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.
Notice that, in all of the above-mentioned works with the exception of @cite_3 , no upper bound on the number of inserted duplicates at each position was assumed, i.e., @math is taken to be @math . We find the assumption @math realistic, especially in the context of the model with synchronization errors that inspired introducing the duplication channel in the first place. Our main results here are a construction of optimal zero-error codes and a characterization of the zero-error capacity of the @math -duplication channel, for any fixed @math and @math . We recover the mentioned results from @cite_7 as a special case when @math , and we also obtain the discrete-time analogs of the results from @cite_3 as a special case when @math . Moreover, and further generalizing our results, we obtain a characterization of the zero-error capacity of the @math - @math -insertion channel.
@cite_7 @cite_3
2888898008 2858362441 2884699236 2170484108
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. In this paper, we study codes correcting @math duplications of @math consecutive symbols. These errors are known as tandem duplication errors, where a sequence of symbols is repeated and inserted directly after its original occurrence. Using sphere packing arguments, we derive non-asymptotic upper bounds on the cardinality of codes that correct such errors for any choice of parameters. Based on the fact that a code correcting insertions of @math zero-blocks can be used to correct @math tandem duplications, we construct codes for tandem duplication errors. We compare the cardinalities of these codes with their sphere packing upper bounds. Finally, we discuss the asymptotic behavior of the derived codes and bounds, which yields insights about the tandem duplication channel. 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. We introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type. These erroneous versions are considered as outputs of repeated transmissions over a channel, either a combinatorial channel defined by the maximum number of permissible errors of a given type, or a discrete memoryless channel. We are interested in the smallest N such that N erroneous versions always suffice to reconstruct a sequence of length n, either exactly or with a preset accuracy and or with a given probability. We are also interested in simple reconstruction algorithms. Complete solutions for combinatorial channels with some types of errors of interest in coding theory, namely, substitutions, transpositions, deletions, and insertions of symbols are given. For these cases, simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found. In general, for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius. A certain sufficient condition for solution of this problem is presented. For a discrete memoryless channel, the asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length n within Hamming distance d with error probability spl epsiv is found when d n and spl epsiv tend to 0 as n spl rarr spl infin . A similar result for the continuous channel with discrete time and additive Gaussian noise is also obtained.