Constructions of biangular tight frames and their relationships with equiangular tight frames

Haas, John I.; Cahill, Jameson; Tremain, Janet C.; Casazza, Peter G.

doi:10.48550/arxiv.1703.01786

Cited by 7 publications

(11 citation statements)

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“…In this regime, the provably optimal packings tend to be tight frames with small angle sets. This has spurred interest in so-called biangular frames [49], in which the off-diagonal Gram matrix entries all have the form ±α and ±β for some α, β ≥ 0. As we will see, biangular packings with α = 0 emerge frequently in practice, in which case we call the packing orthobiangular.…”

Section: Preliminariesmentioning

confidence: 99%

Packings in Real Projective Spaces

Fickus¹,

Jasper²,

Mixon³

2018

SIAM J. Appl. Algebra Geometry

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This paper applies techniques from algebraic and differential geometry to determine how to best pack points in real projective spaces. We present a computer-assisted proof of the optimality of a particular 6-packing in RP 3 , we introduce a linear-time constant-factor approximation algorithm for packing in the so-called Gerzon range, and we provide local optimality certificates for two infinite families of packings. Finally, we present perfected versions of various putatively optimal packings from Sloane's online database, along with a handful of infinite families they suggest, and we prove that these packings enjoy a certain weak notion of optimality.the correlation | x, y |; thanks to this relationship, our representation of points in RP d−1 by unit vectors in R d is not problematic. The coherence of an n-packing Φ = {ϕ i } i∈[n] is given byAn n-packing Φ in RP d−1 is optimal if µ(Φ) ≤ µ(Ψ) for all n-packings Ψ in RP d−1 ; such packings necessarily exist by compactness. Optimal packings are called Grassmannian frames in the finite frames literature [71], but we avoid this terminology here for the sake of clarity. Optimal packings in the d = 2 case are trivial: RP 1 is isometric to the circle, and so optimal packings amount to regularly spaced points by the pigeonhole principle [11]. Packing is more difficult in higher dimensions. In [78], Welch introduced a useful lower bound:

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Section: Preliminariesmentioning

confidence: 99%

Packings in Real Projective Spaces

Fickus¹,

Jasper²,

Mixon³

2018

SIAM J. Appl. Algebra Geometry

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“…Then {M s T t 1 D } s,t∈Z/dZ is a (2, 1, 2d)-equiangular tight frame of d 2 vectors in F d q 2 . The ETF construction in Theorem 20 is a finite field analog of a biangular Gabor frame that was suggested in [8,25]. In the finite field setting, one might view this as a Steiner ETF [18] in which a harmonic ETF [44,43,48,14] plays the role of a "flat" simplex.…”

Section: A Construction For Gerzon Equalitymentioning

confidence: 99%

A note on tight projective 2-designs

Iverson¹,

King²,

Mixon³

2021

Preprint

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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 H d determines an equi-isoclinic tight fusion frame of d(2d − 1) subspaces of R d(2d+1) of dimension 3.

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“…Let h > 0, h < 1. Starting from a spherical 2-distance set X ⊂ R d , one may obtain a family of biangular line systems in R d+1 , where the vectors x ∈ X are rescaled by a factor of √ 1 − h 2 and translated along the (d + 1)th coordinate to height h. In a similar spirit, the 6 diagonals of the icosahedron can be continuously twisted in R 3 , yielding a family of biangular lines [21].…”

Section: Constructions Of Biangular Linesmentioning

confidence: 99%

“…Equiangular lines (i.e., the case m = 1) are classical combinatorial objects [16], [29], [30], receiving considerable recent attention, see e.g., [3], [20]. Biangular lines correspond to the case m = 2, which have also been the subject of several recent studies [7], [8], [21], [25], [37] where in particular engineers investigated them focusing on tight frames [19], [40]. Our motivation for studying these objects is fueled by their intrinsic connection to kissing arrangements [14], [15], [33].…”

Section: Introductionmentioning

confidence: 99%

Biangular lines revisited

Ganzhinov¹,

Szöllősi²

2019

Preprint

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Line systems passing through the origin of the d dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least 2(d−1)(d−2), and this result is sharp for d ∈ {4, 5, 6}. Connections to binary codes, fewdistance sets, and association schemes are explored, along with their multiangular generalization.

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Constructions of biangular tight frames and their relationships with equiangular tight frames

Cited by 7 publications

References 0 publications

Packings in Real Projective Spaces

Packings in Real Projective Spaces

A note on tight projective 2-designs

Biangular lines revisited

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