text: 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.
  @cite_3: We introduce the continuous time asynchronous channel as a model for time jitter in a communication system with no common clock between the transmitter and the receiver. We have obtained a simple characterization for an optimal zero-error self-synchronizable code for the asynchronous channel. The capacity of this channel is determined by both a combinatorial approach and a probabilistic approach. Our results unveil the somewhat surprising fact that it is not necessary for the receiver clock to resynchronize with the transmitter clock within a fixed maximum time in order to achieve reliable communication. This means that no upper limit should be imposed on the run lengths of the self-synchronization code as in the case of run-length limited (RLL) codes which are commonly used in magnetic recording.
  @cite_7: The ability to store data in the DNA of a living organism has applications in a variety of areas including synthetic biology and watermarking of patented genetically modified organisms. Data stored in this medium are subject to errors arising from various mutations, such as point mutations, indels, and tandem duplication, which need to be corrected to maintain data integrity. In this paper, we provide error-correcting codes for errors caused by tandem duplications, which create a copy of a block of the sequence and insert it in a tandem manner, i.e., next to the original. In particular, we present two families of codes for correcting errors due to tandem duplications of a fixed length: the first family can correct any number of errors, while the second corrects a bounded number of errors. We also study codes for correcting tandem duplications of length up to a given constant @math , where we are primarily focused on the cases of @math . Finally, we provide a full classification of the sets of lengths allowed in tandem duplication that result in a unique root for all sequences.
summary: 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 .