Equiangular tight frames with centroidal symmetry |
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Authors: | Matthew Fickus John Jasper Dustin G Mixon Jesse D Peterson Cody E Watson |
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Institution: | 1. Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, United States;2. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, United States |
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Abstract: | An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. Though they arise in many applications, only a few methods for constructing them are known. Motivated by the connection between real ETFs and graph theory, we introduce the notion of ETFs that are symmetric about their centroid. We then discuss how well-known constructions, such as harmonic ETFs and Steiner ETFs, can have centroidal symmetry. Finally, we establish a new equivalence between centroid-symmetric real ETFs and certain types of strongly regular graphs (SRGs). Together, these results give the first proof of the existence of certain SRGs, as well as the disproofs of the existence of others. |
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Keywords: | 42C15 05E30 Equiangular tight frame Strongly regular graph |
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