Tremain equiangular tight frames

Fickus, Matthew; Jasper, John; Mixon, Dustin G.; Peterson, Janice

doi:10.1016/j.jcta.2017.08.005

Cited by 46 publications

(71 citation statements)

References 18 publications

(31 reference statements)

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“…In a similar manner, as summarized in Theorem 4.2, the existing literature provides ETFs of type (5, −1, S) for exactly three values of S, namely ETF (12,45), ETF (19,76) and ETF(63, 280) which have S = 4, 5, 9 and so M = 15, 19, 35, respectively. For these particular values of M , the corresponding necessary conditions (7) on the existence of 5-GDDs of type M U are known to be sufficient [27], meaning we only need U to satisfy (19) and (20) for the corresponding value of S, namely to satisfy U ≥ 5, 1 4 (U − 1) ∈ Z and that…”

Section: Negative Equiangular Tight Framesmentioning

confidence: 85%

“…The ETF (6,16) is type (4, −1, 3), and so we can apply Theorem 3.2 whenever there exists a 4-GDD of type 8 U where U satisfies (20). By Theorem 3.2, the known necessary conditions (7) on the existence of such GDDs reduce to (19):…”

Section: Negative Equiangular Tight Framesmentioning

confidence: 99%

“…In the next section, we introduce a method for constructing ETFs that uses GDDs. This method makes use of a concept from [19], which we now generalize from BIBDs to GDDs: Definition 2.3. Take a K-GDD of type M U where M ≥ 1 and U ≥ K ≥ 2, and define R, B and an incidence matrix X according to (6).…”

Section: Previously Known Constructions Of Etfs Involving Bibds and Molsmentioning

confidence: 99%

“…Harmonic ETFs are equivalent to difference sets in finite abelian groups [54,49,58,18], while Steiner ETFs arise from balanced incomplete block designs (BIBDs) [29,25]. Recent generalizations of Steiner ETFs have led to new infinite families of ETFs arising from projective planes that contain hyperovals [24] as well as from Steiner triple systems [19], dubbed hyperoval ETFs and Tremain ETFs, respectively. Another new family arises by generalizing the SRG construction of [28] to the complex setting, using generalized quadrangles to produce abelian DRACKNs [22].…”

Section: Introductionmentioning

confidence: 99%

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Equiangular tight frames from group divisible designs

Fickus

Jasper

2018

Des. Codes Cryptogr.

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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. We provide combinatorial analogs of their algebraic results, unifying Steiner ETFs with hyperoval ETFs and Tremain ETFs.

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Section: Negative Equiangular Tight Framesmentioning

confidence: 85%

Section: Negative Equiangular Tight Framesmentioning

confidence: 99%

Section: Previously Known Constructions Of Etfs Involving Bibds and Molsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equiangular tight frames from group divisible designs

Fickus

Jasper

2018

Des. Codes Cryptogr.

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“…Meanwhile, Steiner ETFs arise from balanced incomplete block designs [26,25]. This construction has recently been generalized to yield new infinite families of ETFs arising from projective planes that contain hyperovals, Steiner triple systems, and group divisible designs [24,21,19].…”

mentioning

confidence: 99%

Harmonic equiangular tight frames comprised of regular simplices

Fickus

Schmitt

2020

Linear Algebra and its Applications

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An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies.

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A note on tight projective 2‐designs

Iverson

King

Mixon

2021

J of Combinatorial Designs

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We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for d determines an equi-isoclinic tight fusion frame of d d(2 − 1) subspacesThe vertices of the tetrahedron provide a famously nice arrangement of points on the unit sphere in 3 . This highly symmetric configuration was constructed thousands of years ago in Euclid's Elements, and it has since emerged as an optimal arrangement for several fundamental problems in metric geometry [3,18]. For example, it solves the n = 4 case of Tammes's inimical dictators problem [19], which asks to maximize the minimum distance between n dictators on the sphere. It also solves the n = 4 case of Thomson's energy minimization problem [50], and more generally, it minimizes every completely monotonic potential [12], which explains the tetrahedral shape exhibited by the methane molecule [30].

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Tremain equiangular tight frames

Cited by 46 publications

References 18 publications

Equiangular tight frames from group divisible designs

Equiangular tight frames from group divisible designs

Harmonic equiangular tight frames comprised of regular simplices

A note on tight projective 2‐designs

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