Optimal Reverse-Complement-Duplication Error-Correcting Codes

Motivated by DNA storage in living organisms and inspired by biological mutation processes, this study explores the reverse-complement string-duplication system. We commence our investigation by introducing an optimal $q$-ary reverse-complement-duplication code construction for duplication length $1$ and any number of duplications, achieving a size of $\Theta(q^n)$. Subsequently, we establish a fundamental limitation, proving that for duplication lengths greater than $1$, all reverse-complement-duplication codes correcting any number of duplications possess a size of $o(q^n)$. Further, we present a construction of reverse-complement-duplication codes with a duplication length of $2$, demonstrating a redundancy of at most $\log_q(n/2) + \log_q(\log_q(n)+1) + 2 + \log_q(3)$. Finally, we contribute an explicit construction for $q$-ary codes addressing a single classical tandem duplication for any $k$. The redundancy of these codes is $\log_q(n/k) + 1 + (k-1)\log_q(\log_q(2n/k)+1)$.