Maximally Equiangular Frames and Gauss Sums

Fickus, Matthew

doi:10.1007/s00041-009-9064-2

Cited by 13 publications

(10 citation statements)

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“…Other approaches are given in [11,25,29]. In the complex setting, much attention has focused on the maximal case of M 2 vectors in H M [2,15,18,21,23].…”

Section: Introductionmentioning

confidence: 99%

Steiner equiangular tight frames

Fickus

Mixon

Tremain

2012

Linear Algebra and its Applications

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189

242

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We provide a new method for constructing equiangular tight frames (ETFs). The construction is valid in both the real and complex settings, and shows that many of the few previously-known examples of ETFs are but the first representatives of infinite families of such frames. It provides great freedom in terms of the frame's size and redundancy. This method also explicitly constructs the frame vectors in their native domain, as opposed to implicitly defining them via their Gram matrix. Moreover, in this domain, the frame vectors are very sparse. The construction is extremely simple: a tensor-like combination of a Steiner system and a regular simplex. This simplicity permits us to resolve an open question regarding ETFs and the restricted isometry property (RIP): we show that the RIP behavior of some ETFs is unfortunately no better than their coherence indicates.

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“…Other approaches are given in [11,25,29]. In the complex setting, much attention has focused on the maximal case of M 2 vectors in H M [2,15,18,21,23].…”

Section: Introductionmentioning

confidence: 99%

Steiner equiangular tight frames

Fickus

Mixon

Tremain

2012

Linear Algebra and its Applications

Self Cite

189

242

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“…(We caution that Γ and C are distinct: C is V-circulant whereas Γ is not, with the latter having an all-ones first column.) Moreover, combining the above facts gives ∆Γ∆ = C = 2 −M sgn(Q)Γ∆Γ, namely that the Fourier-chirp transform Γ∆ satisfies (Γ∆) 3 = 2 3M sgn(Q)I. Analogous transforms arise in the study of SIC-POVMs; see Section 3.4 of [53], and [22]. Interestingly, this implies that the shifts of c form an equal-norm orthogonal basis of eigenvectors of the DFT, that is, Γ and C orthogonally diagonalize each other:…”

Section: Paired Difference Sets From Quadratic Formsmentioning

confidence: 92%

“…Since C = ∆Γ∆ the ETF of (a) is moreover projectively equivalent to one with Gram matrix 2 M −1 [2 M I + sgn(Q)Γ]. Since the entries in the 0th row and column of Γ are constant, we can apply (22) to this ETF to obtain a subordinate SRG on the V = 2 2M − 1 vertices of F 2M 2 \{0} in which adjacency depends on the value of Γ(y 1 , y 2 ) = (−1) B(y 1 ,y 2 ) . This is a known symplectic graph "Sp 2M (2)" [8].…”

Section: Paired Difference Sets From Quadratic Formsmentioning

confidence: 99%

Grassmannian codes from paired difference sets

Fickus¹,

Iverson²,

Jasper³

et al. 2020

Preprint

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An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional subspaces of a Hilbert space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every ECTFF is a type of optimal Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of a Hilbert space. We construct ECTFFs by exploiting new relationships between known ETFs. Harmonic ETFs equate to difference sets for finite abelian groups. We say that a difference set for such a group is "paired" with a difference set for its Pontryagin dual when the corresponding subsequence of its harmonic ETF happens to be an ETF for its span. We show that every such pair yields an ECTFF. We moreover construct an infinite family of paired difference sets using quadratic forms over the field of two elements. Together this yields two infinite families of real ECTFFs.

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“…Such a WH orbit would then produce the maximal possible number of vectors in C N that are pairwise equiangular, namely N 2 [29]. Yet another terminology that appears for this phenomenon is maximal equiangular tight frame (or maximal ETF for short) [11], which is a special case of the packing problem in the setting of projective spaces. The interest of the algebraic construction of families of ETFs has also increased due to applications to signal processing [12,16,17].…”

Section: Introductionmentioning

confidence: 99%

Spark deficient Gabor frames

Malikiosis

2018

Pacific J. Math.

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Abstract. The theory of Gabor frames of functions defined on finite abelian groups was initially developed in order to better understand the properties of Gabor frames of functions defined over the reals. However, during the last twenty years the topic has acquired an interest of its own. One of the fundamental questions asked in this finite setting is the existence of full spark Gabor frames. The author proved the existence [21], as well as constructed such frames, when the underlying group is finite cyclic. In this paper, we resolve the non-cyclic case; in particular, we show that there can be no full spark Gabor frames of windows defined on finite abelian non-cyclic groups. We also prove that all eigenvectors of certain unitary matrices in the Clifford group in odd dimensions generate spark deficient Gabor frames. Finally, similarities between the uncertainty principles concerning the finite dimensional Fourier transform and the short-time Fourier transform are discussed. IntroductionThe Gabor frame of a function f ∈ L 2 (R) is the set of all time-frequency translates of f , that is, the set of all functions of the form e 2πixy f (x−t), for y, t ∈ R, and it is a fundamental concept in time-frequency analysis and frame theory [25]. The function f usually represents a signal, t the time delay, and the pointwise multiplication by e 2πixy is the frequency "shift". Through sampling and periodization [8] one passes to the finite version of a Gabor frame, namely the shift-frequency translates of a complex function defined on a finite cyclic group. Even though finite dimensional Gabor frames were studied in order to analyze the properties of continuous signals, they later developed an interest of their own.Up to multiplication by roots of unity, a finite dimensional Gabor frame is the same as a Weyl-Heisenberg orbit, and this terminology is much more prevalent in mathematical physics and quantum information theory. A conjecture by Zauner [33] states that for every dimension N there are vectors (called "fiducials") whose WH orbit is equiangular. This means that the expression | u, v | is constant for every pair of distinct vectors u, v within this orbit. This is also known as the SIC-POVM problem which has attracted a lot of attention lately due to the vast connections to scientific areas such as quantum cryptography [26], quantum tomography [27], and algebraic number theory, especially Hilbert's 12th problem for real quadratic fields [2,3,4,5]. Such a WH orbit would then produce the maximal possible number of vectors in C N that are pairwise equiangular, namely N 2 [29]. Yet another terminology that appears for this phenomenon is maximal equiangular tight frame (or maximal ETF for short) [11], which is a special case of the packing problem in the setting of projective spaces. The interest of the algebraic construction of families of ETFs has also increased due to applications to signal processing [12,16,17].A conjecture by Heil, Ramanathan, and Topiwala from 1996 [14] states that any finite set of a Gabor frame of a ...

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Maximally Equiangular Frames and Gauss Sums

Cited by 13 publications

References 13 publications

Steiner equiangular tight frames

Steiner equiangular tight frames

Grassmannian codes from paired difference sets

Spark deficient Gabor frames

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