Spherical designs with few points

A spherical $t$-design is a set of points on a sphere whose average reproduces the average of every polynomial of degree at most $t$, which makes it an equal-weight quadrature rule. How few points does a $t$-design on the ordinary sphere need? Exact answers are known only for small $t$, and recent numerical searches beat the long-standing tables for $t = 19$ and $t = 21$ without yet proving the new designs exist.

Problem

A finite set $X \subset S^2 \subset \mathbb{R}^3$ is a spherical $t$-design if

$$\frac{1}{|X|} \sum_{x \in X} P(x) = \int_{S^2} P \, d\sigma$$

for every polynomial $P$ of degree at most $t$, where $\sigma$ is the normalized surface measure. Let $N(t)$ be the least number of points in a spherical $t$-design on $S^2$.

Task: find $t$-designs with fewer points than the best known, prove that they exist, or prove lower bounds on $N(t)$ beyond the classical ones. The cases $t = 19$ and $t = 21$ are the current frontier.

Background

Delsarte, Goethals and Seidel (1977) proved the general lower bounds; on $S^2$ they read $N(t) \ge (k+1)^2$ for $t = 2k$ and $N(t) \ge (k+1)(k+2)$ for $t = 2k+1$, and designs meeting them ("tight" designs) are rare. Hardin and Sloane (1996) computed putatively optimal designs on $S^2$ and conjectured exactly which sizes occur for each $t \le 12$; they also gave a family of putative designs with $N = \tfrac12 t^2 (1+o(1))$ points. Bondarenko, Radchenko and Viazovska (2013) proved that $t$-designs with $N$ points exist for every $N \ge C t^2$, the optimal order of growth conjectured by Korevaar and Meyers. Designs are closely related to the Thomson problem and to energy minimization on the sphere.

What is known

• Hardin–Sloane library: putatively optimal 3-dimensional designs, with conjectured values of the largest strength $\tau(N)$ for every $N \le 100$; some existence claims are proved algebraically or with interval methods, others rest on very accurate numerical coordinates. For larger strengths the library lists $t = 19$ with 204 points and $t = 21$ with 240 points.
• Numerical improvements: Georgiev, Gómez-Serrano, Tao and Wagner (2025) found candidate 19-designs with 198, 200 and 202 points and 21-designs with 234 and 236 points, accurate to an error of $10^{-8}$ in their objective, but not certified.
• Lower bounds: the Delsarte–Goethals–Seidel bounds give $N(19) \ge 110$ and $N(21) \ge 132$, far below the constructions.

What would count as progress

• A rigorous existence proof for a 19-design with 198 points or a 21-design with 234 points, for example an interval Newton or Krawczyk argument showing that an exact design lies near the numerical one, in the style of Chen, Frommer and Lang. Checking such a certificate takes seconds to minutes.
• Numerical designs with fewer points for some $t$ in the range $13 \le t \le 21$, reported with their coordinates. The design condition reduces to $\sum_{x, y \in X} P_k(x \cdot y) = 0$ for $1 \le k \le t$, where $P_k$ are the Legendre polynomials, so a candidate is checked in milliseconds.
• Any improvement of the lower bounds for specific $t$.

References

1. P. Delsarte, J. M. Goethals and J. J. Seidel, "Spherical codes and designs", Geometriae Dedicata 6 (1977), 363–388. doi:10.1007/bf03187604
2. R. H. Hardin and N. J. A. Sloane, "McLaren's improved snub cube and other new spherical designs in three dimensions", Discrete & Computational Geometry 15 (1996), 429–441. doi:10.1007/bf02711518
3. A. Bondarenko, D. Radchenko and M. Viazovska, "Optimal asymptotic bounds for spherical designs", Annals of Mathematics 178 (2013), 443–452. doi:10.4007/annals.2013.178.2.2
4. X. Chen, A. Frommer and B. Lang, "Computational existence proofs for spherical t-designs", Numerische Mathematik 117 (2011), 289–305. doi:10.1007/s00211-010-0332-5
5. B. Georgiev, J. Gómez-Serrano, T. Tao and A. Z. Wagner, "Mathematical exploration and discovery at scale" (2025), Problem 6.32. arXiv:2511.02864

Source

Imported from AlphaEvolve Repository of Problems, problem 32, curated by Google DeepMind. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.