On the maximum number of non-confusable strings evolving under short tandem duplications

The set of all q-ary strings that do not contain repeated substrings of length $\leqslant3$ (i.e., that do not contain substrings of the form aa, abab,andabcabc) constitutes a code cor recting an arbitrary number of tandem-duplication mutations of length $\leqslant3$. 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 $\leqslant3$. We demonstrate that this code is asymptotically optimal in terms of rate, meaning that it represents the largest set of non-con fusable 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.