Asymptotically Optimal Codes Correcting One Substring Edit

The substring edit error is the operation of replacing a substring
$\boldsymbol{u}$
of
$\boldsymbol{x}$
with another string
$\boldsymbol{v}$
, where the lengths of
$\boldsymbol{u}$
and
$\boldsymbol{v}$
are bounded by a given constant
$k$
. It encompasses localized insertions, deletions, and substitutions within a window. Codes correcting one substring edit have redundancy at least
$\log n+k$
. In this paper, we construct codes correcting one substring edit with redundancy
$\log n+O(\log\log n)$
, which is asymptotically optimal. The full version of this paper is available online.
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