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Full-Text Articles in Physical Sciences and Mathematics
Equiangular Tight Frames From Group Divisible Designs, Matthew C. Fickus, John Jasper
Equiangular Tight Frames From Group Divisible Designs, Matthew C. Fickus, John Jasper
Faculty Publications
An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known …
Equiangular Tight Frames With Centroidal Symmetry, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson, Cody E. Watson
Equiangular Tight Frames With Centroidal Symmetry, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson, Cody E. Watson
Faculty Publications
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 …
Tremain Equiangular Tight Frames, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson
Tremain Equiangular Tight Frames, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson
Faculty Publications
Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.