Covering codes for insertions and deletions arise naturally in the study of synchronization errors and differ substantially from their classical counterparts in the Hamming metric. In this paper, we study covering codes under insertion and deletion operations. We first show that, in contrast to the equivalence between insertion and deletion correction, insertion covering and deletion covering are not equivalent. We then develop bounds and constructions for insertion and deletion covering codes, with particular emphasis on the large-alphabet regime. For insertion covering codes, we extend a recent combinatorial approach for single insertions and establish a new lower bound for arbitrary fixed insertion radius. For deletion covering codes, we relate the problem to hypergraph covering and prove that the elementary counting lower bound is asymptotically tight when the alphabet size tends to infinity. We further provide a construction of asymptotically optimal non-binary single-deletion covering codes by using differential Varshamov--Tenengolts (VT) codes together with a completion argument. In addition, we study covering codes for burst deletions. We prove that binary differential VT codes are not only capable of correcting two-burst deletions but also have the corresponding covering property, and hence form binary perfect codes for two-burst deletions. Finally, we extend this construction to non-binary alphabets and obtain explicit $q$-ary two-burst-deletion covering codes.

翻译：插入和删除操作下的覆盖码自然产生于同步错误的研究中，且与汉明度量下的经典覆盖码有显著差异。本文研究了插入与删除操作下的覆盖码。我们首先证明，与插入纠正和删除纠正之间的等价性不同，插入覆盖与删除覆盖并不等价。随后，我们发展了插入覆盖码与删除覆盖码的界及构造方法，特别关注大字母表情形。对于插入覆盖码，我们扩展了近期关于单次插入的组合方法，并为任意固定插入半径建立了新的下界。对于删除覆盖码，我们将该问题与超图覆盖关联，并证明当字母表大小趋于无穷时，初等计数下界是渐近紧的。进一步地，我们利用微分Varshamov–Tenengolts (VT)码结合补全论证，构造了渐近最优的非二进制单次删除覆盖码。此外，我们研究了突发删除的覆盖码。证明二进制微分VT码不仅能纠正两次突发删除，还具有相应的覆盖性质，从而构成二进制两次突发删除的完美码。最后，我们将此构造推广至非二进制字母表，得到了显式的$q$元两次突发删除覆盖码。