Asymptotically Optimal Codes Correcting Fixed-Length Duplication Errors in DNA Storage Systems

A (tandem) duplication of length $ k $ is an insertion of an exact copy of a substring of length $ k $ next to its original position. This and related types of impairments are of relevance in modeling communication in the presence of synchronization errors, as well as in several information storage applications. We demonstrate that Levenshtein's construction of binary codes correcting insertions of zeros is, with minor modifications, applicable also to channels with arbitrary alphabets and with duplication errors of arbitrary (but fixed) length $ k $. Furthermore, we derive bounds on the cardinality of optimal $ q $-ary codes correcting up to $ t $ duplications of length $ k $, and establish the following corollaries in the asymptotic regime of growing block-length: 1.) the presented family of codes is optimal for every $ q, t, k $, in the sense of the asymptotic scaling of code redundancy; 2.) the upper bound, when specialized to $ q = 2 $, $ k = 1 $, improves upon Levenshtein's bound for every $ t \geq 3 $; 3.) the bounds coincide for $ t = 1 $, thus yielding the exact asymptotic behavior of the size of optimal single-duplication-correcting codes.