Codes and Expansions (CodEx) Seminar

Ryutaro Misawa (Tohoku University)
Construction of Spherical Designs via Tight \(t\)-Fusion Frames

A spherical \(t\)-design is a finite set of points on a sphere that exactly reproduces the spherical averages of all polynomials of degree at most \(t\). In this talk, I present a local-to-global construction of spherical designs using tight \(t\)-fusion frames. The main theorem states that if a family of subspaces forms a tight \(t\)-fusion frame and each subspace contains a spherical \(s\)-design of the same size, then their labeled union is a spherical \(\min\{s,2t+1\}\)-design on the ambient sphere. This separates the construction into two exact averaging problems: one inside each subspace and one over the family of subspaces. I will illustrate the method using the cross-polytope and the \(24\)-cell. I will also describe explicit spherical \(5\)-designs with \(O(D^3)\) points in every ambient dimension \(D\), and spherical \(7\)-designs with \(O(D^6)\) points when \(D=2n\) and \(n-1\) is a prime power.