Codes Correcting Long Duplication Errors

We consider the problem of constructing codes capable of correcting long tandem duplication errors of variable length. We present a subquadratic-complexity algorithm that uses only one symbol of redundancy to encode q-ary length-n words into codewords, which can correct a single duplication of length at least
$K=4\cdot \lceil \log _{q} n\rceil +1$
. We enhance the error-correcting capability by introducing codes without efficient encoding, leading to an improved value of
$K= \lceil \log _{q} n\rceil +\phi (n)$
, where
$\phi (n)$
is an arbitrary function such that
$\phi (n)\to \infty $
as
$n\to \infty $
. In the class of codes correcting a single long duplication with redundancy 1, the value K in our constructions is order-optimal. Finally, k-repeat-free codes, in which every codeword contains any k-tuple at most once, are shown to correct any number of independent long duplications, each of length at least
${K} = 2{k}$
, occurring simultaneously without any mutual interference.