On Levenshtein Balls with Radius One

TL;DR

This paper analyzes the size of Levenshtein balls with radius one under the Fixed Length Levenshtein metric, which is relevant for DNA storage error correction, by determining their minimum, maximum, and average sizes.

Contribution

It provides the first detailed analysis of the minimal, maximal, and average sizes of Levenshtein balls with radius one in the FLL metric, including related anticode sizes.

Findings

Minimum, maximum, and average sizes of Levenshtein balls with radius one are characterized.

The sizes depend on the number of runs and alternating segment lengths of the words.

Related sizes of maximal anticodes with diameter one are also computed.

Abstract

The rapid development of DNA storage has brought the deletion and insertion channel, once again, to the front line of research. When the number of deletions is equal to the number of insertions, the Fixed Length Levenshtein (FLL) metric is the right measure for the distance between two words of the same length. The size of a ball is one of the most fundamental parameters in any metric. The size of the ball with radius one in the FLL metric depends on the number of runs and the length of the alternating segments of the given word. In this work, we find the minimum, maximum, and average size of a ball with radius one, in the FLL metric. The related minimum and maximum sizes of a maximal anticode with diameter one are also calculated.