Spherical codes
67 points in 23 dimensions
The best known spherical code of 67 points in 23 dimensions has largest cosine 0.076785993943 between any two points, a minimal angle of 85.5962°. Quantavane found it on 26 October 2026, lowering the 0.076786919884 listed before.
Listed on spherical-codes.org since 26 October 2026; its data file serves this code.
The record
| Ours | 0.076785993943 |
|---|---|
| Previous best | 0.076786919884 |
| Previous source | Henry Cohn's table at spherical-codes.org, which marked the entry as needing more optimization |
| Found | 26 October 2026 |
| Status | Listed on spherical-codes.org since 26 October 2026; its data file serves this code. |
| Check | 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. |
No two of the 67 unit vectors in R^23 have an inner product above 0.076785993943; the table listed 0.076786919884 for an entry marked as needing more optimization. The gain is 9.26 × 10⁻⁷.
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.
See the dataA 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 67 points in 23 dimensions?
Ours: largest cosine 0.076785993943, minimal angle 85.5962°, listed on spherical-codes.org.
How much better is it?
The largest cosine drops from 0.076786919884 to 0.076785993943, and the minimal angle grows from 85.5961° to 85.5962°.
Where are the coordinates?
At https://spherical-codes.org/data/23/67, as served by spherical-codes.org.