# 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

#### Paper Mentions

#### 59 Citations

Equiangular lines and subspaces in Euclidean spaces

- Mathematics, Computer Science
- Electron. Notes Discret. Math.
- 2017

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

- Mathematics
- 2016

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 its… Expand

Computing Upper Bounds for Equiangular Lines in Euclidean Spaces

- Mathematics
- 2016

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

- Computer Science, Mathematics
- SIAM J. Discret. Math.
- 2019

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 and… Expand

Equiangular Lines in Low Dimensional Euclidean Spaces

- Mathematics
- Combinatorica
- 2021

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 on… Expand

Equiangular lines in Euclidean spaces: dimensions 17 and 18

- Mathematics
- 2021

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 the… Expand

Equiangular lines and the Lemmens-Seidel conjecture

- Mathematics, Computer Science
- Discret. Math.
- 2020

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 of… Expand

A new relative bound for equiangular lines and nonexistence of tight spherical designs of harmonic index 4

- Computer Science, Mathematics
- Eur. J. Comb.
- 2016

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 on… Expand

Maximal Sets of Equiangular Lines

- Computer Science, Physics
- 2020

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

- Mathematics
- 2020

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 certain… Expand

#### References

SHOWING 1-10 OF 95 REFERENCES

New bounds for equiangular lines

- Mathematics, Computer Science
- Discrete Geometry and Algebraic Combinatorics
- 2013

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

- Mathematics
- 2004

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-graphs… Expand

SIC-POVMs: A new computer study

- Physics, Mathematics
- 2009

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 are… Expand

Equiangular lines, mutually unbiased bases, and spin models

- Computer Science, Mathematics
- Eur. J. Comb.
- 2009

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

- Mathematics
- 2010

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 the… Expand

Nonexistence of tight spherical design of harmonic index 4

- Mathematics
- 2014

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 on… Expand

Large Equiangular Sets of Lines in Euclidean Space

- Mathematics, Computer Science
- Electron. J. Comb.
- 2000

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

- Mathematics, Physics
- 2011

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 any… Expand

Discrete Geometry and Algebraic Combinatorics

- Mathematics, Computer Science
- Discrete Geometry and Algebraic Combinatorics
- 2014

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

- Computer Science, Mathematics
- Eur. J. Comb.
- 2016

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