Uniqueness (up to isometry) of the 288‑point spherical 5‑design in R16

Determine whether the 288‑point spherical 5‑design in R16 (arising from the Nordstrom–Robinson spherical code) is unique up to isometry among all 288‑point spherical 5‑designs in R16.

Background

The authors prove universal optimality for the Nordstrom–Robinson spherical code and discuss uniqueness for energy minimization, noting that uniqueness fails for absolutely monotonic polynomials of degree ≤ 4 and that uniqueness for degree 5 would hinge on uniqueness of the underlying 5‑design.

They explicitly state they do not know whether the 288‑point spherical 5‑design is unique up to isometry.

References

We do not know whether C is the unique 288-point spherical 5-design in R16 up to isometry, but the existence of another 5-design is the only way uniqueness could fail for polynomials of degree 5 in Theorem 4.1.

— Optimality of spherical codes via exact semidefinite programming bounds (2403.16874 - Cohn et al., 25 Mar 2024) in Section 4