x_j$ for $i \neq j$, with $n \geq t+ \ell-1$ and $t \geq 1$.