On Unique Error Patterns in the Levenshtein's Sequence Reconstruction Model

TL;DR

This paper investigates error patterns in Levenshtein's sequence reconstruction, focusing on cases where channels may produce identical outputs, and introduces an efficient decoding algorithm for complex errors in DNA-related applications.

Contribution

It extends the Levenshtein sequence reconstruction model to scenarios with non-unique outputs and develops a Las Vegas decoding algorithm for insertion, deletion, and substitution errors in DNA-inspired alphabets.

Findings

Properties and optimal results for non-unique output error patterns

Introduction of an efficient decoding algorithm for complex errors

Relevance to DNA-memory applications

Abstract

In the Levenshtein's sequence reconstruction problem a codeword is transmitted through $N$ channels and in each channel a set of errors is introduced to the transmitted word. In previous works, the restriction that each channel provides a unique output word has been essential. In this work, we assume only that each channel introduces a unique set of errors to the transmitted word and hence some output words can also be identical. As we will discuss, this interpretation is both natural and useful for deletion and insertion errors. We give properties, techniques and (optimal) results for this situation. Quaternary alphabets are relevant due to applications related to DNA-memories. Hence, we introduce an efficient Las Vegas style decoding algorithm for simultaneous insertion, deletion and substitution errors in $q$-ary Hamming spaces for $q \geq 4$.