Toward the classification of biangular harmonic frames

Casazza, Peter G.; Farzannia, Amineh; Haas, John I.; Tran, Tin T.

doi:10.1016/j.acha.2017.06.004

Cited by 13 publications

(7 citation statements)

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“…For instance, it is well-known that equiangular tight frames can be constructed from difference sets or Steiner systems, see [8,22]. Likewise, divisible difference sets and partial difference sets are used to construct biangular tight frames [4]. We will continue to exploit some families of block designs to construct desired sets.…”

Section: Construction Regular Two-distance Setsmentioning

confidence: 99%

Regular two distance sets

Casazza¹,

Tran²,

Tremain³

2019

Preprint

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This paper makes a deep study of regular two-distance sets. A set of unit vectors X in Euclidean space R n is said to be regular twodistance set if the inner product of any pair of its vectors is either α or β, and the number of α (and hence β) on each row of the Gram matrix of X is the same. We present various properties of these sets as well as focus on the case where they form tight frames for the underling space. We then give some constructions of regular two-distance sets, in particular, two-distance frames, both tight and non-tight cases. It has been seen that every known example of maximal two-distance sets are tight frames. However, we supply for the first time an example of a non-tight maximal two-distance frame. Connections among twodistance sets, equiangular lines and quasi-symmetric designs are also discussed. For instance, we give a sufficient condition for constructing sets of equiangular lines from regular two-distance sets, especially from quasi-symmetric designs satisfying certain conditions.

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Section: Construction Regular Two-distance Setsmentioning

confidence: 99%

Regular two distance sets

Casazza¹,

Tran²,

Tremain³

2019

Preprint

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“…Manipulating (20) gives an expression for n − d = dim(span{ψ j } n j=1 ) in terms of n and k, namely (18). Moreover, (20) implies that the inverse Welch bound (5) for {ϕ j } n j=1 is…”

Section: Constructing Naimark Complements From Binders With Combinatomentioning

confidence: 99%

Equiangular tight frames that contain regular simplices

Fickus

Jasper

King

et al. 2018

Linear Algebra and its Applications

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An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. A regular simplex is a special type of ETF in which the number of vectors is one more than the dimension of the space they span. In this paper, we consider ETFs that contain a regular simplex, that is, have the property that a subset of its vectors forms a regular simplex. As we explain, such ETFs are characterized as those that achieve equality in a certain well-known bound from the theory of compressed sensing. We then consider the so-called binder of such an ETF, namely the set of all regular simplices that it contains. We provide a new algorithm for computing this binder in terms of products of entries of the ETF's Gram matrix. In certain circumstances, we show this binder can be used to produce a particularly elegant Naimark complement of the corresponding ETF. Other times, an ETF is a disjoint union of regular simplices, and we show this leads to a certain type of optimal packing of subspaces known as an equichordal tight fusion frame. We conclude by considering the extent to which these ideas can be applied to numerous known constructions of ETFs, including harmonic ETFs. Necessary integrality conditions on the existence of various types of ETFs are given in [58].Conference matrices, Hadamard matrices, Paley tournaments and quadratic residues are related, and lead to infinite families of ETFs whose redundancy n d is either nearly or exactly two [57,43,53,56]. Harmonic ETFs and Steiner ETFs offer much more freedom in choosing d and n. Harmonic ETFs are equivalent to difference sets in finite abelian groups [60,57,63,27]. Steiner ETFs arise from particular types of balanced incomplete block designs (BIBDs) [39,36]. Recent generalizations of Steiner ETFs have led to new infinite families of ETFs arising from projective planes that contain hyperovals [35] as well as from Steiner triple systems [31]. Another new family arises by generalizing the SRG construction of [38] to the complex setting [32].Many of these known constructions give ETFs that contain a regular simplex, and thus achieve equality in both (2) and (3). For example, every Steiner ETF is a disjoint union of regular simplices by construction. This property is also enjoyed by harmonic ETFs arising from McFarland difference sets, since it is known that they can be obtained by applying unitary operators to certain Steiner ETFs [45]. We study ETFs that contain regular simplices in general, and then explore the degree to which the ETFs constructed in [63,27,35,31,32] have this property.In particular, in the next section, we establish notation, discuss some known results that we will need later on, and elaborate on the connections between these ideas and compressed sensing. In Section 3, we show that an ETF achieves equality in (3) if and only if it contains a regular simplex (Theorem 3.1), and give a strong necessary condition on the existence of real ETFs that are full spark (Theorem 3.2). In the fourth section, we characterize regular simplices that are containe...

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“…The claim follows by equating Equation (7) with Equation ( 10), replacing the inner product with the value given in Equation ( 15), and then rearranging to the desired form.…”

Section: Hsmentioning

confidence: 99%

Low frame coherence via zero-mean tensor embeddings

Bodmann¹,

Haas²

2017

Preprint

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This paper is concerned with achieving optimal coherence for highly redundant real unit-norm frames. As the redundancy grows, the number of vectors in the frame becomes too large to admit equiangular arrangements. In this case, other geometric optimality criteria need to be identified. To this end, we use an iteration of the embedding technique by Conway, Hardin and Sloane. As a consequence of their work, a quadratic mapping embeds equiangular lines into a simplex in a real Euclidean space. Here, higher degree polynomial maps embed highly redundant unit-norm frames to simplices in high-dimensional Euclidean spaces. We focus on the lowest degree case in which the embedding is quartic.

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Toward the classification of biangular harmonic frames

Cited by 13 publications

References 50 publications

Regular two distance sets

Regular two distance sets

Equiangular tight frames that contain regular simplices

Low frame coherence via zero-mean tensor embeddings

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