On the Maximum Number of Non-Confusable Strings Evolving under Short Tandem Duplications

Abstract

The set of all \(q\)-ary strings that do not contain repeated substrings of length \({\le\! 3}\) (i.e., that do not contain substrings of the form \(a a\), \(a b a b\), and \(a b c a b c\)) constitutes a code correcting an arbitrary number of tandem-duplication mutations of length \({\le\! 3}\). In other words, any two such strings are non-confusable in the sense that they cannot produce the same string while evolving under tandem duplications of length \({\le\! 3}\). We demonstrate that this code is asymptotically optimal in terms of rate, meaning that it represents the largest set of non-confusable strings up to subexponential factors. This result settles the zero-error capacity problem for the last remaining case of tandem-duplication channels satisfying the “root-uniqueness” property.