Spherical codes: maximize the minimum angle

Formal definition

• ContainerThe unit sphere S² in three-dimensional Euclidean space
• SubmissionExactly n stereographic pairs [u,v] in [-20,20]²
• MapStereographic projection maps [u,v] exactly to the unit-sphere point below$$(u,v)\mapsto \frac{(2u,2v,1-u^2-v^2)}{1+u^2+v^2}$$
• Scorepᵢ is the mapped unit vector, and μ(P) is the largest inner product over all pairs; the verifier compares exact rationals by cross-multiplication and records ceil(μ·10¹⁸)$$\mu \left(P\right)=\underset{1\le i<j\le n}{\max }{p}_i·p_j$$
• Minimum angleBecause arccos is strictly decreasing on [-1, 1], the largest inner product corresponds to the smallest central angle$$\theta \left(P\right)=\arccos \left(\mu \right(P\left)\right)$$
• ObjectiveMinimize μ(P), equivalently maximize the minimum angle θ(P)$$\underset{P}{\min }\mu \left(P\right)\iff \underset{P}{\max }\theta \left(P\right)$$