86 points in 31 dimensions

The record

No two of the 86 unit vectors in R^31 have an inner product above 0.064362071287; the table listed 0.064362086420 for an entry marked as needing more optimization. The gain is 1.51 × 10⁻⁸.

How it was found and checked

Repeated annealing from the table's code (L-BFGS on a smooth log-sum-exp surrogate of the largest cosine), then polishing by sequential linear programming. The largest cosine was computed in 50-digit arithmetic by a separate program before upload, and spherical-codes.org recomputes every uploaded code before listing it.

A spherical code is a set of N points on the unit sphere in d dimensions, spread out so that no two are close. The aim is to make the largest cosine between two of the points, the cosine of the smallest angle between them, as small as possible. Henry Cohn's table at spherical-codes.org keeps the best code known for every dimension from 3 to 32 and up to 1,024 points.

Questions

What is the best known spherical code of 86 points in 31 dimensions?

Ours: largest cosine 0.064362071287, minimal angle 86.3098°, listed on spherical-codes.org.

How much better is it?

The largest cosine drops from 0.064362086420 to 0.064362071287, and the minimal angle grows from 86.3098° to 86.3098°.

Where are the coordinates?

At https://spherical-codes.org/data/31/86, as served by spherical-codes.org.

More spherical codes

• 109 points in 18 dimensions
• 110 points in 18 dimensions
• 111 points in 18 dimensions
• 77 points in 19 dimensions
• 86 points in 20 dimensions
• 87 points in 20 dimensions