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.
Applied and Computational Harmonic Analysis
Fickus, M. C., Jasper, J., Mixon, D. G., Peterson, J. D., & Watson, C. E. (2019). Polyphase equiangular tight frames and abelian generalized quadrangles. Applied and Computational Harmonic Analysis, 47(3), 628–661. https://doi.org/10.1016/j.acha.2017.11.007
The published version of record for this work is hosted at ScienceDirect as a subscription article. The citation to the version of record is noted below under "Recommended citation."
This posting provides the preprint version from the arXiv eprint server. arXiv:1604.07488 [math.FA].
Date of arXiv submission: 26 Apr 2016, updated 1 Jul 2017.
Reviewed at MR3994989.