Error-Correcting Codes for Two Bursts of t1-Deletion-t2-Insertion with Low Computational Complexity

Burst errors involving simultaneous insertions, deletions, and substitutions occur in practical scenarios, including DNA data storage and document synchronization, motivating developments of channel codes that can correct such errors. In this paper, we address the problem of constructing error-correcting codes (ECCs) capable of handling multiple bursts of $t_1$-deletion-$t_2$-insertion ($(t_1,t_2)$-DI) errors, where each burst consists of $t_1$ deletions followed by $t_2$ insertions in a binary sequence. We make three key contributions: Firstly, we establish the fundamental equivalence of (1) two bursts of $(t_1,t_2)$-DI ECCs, (2) two bursts of $(t_2,t_1)$-DI ECCs, and (3) one burst each of $(t_1,t_2)$-DI and $(t_2,t_1)$-DI ECCs. Then, we derive lower and upper bounds on the code size of two bursts of $(t_1,t_2)$-DI ECCs, which can naturally be extended to the case of multiple bursts. Finally, we present constructions of two bursts of $(t_1,t_2)$-DI ECCs. Compared to the codes obtained by the syndrome compression technique, the resulting codes achieve significantly lower computational complexity.