In recent years, the emergence of DNA storage systems has led to a widespread interest in codes correcting insertions, deletions, and classic substitutions. Levenshtein discovered that the VT codes are capable of correcting single insertion/deletion and then extended the VT construction to single-insertion/deletion/substitution correcting codes. Inspired by this, we employ the higher-order VT syndromes, which were initially introduced for 2-insertion/deletion correction, to construct 2-insertion/deletion/substitution correcting codes with redundancy 6 log2n+8. Our key technical contributions include the formalization of sign-preserving number as a core concept in applying higher-order VT syndromes and the development of its analytical framework.

In this paper, we provide non-asymptotic upper bounds on the size and average size of s-insertion s-deletion balls. As a corollary, we conclude an explicit non-asymptotic lower bound on the size of optimal s-insertionldeletion correcting code which is a strict improvement of Levenshtein's lower bound given in 2002. Particularly, in the case of single insertionldeletion, comparing with Levenshtein's bound, we reduce the gap to the maximum potential lower bound by at least 1/3.