Asymptotically Optimal Codes for Correcting Burst Deletions and Insertions in Labeled DNA Sequences

Fluorescent labeling is a cornerstone of DNA visualization and a key enabler of random access in DNA-based data storage. However, the stochastic nature of biochemical processes, including synthesis, hybridization, and optical readout, induces \emph{burst} synchronization errors within the resulting labeling sequences. To address this critical challenge, we formally introduce \emph{burst $t$-deletion/insertion $\mathcal{A}$-labeling codes,} designed to correct a single burst of $t$ deletions or insertions in the label domain. Our contributions are threefold. \begin{itemize} \item $\mathbf{\text{Fundamental}}\mathbf{\text{limit.}}$ We establish an information-theoretic lower bound of ${\log }_{4}n+\mathcal{O}(1)$ on the redundancy of any such code for all $t\ge 1$ with $t\mid n$. To the best of our knowledge, this resolves the first information-theoretic lower bound even for the single-error case $t=1$. \item $\mathbf{\text{Explicit}}\mathbf{\text{construction.}}$ For $t\ge 2$, $t\mid n$, and $n\ge 7t+3$, we propose explicit encoding and decoding algorithms, both running in $\mathcal{O}(n^2)$ time. A novel generalized Run-Length Limited (RLL) constraint is introduced to bridge the structural mismatch between the DNA encoding domain and the label error domain. \item $\mathbf{\text{Asymptotic}}\mathbf{\text{optimality.}}$ The proposed scheme achieves redundancy ${\log }_{4}n+(t-1){\log }_4{\log }_{8\mathrm{/}3}n+\mathcal{O}(1)$, matching the dominant term of the lower bound up to a small $\mathcal{O}(\log \log n)$ overhead, rendering the construction asymptotically optimal for fixed $t$. \end{itemize}