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

TL;DR

This paper extends Levenshtein's binary insertion-correcting codes to channels with fixed-length duplication errors over arbitrary alphabets, deriving bounds and establishing optimality in the asymptotic regime for various parameters.

Contribution

It generalizes Levenshtein's construction to arbitrary alphabets and duplication lengths, providing bounds and proving asymptotic optimality of the codes.

Findings

Codes are optimal for all parameters in the asymptotic limit.

Improved upper bounds over previous results for certain parameters.

Exact asymptotic behavior established for single-duplication correction.

Abstract

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.