In recent years, DNA-based systems have become a promising medium for long-term data storage. There are two layers of errors in DNA-based storage systems. The first is the dropouts of the DNA strands, which has been characterized in the shuffling-sampling channel. The second is insertions, deletions, and substitutions of nucleotides in individual DNA molecules. In this paper, we describe a DNA noisy synchronization error channel to characterize the errors in individual DNA molecules. We derive non-trivial lower and upper capacity bounds of the DNA noisy synchronization error channel based on information theory. By cascading these two channels, we provide theoretical capacity limits of the DNA storage system. These results reaffirm that DNA is a reliable storage medium with high storage density potential.

With the development of DNA synthesis and sequencing technologies, DNA becomes a promising medium forlong-term data storage. Three types of errors may occur in the DNA strand, insertions, deletions and substitutions,which we collectively call edit errors. It is still challenging to design a code that can correct multiple edit errors onnon-binary alphabets. In this paper, we propose a new coding schema for correcting multiple edit errors on DNAstrands by splitting the whole strand into consecutive blocks with appropriate length and correcting a single editerror in each block. Our method, called theDNA-LMcode, could be considered a generalization of the Levenshteincode combined with the marker code. We provide a linear encoding and decoding algorithm for ourDNA-LMcode.Compared to other encoding methods for DNA strands of several hundred base-pairs, ourDNA-LMcode achievedsimilar code rates and a much lower average nucleotide error rate in decoding.