CRII: CIF: Locality in Error Correcting Codes: Fundamental Trade-Offs

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This project investigates the fundamental trade-offs surrounding locality in error correcting codes. In error correcting codes, "locality" refers to several ways of quantifying how easily a small amount of information can be recovered from encoded data. Notions of locality have been around almost as long as coding theory itself, and today the study of locality in error correcting codes is fundamental to applications ranging from the very theoretical (complexity theory) to the very practical (distributed storage schemes). This research improves our basic scientific understanding of locality, and in doing so it makes progress on these applications. Beyond its scientific impact, this project has broader educational impact, through the development of course materials and by providing opportunities for graduate and undergraduate research.The approach of this research is to bridge a gap between two different bodies of literature. In Theoretical Computer Science, Locally Decodable and Locally Correctable Codes have been studied for over 15 years, and have been linked to many fundamental problems in complexity theory and cryptography. In the past five years, the study of locality has also taken off in distributed storage, in the form of Locally Recoverable Codes and related notions. These two lines of research operate in different parameter regimes and are motivated by different applications. The current research ties the two together by studying intermediate parameter regimes, and explores the fundamental mathematical trade-offs surrounding locality. By extending our understanding of locality in coding theory, this research can make progress on important problems in coding, complexity theory, cryptography, and distributed storage; it also develops a theory which could find many more applications going forward.