Existence of rational spherical 4-designs on the unit circle

Determine whether there exists a rational spherical 4-design on the unit circle S^1.

Background

The paper studies ellipsoidal designs as a generalization of spherical designs and establishes classification and construction results for tight designs. In the case D=1, ellipsoidal designs coincide with spherical designs on S1. The authors note that, unlike the existence of infinitely many rational ellipsoidal 5-designs for D=3, the existence of a rational spherical 4-design on S1 is unresolved; this question is related to the study of designs with rational inner products. The problem is not addressed or resolved in the paper.

References

It is noteworthy (see \cref{Dstr}) that there exist infinitely many rational ellipsoidal $5$-designs for $D=3$, i.e. designs for which the components of each point are all rational numbers, whereas the existence of rational $4$-designs over $S1$ is still an open problem; see \cref{rat} and \\S 1.3 for details.

— Ellipsoidal designs and the Prouhet--Tarry--Escott problem  (2502.17106 - Matsumura et al., 24 Feb 2025) in Section 1, Introduction