Codes Correcting Long Duplication Errors

Abstract

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.