arXiv:2611.16835 · v1

Explicit Edit Codes via Multicoloring

首次提交 2026-11-12 22:08

版本更新 2026-11-12 22:08

雷达观测 2026-11-13 14:15

摘要

For any fixed integers $q\ge2$ and $t\ge1$, we construct explicit $q$-ary codes of length $n$ that correct any combination of at most $t$ insertions, deletions, and substitutions, with redundancy at most $2t\log_q n+O_{q,t}(\log\log n)$ $q$-ary symbols. This matches the leading redundancy term of Levenshtein's classical existential bound. Building on the standard graph-coloring approach, we use a prime-based multicoloring of the edit-confusion graph to assign multiple colors to each $q$-ary word of length $m<n$. Appending a short, edit-protected representation of an assigned color to the length-$m$ word yields a length-$n$ codeword. The choice among the assigned colors carries additional message information, thereby reducing the redundancy. The construction admits deterministic encoding and decoding in $\widetilde O_{q,t}(n^{6t+2})$ and $\widetilde O_{q,t}(n^{7t+2})$ time, respectively.