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.

翻译：在实际场景中，例如DNA数据存储和文档同步，会出现同时包含插入、删除和替换的突发错误，这推动了能够纠正此类错误的信道码的发展。本文致力于构建能够处理多段$t_1$删除-$t_2$插入（$(t_1,t_2)$-DI）突发错误的纠错码（ECCs），其中每段突发错误由二进制序列中的$t_1$个删除后接$t_2$个插入组成。我们做出了三个关键贡献：首先，我们确立了（1）两段$(t_1,t_2)$-DI ECCs、（2）两段$(t_2,t_1)$-DI ECCs以及（3）一段$(t_1,t_2)$-DI和一段$(t_2,t_1)$-DI ECCs之间的基本等价关系。接着，我们推导了两段$(t_1,t_2)$-DI ECCs码尺寸的下界和上界，这些界可以自然地推广到多段突发错误的情况。最后，我们提出了两段$(t_1,t_2)$-DI ECCs的构造方法。与通过伴随式压缩技术获得的码相比，所构造的码实现了显著降低的计算复杂度。