Sequence Reconstruction for the Single-Deletion Single-Substitution Channel

TL;DR

This paper analyzes the minimum number of outputs needed to uniquely reconstruct sequences transmitted over a channel with single-deletion and single-substitution errors, providing bounds on error ball intersections for q-ary codes.

Contribution

It establishes an upper bound on the intersection size of error balls for q-ary sequences with minimum Hamming distance at least 2, and proves the bound's tightness.

Findings

Bound on intersection size: 2qn - 3q - 2 - δ_{q,2}

Constructed sequences achieving the bound

Applicable to sequences with minimum Hamming distance ≥ 2

Abstract

The central problem in sequence reconstruction is to find the minimum number of distinct channel outputs required to uniquely reconstruct the transmitted sequence. According to Levenshtein's work in 2001, this number is determined by the size of the maximum intersection between the error balls of any two distinct input sequences of the channel. In this work, we study the sequence reconstruction problem for single-deletion single-substitution channel, assuming that the transmitted sequence belongs to a $q$-ary code with minimum Hamming distance at least $2$, where $q\geq 2$ is any fixed integer. Specifically, we prove that for any two $q$-ary sequences of length $n$ and with Hamming distance $d\geq 2$, the size of the intersection of their error balls is upper bounded by $2qn-3q-2-\delta_{q,2}$, where $\delta_{i,j}$ is the Kronecker delta. We also prove the tightness of this bound by constructing two sequences the intersection size of whose error balls achieves this bound.