Equiangular lines with a fixed angle

Equiangular lines in Euclidean spaces

On Weyl-Heisenberg orbits of equiangular lines

An element z ∈CPd−1 is called fiducial if {gz : g ∈G} is a set of lines with only one angle between each pair, where G∼= Zd × Zd is the one-dimensional finite Weyl-Heisenberg group modulo its centre. We give a new characterization of fiducial vectors. Using this characterization, we show that the existence of almost flat fiducial vectors implies the existence of certain cyclic difference sets. ...

Isoperimetric Triangular Enclosure with a Fixed Angle

Given a set S of n > 2 points in the plane (in general position), we show how to compute in O(n) time, a triangle T with maximum (or minimum) area enclosing S among all enclosing triangles with fixed perimeter P and one fixed angle ω. We show that a similar approach can be used to compute a triangle T with maximum (or minimum) perimeter enclosing S among all enclosing triangles with fixed area ...

New Bounds for Equiangular Lines and Spherical Two-Distance Sets

The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products {α,−α}, α ∈ [0, 1), are called equiangular. The problem of determining the maximal size of s-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding th...

Equiangular lines, mutually unbiased bases, and spin models

We use difference sets to construct interesting sets of lines in complex space. Using (v, k, 1)-difference sets, we obtain k2−k+1 equiangular lines in Ck when k − 1 is a prime power. Using semiregular relative difference sets with parameters (k, n, k, λ) we construct sets of n + 1 mutually unbiased bases in Ck. We show how to construct these difference sets from commutative semifields and that ...