Optimal codes on spheres and platonic solids

A code is a configuration of points. An optimal code is a code that maximizes the minimal distance between the points in the code, given the number of points. For example, the 2 points (0,0,0,0,0,0) and (1,1,1,1,1,1) are an optimal code in {0,1}6 under the Hamming distance.

For a code C on the unit sphere Sn-1, the energy Ef(C) is defined by E f(C) = Sumx,y in C, x != y f(d(x,y)2), where d(x,y) is the distance between x and y and f is considered to be a potential function. A code C0 is called universally optimal if Ef(C0) <= Ef(C) for all completely monotonic functions f and all codes C on the unit sphere S n-1.

The Lp-norm of the point (x1, ..., xn) is defined by ||x||p = (|x_1|p + ... + |x_n|p)1/p. The Lp unit sphere in Rn is defined to be the set of points in R n with L p-norm equal to 1.

Open Problems
Find every optimal code and every universally optimal code under the Euclidean distance such that the points in the code lie on the surface of a Platonic solid.

Find every optimal code and every universally optimal code under the Euclidean distance such that the points in the code lie on the surface of the L p unit sphere in R n.

Code for finding optimal codes
This page has Python code for finding optimal codes and universally optimal codes on the surface of the  L p  unit sphere in  R n. The code at the top of the page works with Python 2, and the code in post #18 works with Python 3.

This page has Python code for finding optimal binary codes. Only bounds are known in many cases, and the same page has links to tables of bounds.