Packings in Real Projective Spaces

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

doi:10.1137/17m1137528

Cited by 28 publications

(34 citation statements)

References 74 publications

(147 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These features can be used to isolate several contact graph candidates, most of which can then be ruled out by (spherical) geometric arguments. Similar techniques have been leveraged to find the optimal packing of n = 5 points in RP 2 (see [8,16]), of n = 8 points in RP 2 (see [51]), and of n = 6 points in RP 3 (see [20]); in this last case, techniques from spherical geometry are no longer applicable, and so the authors resorted to techniques from algebraic geometry.…”

Section: Proving Optimality Of a Packingmentioning

confidence: 99%

Game of Sloanes: best known packings in complex projective space

Jasper

King

Mixon

2019

Wavelets and Sparsity XVIII

Self Cite

View full text Add to dashboard Cite

It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield representations of data that are optimally robust to noise and erasures. The minimum distance of an optimal configuration not only depends on the number of points and the dimension of the projective space, but also on whether the space is real or complex. For decades, Neil Sloane's online Table of Grassmannian Packings has been the go-to resource for putatively or provably optimal packings of points in real projective spaces. Using a variety of numerical algorithms, we have created a similar table for complex projective spaces. This paper surveys the relevant literature, explains some of the methods used to generate the table, presents some new putatively optimal packings, and invites the reader to competitively contribute improvements to this table.

show abstract

Section: Proving Optimality Of a Packingmentioning

confidence: 99%

Game of Sloanes: best known packings in complex projective space

Jasper

King

Mixon

2019

Wavelets and Sparsity XVIII

Self Cite

View full text Add to dashboard Cite

show abstract

“…Letting n = v(s+1), note that by Theorem 3.1, the value of s is necessarily (5). Squaring (5) and then solving for d gives (30). Continuing, for any i = 1, .…”

Section: Equiangular Tight Frames That Are a Disjoint Union Of Regulamentioning

confidence: 99%

“…Indeed, (28) has k = 3+4−1 4 = 3 2 . It is therefore remarkable that b is an integer in general: by (30), b = v k r = vr 2 v+r−1 = vs 2 v+s−1 = d. This derived parameter k is useful in characterizing when {ψ i,j } v i=1, s+1 j=1 given by Theorem 6.1 is itself an ETF: by (32), this occurs if and only if s v−1 = 1, namely if and only if k = 2. In particular,…”

Section: Equiangular Tight Frames That Are a Disjoint Union Of Regulamentioning

confidence: 99%

Equiangular tight frames that contain regular simplices

Fickus

Jasper

King

et al. 2018

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…The graph generation is required to gather combinatorial information on the structure of the putative few-distance set X , while studying its algebraic properties is necessary to gain information on A(X ) and control the size of the ambient dimension d. This approach builds upon, and considerably extends the earlier work [26], where maximum 2-distance sets were studied by means of computers. We remark that the use of computational commutative algebra recently lead to the resolution of several challenging problems in metric geometry, such as finding a unit-distance planar embedding of the Heawood graph [15], determining optimal packings in real projective spaces [14], or showing the nonexistence of certain complex equiangular tight frames [38].…”

Section: Introductionmentioning

confidence: 99%

Constructions of Maximum Few-Distance Sets in Euclidean Spaces

Szöllősi

Östergård

2020

Electron. J. Combin.

View full text Add to dashboard Cite

A finite set of distinct vectors X in the d-dimensional Euclidean space R d is called an s-distance set if the set of mutual distances between distinct elements of X has cardinality s. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest 3-distance sets in R 4 , the largest 4-distance sets in R 3 , and the largest 6-distance sets in R 2 . We also construct new examples of large s-distance sets for d ≤ 8 and s ≤ 6, and independently verify several earlier results from the literature.April 18, 2018, Preprint.

show abstract

Packings in Real Projective Spaces

Cited by 28 publications

References 74 publications

Game of Sloanes: best known packings in complex projective space

Game of Sloanes: best known packings in complex projective space

Equiangular tight frames that contain regular simplices

Constructions of Maximum Few-Distance Sets in Euclidean Spaces

Contact Info

Product

Resources

About