A point balance algorithm for the Spherical Code problem

The Spherical Code (SC) problem has many important applications in such fie lds as physics, molecular biology, signal transmission, chemistry, engineer ing and mathematics. This paper presents a bilevel optimization formulation of the SC problem. Based on this formulation, the concept of balanced sphe rical code is introduced and a new approach, the Point Balance Algorithm (P BA), is presented to search for a 1-balanced spherical code. Since an optim al solution of the SC problem (an extremal spherical code) must be a 1-bala nced spherical code, PBA can be applied easily to search for an extremal sp herical code. In addition, given a certain criterion, PBA can generate effi ciently an approximate optimal spherical code on a sphere in the n-dimensio nal space R-n. Some implementation issues of PBA are discussed and putative global optimal solutions of the Fekete problem in 3, 4 and 5-dimensional s pace are also reported. Finally, an open question about the geometry of Fek ete points on the unit sphere in the 3-dimensional space is posed.