Codes Correcting All Patterns of Tandem-Duplication Errors of Maximum Length 3

11/15/2019

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by   Mladen Kovačević, et al.

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The set of all q-ary strings that do not contain repeated substrings of length ≤ℓ forms a code correcting all patterns of tandem-duplication errors of length ≤ℓ, when ℓ∈{1, 2, 3}. For ℓ∈{1, 2}, this code is also known to be optimal in terms of asymptotic rate. The purpose of this paper is to demonstrate asymptotic optimality for the case ℓ = 3 as well, and to give the corresponding characterization of the zero-error capacity of the (≤ 3)-tandem-duplication channel. This settles the zero-error problem for (≤ℓ)-tandem-duplication channels in all cases where duplication roots of strings are unique.