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 $\leqslant! 3$ (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 $\leqslant! 3$. 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 $\leqslant! 3$. 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.