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 4dimensional second hypersimplex or the midpoints of the edges of the regular simplex in dimension 4) is the unique (4, 10, 1/6) spherical code.