Combinatorial design theory and digital communications

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Modern society would be unrecognisable 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 and highly dependable is the critical support provided by mathematical structures and algorithms that remain invisible to the user. The mathematics underlying these technologies arises from a combination of physical requirements, for example the desire to: send information from one device to another using energy efficiently; pass information securely between two parties even though outsiders can monitor the exchange; or recover transmitted information correctly despite corruption of the original signal. These combinations of physical requirements correspond to mathematical problems of arranging objects subject to multiple constraints; such problems are solved by combinatorial designs. The long-term vision of this research program is to combine combinatorial, algebraic, analytical, and computational techniques to revolutionise the study of classical and emerging problems involving combinatorial designs, by identifying the simplest possible structure explaining them. The short-term objectives of this research program are to solve three notoriously challenging problems of combinatorial design theory: 1. The existence pattern for Hadamard matrices 2. The existence of transversals and near-transversals in Latin squares 3. The existence pattern for difference sets in groups of order 4d. Seeking a complete solution to these problems is highly ambitious: each has perplexed researchers for sixty years or more. These problems are important practically, as well as theoretically. Problem 3 arises in signal design, where one wishes to determine when two signals are synchronised or when two images are optically aligned. Problems 1 and 2 arise, for example, in experimental design, where one wishes to test the statistical association between two quantities such as the frequency of cell phone use and the development of cancer, after accounting for other possible influences including lifestyle and genetic predisposition. The originality of this research program lies in the careful choice of which techniques to develop and which to discard as being unlikely to lead to a resolution. The anticipated outcomes are new insights into the mathematical structures underlying these problems, and greatly simplified ways of viewing them. Solving any of these problems would be a tremendous theoretical advance of great importance to mathematicians and computer scientists, and would stimulate interest in resolving other longstanding open problems using similar methods. Additionally, the wider Canadian and international scientific communities would take significant interest in the resulting deeper understanding of the possibilities and limitations of future digital communications technologies.