Sequence Reconstruction over the Deletion Channel

TL;DR

This paper investigates the minimum number of distinct outputs needed from a deletion channel to reliably reconstruct the original binary sequence among multiple candidates, extending understanding of sequence reconstruction limits.

Contribution

It precisely characterizes the maximum intersection size of multiple deletion balls, advancing the theoretical understanding of sequence reconstruction over deletion channels.

Findings

Determines the maximum intersection size of deletion balls for multiple sequences.

Provides bounds on the number of outputs needed for reliable reconstruction.

Extends previous results to the case of multiple candidate sequences.

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 $t \geq 1$.