Equiangular lines in Euclidean spaces

@article{Greaves2016EquiangularLI,
  title={Equiangular lines in Euclidean spaces},
  author={Gary R. W. Greaves and Jacobus H. Koolen and Akihiro Munemasa and Ferenc Sz{\"o}ll{\"o}si},
  journal={J. Comb. Theory, Ser. A},
  year={2016},
  volume={138},
  pages={208-235}
}
We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d = 14 and d = 16 . Additionally… Expand

Tables and Topics from this paper

Paper Mentions

Equiangular lines and subspaces in Euclidean spaces
TLDR
It is proved that for every fixed angle θ and n sufficiently large, there are at most 2 n − 2 lines in R n with common angleθ, and this is achievable only for θ = arccos ⁡ 1 3 . Expand
Equiangular line systems and switching classes containing regular graphs
Abstract We develop the theory of equiangular lines in Euclidean spaces. Our focus is on the question of when a Seidel matrix having precisely three distinct eigenvalues has a regular graph in itsExpand
Computing Upper Bounds for Equiangular Lines in Euclidean Spaces
We develop a computable upper bound of the number of equiangular lines in various Euclidean vector spaces by combining the classical pillar decomposition and the semidefinite programming (SDP)Expand
New Upper Bounds for Equiangular Lines by Pillar Decomposition
We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by (Lemmens andExpand
Equiangular Lines in Low Dimensional Euclidean Spaces
We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds onExpand
Equiangular lines in Euclidean spaces: dimensions 17 and 18
We show that the maximum cardinality of an equiangular line system in 17 dimensions is 48, thereby solving a longstanding open problem. Furthermore, by giving an explicit construction, we improve theExpand
Equiangular lines and the Lemmens-Seidel conjecture
In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle $1/5$ are proved by carefully analyzing pillar decompositions, with the aid of the uniqueness ofExpand
A new relative bound for equiangular lines and nonexistence of tight spherical designs of harmonic index 4
We give a new upper bound for the cardinality of a set of equiangular lines in R n with a fixed common angle ? for each ( n , ? ) satisfying certain conditions. Our techniques are based onExpand
Maximal Sets of Equiangular Lines
TLDR
The problem of finding maximal sets of equiangular lines, in both its real and complex versions, is introduced, attempting to write the treatment that the author would have wanted when he first encountered the subject. Expand
On tetrahedrally closed line systems and a generalization of the Haemers-Roos inequality
Abstract We generalize the well-known Haemers-Roos inequality for generalized hexagons of order ( s , t ) to arbitrary near hexagons S with an order. The proof is based on the fact that a certainExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 95 REFERENCES
New bounds for equiangular lines
  • A. Barg, Wei-Hsuan Yu
  • Mathematics, Computer Science
  • Discrete Geometry and Algebraic Combinatorics
  • 2013
TLDR
The question of determining the maximum size of equiangular line sets in R, using semidefinite programming to improve the upper bounds on this quantity is addressed, providing a partial resolution of the conjecture set forth by Lemmens and Seidel (1973). Expand
Regular Two-Graphs and Equiangular Lines
Regular two-graphs are antipodal distance-regular double coverings of the complete graph, and they have many interesting combinatorial properties. We derive a construction for regular two-graphsExpand
SIC-POVMs: A new computer study
We report on a new computer study into the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and areExpand
Equiangular lines, mutually unbiased bases, and spin models
TLDR
It is shown how to construct difference sets from commutative semifields and that all known maximal sets of mutually unbiased bases can be obtained in this way, resolving a conjecture about the monomiality of maximal sets. Expand
Symmetric informationally complete positive-operator-valued measures: A new computer study
We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are theExpand
Nonexistence of tight spherical design of harmonic index 4
We give a new upper bound of the cardinality of a set of equiangular lines in $\R^n$ with a fixed angle $\theta$ for each $(n,\theta)$ satisfying certain conditions. Our techniques are based onExpand
Large Equiangular Sets of Lines in Euclidean Space
  • D. D. Caen
  • Mathematics, Computer Science
  • Electron. J. Comb.
  • 2000
TLDR
This is the first known constructive lower bound of order $d^2$ of order Euclidean-space, and compares with the well known "absolute" upper bound of d(d+1)$ lines in any equiangular set. Expand
MUBs inequivalence and affine planes
There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The number of such sets is not bounded above by anyExpand
Discrete Geometry and Algebraic Combinatorics
  • A. Barg, O. Musin
  • Mathematics, Computer Science
  • Discrete Geometry and Algebraic Combinatorics
  • 2014
TLDR
A short survey on the status of the affine plank conjecture of Bang (1950) is given and some new partial results for the successive inradii of the convex bodies involved are proved. Expand
Biregular graphs with three eigenvalues
TLDR
The focus is mainly on the case of graphs having two distinct valencies and the results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. Expand
...
1
2
3
4
5
...