Optimality and uniqueness of the (4,10,1/6) spherical code

Traditionally, optimality and uniqueness of an (n,N,t) spherical code is proved using linear programming bounds. However, this approach does not apply to the parameter (4,10,1/6). We use semidefinite programming bounds instead to show that the Petersen code (which are the vertices of the 4-dimensional second hypersimplex or the midpoints of the edges of the regular simplex in dimension 4) is the unique (4,10,1/6) spherical code.