Construction of optimal codes in deletion and insertion metric

TL;DR

This paper improves the upper bounds for single-deletion correcting codes of length four over even-sized alphabets and constructs an optimal perfect code, demonstrating the bounds' sharpness.

Contribution

It provides a tighter upper bound for such codes and explicitly constructs an optimal perfect code, advancing the understanding of deletion-correcting code limits.

Findings

Improved Levenshtein's upper bound for length-four codes

Constructed an optimal perfect single-deletion correcting code

Showed the new upper bound is sharp

Abstract

We improve Levenshtein's upper bound for the cardinality of a code of length four that is capable of correcting single deletions over an alphabet of even size. We also illustrate that the new upper bound is sharp. Furthermore we construct an optimal perfect code that is capable of correcting single deletions for the same parameters.