Combinatorial designs in quantum information theory and digital communications

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Modern society would be unrecognizable without digital communications technologies such as satellite communication, cell phones, portable music players, flash drives, GPS (global positioning system) navigation, and movies on demand. A key reason these technologies have become ubiquitous, widely accepted, and highly dependable is the critical support provided by embedded mathematical structures and algorithms that remain invisible to the user.******The requirement for embedded mathematics in these technologies derives from a combination of physical constraints, for example the desire to send information from one device to another using energy efficiently, or to pass information securely between two parties even though outsiders can monitor the exchange, or to recover transmitted information correctly despite corruption of the original signal by noise. These combinations of physical constraints correspond to mathematical problems of arranging objects subject to multiple constraints; such problems are solved by combinatorial designs. ******The short-term objectives of this proposal involve the use of combinatorial designs to solve three notoriously challenging problems of digital communications:***(1) Construct large sets of complex equiangular lines.***(2) Constrain the largest number of complex mutually unbiased bases.***(3) Construct families of binary sequences with large asymptotic merit factor.***Seeking a complete solution to these problems is highly ambitious: each has resisted considerable effort over decades by numerous mathematicians, physicists, and engineers.******Problems 1 and 2 arise in quantum information theory, a dynamic research area dealing with the processing of information at very small scales where conventional physics no longer applies. Solving these problems would benefit the quantum version of cryptography (where one wishes to make messages difficult for an adversary to read) and the quantum version of error-correcting codes (where, on the contrary, one wishes to make messages easy for others to read even in the presence of noise). The importance of combinatorial designs in solving these problems has not yet been established, and is a major innovative aspect of this proposal.******Problem 3 arises in conventional (non-quantum) digital communications and in statistical mechanics. Its solution would allow the design of more efficient radar and wireless communications, and the creation of patentable technology.******The aim is not only to solve these specific problems, but in doing so to create new tools and methods for analyzing a wide class of combinatorial structures related to digital communications. This links to the long-term goal of my research program: to combine combinatorial, algebraic, analytical, and computational techniques to transform radically the study of classical and emerging problems that are important in practical and theoretical digital communications.