Limits and Efficiency of Coding Against Synchronization Errors

forthcoming

Understanding the limits of reliable information transmission over noisy communication channels and designing efficient codes for such channels are major cornerstones of information and coding theory. Most techniques developed in this area in the last 75 years have been targeted at discrete memoryless channels, which have the property that the i-th received symbol is an independent noisy function of the i-th transmitted symbol only. As a result, sender and receiver are synchronized, greatly simplifying their analysis. The study of such channels has given rise to a rich theory with many important applications beyond their original motivation. The project's goal is to tackle fundamental problems in the theory of information and efficient coding for channels which cause a loss of synchronization between sender and receiver. Besides theoretical interest, these channels capture important properties of modern data storage systems, such as DNA-based data storage. Almost all techniques designed for discrete memoryless channels break down when applied to channels with loss of synchronization. Therefore, studying even the simplest such channels (like the Binary Deletion Channel, which independently deletes each input bit with some probability) requires developing conceptually new techniques. I expect these techniques to have groundbreaking influence in other areas, like the techniques developed for discrete memoryless channels did. In this project, I aim to characterize the capacity of channels with synchronization errors and to design reliable codes with nearly-optimal rate and efficient encoding and error-correction procedures for these channels. This includes multi-trace channels with synchronization errors (that produce multiple corrupted outputs on a given input) and channels with correlated synchronization errors, both motivated by applications to DNA-based data storage.