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 center dot [log(q) n ] + 1. We enhance the error-correcting capability by introducing codes without efficient encoding, leading to an improved value of K = [log(q) n] + phi(n), where phi(n) is an arbitrary function such that phi(n) -> infinity as n -> infinity. 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 = 2k, occurring simultaneously without any mutual interference.