This paper is motivated by the problem of error-free communication through the i.i.d. duplication channel which acts on the transmitted sequence by inserting a random number of duplicates of each symbol next to the original symbol. A construction of optimal codes correcting all patterns of errors of this type is described, and the zero-error capacity of the duplication channel is determined. A generalization of these results to channel models in which blocks of @math consecutive symbols are being duplicated, and which are inspired by DNA-based data storage systems where the stored molecules are subject to tandem-duplication mutations, is also given.
The capacity of the @math -ary @math -duplication channel was determined in @cite_7 . We should note, however, that for @math , which is the main motivating model in the present paper, the zero-error capacity of the @math -duplication channel is trivially @math , so the mentioned results from @cite_7 are only interesting for @math . The zero-error capacity of a model that can be seen as the version of the @math -duplication channel was determined in @cite_3 .
@cite_3 @cite_7
2884699236 2888898008
Motivated by the error behavior in the DNA storage channel, in this paper, we extend the previously studied sequence reconstruction problem by Levenshtein. The reconstruction problem studies the model in which the information is read through multiple noisy channels, and the decoder, which receives all channel estimations, is required to decode the information. For the combinatorial setup, the assumption is that all the channels cause at most some @math errors. Levenshtein considered the case in which all the channels have the same behavior, and we generalize this model and assume that the channels are not identical. Thus, different channels may cause different maximum numbers of errors. For example, we assume that there are @math channels, which cause at most @math or @math errors, where @math t_ 1 @math pN @math 0 . If the information codeword belongs to a code with minimum distance @math , the problem is then to find the minimum number of channels that guarantees successful decoding in the worst case. A different problem we study in this paper is where the number of channels is fixed, and the question is finding the minimum distance @math that provides exact reconstruction. We study these problems and show how to apply them for the cases of substitutions and transpositions. A (tandem) duplication of length @math is an insertion of an exact copy of a substring of length @math 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 @math . Furthermore, we derive bounds on the cardinality of optimal @math -ary codes correcting up to @math duplications of length @math and establish the following corollaries in the asymptotic regime of growing block length: 1) the presented family of codes is optimal for every @math , in the sense of the asymptotic scaling of code redundancy; 2) the upper bound, when specialized to @math , @math , improves upon Levenshtein’s bound for every @math ; and 3) the bounds coincide for @math , thus yielding the exact asymptotic behavior of the size of optimal single-duplication-correcting codes.