Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions

We consider the problem of designing low-redundancy codes in settings where one must correct deletions conjunction with substitutions or adjacent transpositions; a combination errors that is usually observed DNA-based data storage. One most basic versions this was settled more than 50 years ago by Levenshtein, who proved binary Varshamov-Tenengolts arbitrary edit error, i.e., deletion substitution, nearly optimal redundancy. However, approach fails to extend many simple and natural variations single-edit error setting. In work, we make progress on code design above three such variations: 1) construct linear-time encodable decodable length- $n$ non-binary correcting single redundancy notation="LaTeX">$\log n+O(\log \log n)$ , providing an alternative simpler proof result Cai et al. (IEEE Trans. Inf. Theory 2021). This achieved employing what call weighted VT sketches, new notion may be independent interest. 2) show existence transposition . 3) list-decodable list-size 2 for substitution notation="LaTeX">$4\log matches Gilbert-Varshamov existential bound up notation="LaTeX">$O(\log additive term.