Polyphase equiangular tight frames and abelian generalized quadrangles

An equiangular tight frame (ETF) is a type of optimal packing of lines in a finite-dimensional Hilbert space. ETFs arise in various applications, such as waveform design for wireless communication, compressed sensing, quantum information theory and algebraic coding theory. In a recent paper, signatu...

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Bibliographic Details
Published in:Applied and computational harmonic analysis 2019-11, Vol.47 (3), p.628-661
Main Authors: Fickus, Matthew, Jasper, John, Mixon, Dustin G., Peterson, Jesse D., Watson, Cody E.
Format: Article
Language:English
Subjects:
Equiangular tight frame
Filter bank
Generalized quadrangle
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Summary:An equiangular tight frame (ETF) is a type of optimal packing of lines in a finite-dimensional Hilbert space. ETFs arise in various applications, such as waveform design for wireless communication, compressed sensing, quantum information theory and algebraic coding theory. In a recent paper, signature matrices of ETFs were constructed from abelian distance regular covers of complete graphs. We extend this work, constructing ETF synthesis operators from abelian generalized quadrangles, and vice versa. This produces a new infinite family of complex ETFs as well as a new proof of the existence of certain generalized quadrangles. This work involves designing matrices whose entries are polynomials over a finite abelian group. As such, it is related to the concept of a polyphase matrix of a finite filter bank.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2017.11.007