Doubly transitive lines I: Higman pairs and roux

Iverson, Joseph W.; Mixon, Dustin G.

doi:10.48550/arxiv.1806.09037

Cited by 8 publications

(29 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some of the ideas in the paper may have interesting applications elsewhere. For example, there has been a lot of work to develop symmetric arrangements of points in the Grassmannian [62,11,8,65,61,10,35,36,37,45,7,42,38,29,19]. What are the projection and cross Gramian algebras of these arrangements?…”

Section: Discussionmentioning

confidence: 99%

“…First, it is natural to consider highly symmetric arrangements of points. Such arrangements were extensively studied in [62,11,8,65,61,10] in the context of designs, and later, symmetry was used to facilitate the search for optimal codes [35,36,37,45,7,42,38]. In many cases, the symmetries that underly optimal codes can be abstracted to weaker combinatorial structures that produce additional codes.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Testing isomorphism between tuples of subspaces

King¹,

Mixon²,

Waldron³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Given two tuples of subspaces, can you tell whether the tuples are isomorphic? We develop theory and algorithms to address this fundamental question. We focus on isomorphisms in which the ambient vector space is acted on by either a unitary group or general linear group. If isomorphism also allows permutations of the subspaces, then the problem is at least as hard as graph isomorphism. Otherwise, we provide a variety of polynomial-time algorithms to test for isomorphism.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Testing isomorphism between tuples of subspaces

King¹,

Mixon²,

Waldron³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this section we demonstrate some constructions of ETFs in unitary geometries, focusing especially on those derived from modular difference sets. We have not investigated finite field analogs of other sources of complex ETFs, such as Steiner systems [21], hyperovals [20], graph coverings [10,18,34], the Tremain construction [17], association schemes [12,33], or Gelfand pairs [32]. We leave these topics for future research.…”

Section: First Examplesmentioning

confidence: 99%

Frames over finite fields: Basic theory and equiangular lines in unitary geometry

Greaves¹,

Iverson²,

Jasper³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension d = 2 2l+1 over the field F 3 2 . We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.

show abstract

“…By virtue of this optimality, ETFs find applications in wireless communication [43], compressed sensing [2], and digital fingerprinting [35]. Motivated by these applications, many ETFs were recently constructed using various mixtures of algebra and combinatorics [43,48,14,18,16,7,27,26,28,29]; see [17] for a survey. Despite this flurry of work, several problems involving ETFs (such as Zauner's conjecture) remain open, and a finite field model was recently proposed to help study these remaining problems [23,24].…”

Section: Introductionmentioning

confidence: 99%

A note on tight projective 2-designs

Iverson¹,

King²,

Mixon³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Doubly transitive lines I: Higman pairs and roux

Cited by 8 publications

References 48 publications

Testing isomorphism between tuples of subspaces

Testing isomorphism between tuples of subspaces

Frames over finite fields: Basic theory and equiangular lines in unitary geometry

A note on tight projective 2-designs

Contact Info

Product

Resources

About