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

TL;DR

This paper characterizes the largest set of q-ary strings avoiding certain repeated substrings, proving its asymptotic optimality and solving the zero-error capacity problem for a specific tandem-duplication channel case.

Contribution

It introduces a maximal code of non-confusable strings under tandem duplications of length ≤ 3 and proves its asymptotic optimality, resolving a key capacity problem.

Findings

The code avoids substrings a a, a b a b, a b c a b c.

The code is asymptotically optimal in rate.

It solves the zero-error capacity problem for the root-uniqueness tandem-duplication channel.

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.