Recently, codes for correcting a burst of errors have attracted significant attention. One of the most important reasons is that bursts of errors occur in certain emerging techniques, such as DNA storage. In this paper, we investigate a type of error, called a $(t,s)$-burst, which deletes $t$ consecutive symbols and inserts $s$ arbitrary symbols at the same coordinate. Note that a $(t,s)$-burst error can be seen as a generalization of a burst of insertions ($t=0$), a burst of deletions ($s=0$), and a burst of substitutions ($t=s$). Our main contribution is to give explicit constructions of $q$-ary $(t,s)$-burst correcting codes with $\log n + O(1)$ bits of redundancy for any given non-negative integers $t$, $s$, and $q \geq 2$. These codes have optimal redundancy up to an additive constant. Furthermore, we apply our $(t,s)$-burst correcting codes to combat other various types of errors and improve the corresponding results. In particular, one of our byproducts is a permutation code capable of correcting a burst of $t$ stable deletions with $\log n + O(1)$ bits of redundancy, which is optimal up to an additive constant.

翻译：近期，纠正突发错误的编码技术引起了广泛关注。其中一个重要原因是某些新兴技术（如DNA存储）中会频繁出现突发错误。本文研究了一种称为$(t,s)$-突发的新型错误类型，该错误会在同一坐标位置连续删除$t$个符号并插入$s$个任意符号。值得注意的是，$(t,s)$-突发错误可视为插入突发（$t=0$）、删除突发（$s=0$）和替换突发（$t=s$）的泛化形式。我们的主要贡献是：对于任意给定非负整数$t$、$s$及$q \geq 2$，显式构造了冗余度为$\log n + O(1)$比特的$q$进制$(t,s)$-突发纠错码。这些编码的冗余度在加法常数意义下达到最优。此外，我们将所提$(t,s)$-突发纠错码应用于对抗其他多种类型错误，并改进了相应结果。特别地，我们的一个副产品是构建了一种能够纠正$t$次稳定删除突发的置换码，其冗余度为$\log n + O(1)$比特（在加法常数意义下达到最优）。