An Improvement to Levenshtein's Upper Bound on the Cardinality of Deletion Correcting Codes

TL;DR

This paper generalizes Levenshtein's upper bound on the size of deletion-correcting codes over q-ary alphabets by considering mixed insertion and deletion channels, resulting in improved bounds for certain parameters.

Contribution

It introduces a new asymptotic upper bound for deletion-correcting codes that accounts for mixed insertions and deletions, extending prior bounds to broader scenarios.

Findings

The new bound improves upon Levenshtein's bound when deletions exceed the alphabet size.

The bounds vary depending on the mixture of insertions and deletions considered.

The work recovers known bounds as special cases for pure insertion or deletion channels.

Abstract

We consider deletion correcting codes over a q-ary alphabet. It is well known that any code capable of correcting s deletions can also correct any combination of s total insertions and deletions. To obtain asymptotic upper bounds on code size, we apply a packing argument to channels that perform different mixtures of insertions and deletions. Even though the set of codes is identical for all of these channels, the bounds that we obtain vary. Prior to this work, only the bounds corresponding to the all insertion case and the all deletion case were known. We recover these as special cases. The bound from the all deletion case, due to Levenshtein, has been the best known for more than forty five years. Our generalized bound is better than Levenshtein's bound whenever the number of deletions to be corrected is larger than the alphabet size.