  The first 195 spherical codes for the global minima of 1 to 65 points on S2
have been obtained for 3 types of potentials: logarithmic, Coulomb, called the
Thomson problem, and the inverse square law, with 77, 38, and 38 digits
precision respectively. It was discovered that certain point sets have embedded
polygonal structures, constraining the points, enabling them to be
parameterized and to successfully recover the algebraic polynomial. So far 49
algebraic number sets have been recovered, but 109 more remain to be recovered
from their 1,622 parameters, 983 known to 50,014 digit precision. The very high
algebraic degree of these minimal polynomials eludes finding the algebraic
numbers from the spherical codes and requires new mathematical tools to meet
this challenge.
