Levenshtein's Sequence Reconstruction Problem and Results for Larger Alphabet Sizes

The problem of storing large amounts of information safely for a long period of time has become essential. One of the most promising new data storage mediums are the polymer-based data storage systems, like the DNA-storage system. These storage systems are highly durable and they consume very little energy to store the data. When information is retrieved from a storage, however, several different types of errors may occur in the process. It is known that the Levenshtein's sequence reconstruction framework is well-suited to overcome such errors and to retrieve the original information. Many of the previous results regarding Levenshtein's sequence reconstruction method are so far given only for the binary alphabet. However, larger alphabets are natural for the polymer-based data storage. For example, the quaternary alphabet is suitable for DNA-storage due to the four amino-acids in DNA. The results for larger alphabets often require, as we will see in this work, different and more complicated techniques compared to the binary case. Moreover, we show that an increase in the alphabet size makes some error types behave rather surprisingly.