Heisenberg-Weyl Groups, Sequences, and Codes

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This proposal develops a new mathematical framework for detection and estimation in applications as diverse as remote sensing and quantum computing. The application domain defines one Heisenberg-Weyl group, the signal design domain defines a second group, and the new mathematical framework results from expressing one group in terms of the other. Classical algebraic error correcting codes find new application as phase coded waveforms and the geometry of codewords provides fundamental limits on detection and estimation. In the second order Reed Muller code, pairs of phase coded waveforms associated with Rudin-Shapiro sequences enable instantaneous radar polarimetry which provides concurrent and coherent access to all four dimensions of the polarization scattering matrix rather than serial and non-coherent access as is the case today. The value of this new primitive is detection based on simple statistics in four dimensions rather than detection based on complicated statistics applied to lossy one-dimensional projections.As a training program, this proposal provides graduate students with an opportunity to create new mathematics that brings fundamental change to an important application domain. It will increase the pool of mathematicians who appreciate the power of mathematics and the important role that it plays in engineering disciplines. Graduate students will learn to flourish in the different cultures associated with the different academic disciplines, develop an ability to bridge different worlds and this experience will advantage them in both academic and non-academic careers. This proposal will likely result in technology transfer to industrial partners, including Raytheon, who have confirmed the value of instantaneous radar polarimetry in independent analysis. This engagement and others may lead to additional internship opportunities for US citizens and permanent residents.