A spherical design, part of
combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and/or ''symmetry''. These ...
theory in
mathematics, is a finite set of ''N'' points on the ''d''-dimensional unit
''d''-sphere ''S
d'' such that the average value of any polynomial ''f'' of degree ''t'' or less on the set equals the average value of ''f'' on the whole sphere (that is, the integral of ''f'' over ''S
d'' divided by the area or
measure of ''S
d''). Such a set is often called a spherical ''t''-design to indicate the value of ''t'', which is a fundamental parameter. The concept of a spherical design is due to Delsarte, Goethals, and Seidel, although these objects were understood as particular examples of
cubature formulas earlier.
Spherical designs can be of value in
approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wil ...
, in
statistics for
experimental design
The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
, in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, and in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. The main problem is to find examples, given ''d'' and ''t'', that are not too large; however, such examples may be hard to come by.
Spherical t-designs have also recently been appropriated in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
in the form of
quantum t-designs with various applications to
quantum information theory
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both ...
and
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thou ...
.
Existence of spherical designs
The existence and structure of spherical designs on the circle were studied in depth by Hong. Shortly thereafter, Seymour and Zaslavsky proved that such designs exist of all sufficiently large sizes; that is, given positive integers ''d'' and ''t'', there is a number ''N''(''d'',''t'') such that for every ''N'' ≥ ''N''(''d'',''t'') there exists a spherical ''t''-design of ''N'' points in dimension ''d''. However, their proof gave no idea of how big ''N''(''d'',''t'') is.
Mimura constructively found conditions in terms of the number of points and the dimension which characterize exactly when spherical 2-designs exist. Maximally sized collections of
equiangular lines (up to identification of lines as antipodal points on the sphere) are examples of minimal sized spherical 5-designs. There are many sporadic small spherical designs; many of them are related to finite
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphi ...
s on the sphere.
In 2013, Bondarenko, Radchenko, and Viazovska obtained the asymptotic upper bound