Polynomially larger deletion codes by linear hashing of substring counts

Abstract

We show that binary codes of length $n$ correcting two deletions exist with redundancy $3\log_2n+O(\log_2\log_2n)$. The previous best upper bound had leading coefficient $4$, unchanged since 1965, while the best known lower bound has coefficient $2$. More generally, codes correcting $t\ge2$ deletions exist with redundancy $(2t-1)\log_2n+O_t(\log_2\log_2n)$, improving the coefficient $2t$. We extract a code from one label class of a random linear hash of substring counts, with about $n$ times fewer labels than a direct construction. Confusable words that still share a label are separated by a two-colouring after discarding the words in components with odd cycles, and these are few because an odd cycle forces the edits along it to overlap.