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Equiangular tight frames from group divisible designs

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Abstract

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 families of ETFs and GDDs, we obtain several new infinite families of ETFs. The real instances of these ETFs correspond to several new infinite families of strongly regular graphs. Our approach was inspired by a seminal paper of Davis and Jedwab which both unified and generalized McFarland and Spence difference sets. Our main result is a combinatorial analog of their algebraic results.

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Acknowledgements

We thank the editors and the three anonymous reviewers for their comments, all of which were helpful. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. This work was partially supported by the Summer Faculty Fellowship Program of the United States Air Force Research Laboratory.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH, 45433, USA

    Matthew Fickus

  2. Department of Mathematics and Statistics, South Dakota State University, Brookings, SD, 57007, USA

    John Jasper

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  1. Matthew Fickus
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Correspondence to Matthew Fickus.

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Communicated by J. Jedwab.

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Fickus, M., Jasper, J. Equiangular tight frames from group divisible designs. Des. Codes Cryptogr. 87, 1673–1697 (2019). https://doi.org/10.1007/s10623-018-0569-z

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