Spherical design

A spherical design, part of combinatorial design theory in mathematics, is a finite set of N points on the d-dimensional unit d-sphereS 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 divided by the area or measure of S). Such a set is often called a sphericalt-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 statistics for experimental design, in combinatorics, and in geometry. 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 in the form of quantum t-designs with various applications to quantum information theory and quantum computing.