Detecting Deletions and Insertions in Concatenated Strings with Optimal Redundancy

TL;DR

This paper develops optimal coding schemes for detecting deletions and insertions in concatenated binary strings, providing theoretical bounds and constructions that are asymptotically optimal for deletion detection and effective for small insertion counts.

Contribution

It introduces new optimal codes for deletion detection and constructs insertion-detecting codes, with proofs of optimality and practical decoding capabilities.

Findings

Optimal deletion-detecting codes with asymptotic redundancy bounds

Construction of insertion-detecting codes for up to 2 insertions

Proofs of optimality among block-by-block decodable codes

Abstract

We study codes that can detect the exact number of deletions and insertions in concatenated binary strings. We construct optimal codes for the case of detecting up to $\del$ deletions. We prove the optimality of these codes by deriving a converse result which shows that the redundancy of our codes is asymptotically optimal in $\del$ among all families of deletion detecting codes, and particularly optimal among all block-by-block decodable codes. For the case of insertions, we construct codes that can detect up to $2$ insertions in each concatenated binary string.