Uniqueness of 277-point two-distance sets in dimension 23
A two-distance set in $\mathbb{R}^d$ is a finite set $X$ whose distances $\|x - y\|_2$, $x, y \in X$, $x \ne y$, take exactly two values.
Ge, Koolen and Munemasa constructed a two-distance set of $277$ points in $\mathbb{R}^{23}$, with distances $2$ and $\sqrt{6}$, and showed that it is not contained in a larger two-distance set in $\mathbb{R}^{23}$. Its size is $\binom{24}{2} + 1$, one more than the midpoints of the edges of a regular $23$-simplex. Lisoněk proved that a two-distance set of $29 = \binom{8}{2} + 1$ points in $\mathbb{R}^7$ is unique up to isometry and scaling.
Problem. Are any two two-distance sets $X, Y \subset \mathbb{R}^{23}$ of $277$ points related by a similarity, that is, is $Y = \{t + \lambda Q x : x \in X\}$ for some $\lambda > 0$, some orthogonal $23 \times 23$ matrix $Q$ and some $t \in \mathbb{R}^{23}$? No distance ratio is prescribed, the sets need not lie on a sphere, and they need not contain any particular smaller configuration.
Source: Ge, Koolen and Munemasa (2026), Section 3 (arXiv 2025), after Lisoněk's uniqueness theorem for 29 points in dimension 7 (1997).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, distance geometry, two-distance sets
References
@article{ge2026twodistance,
author = {Ge, Hong-Jun and Koolen, Jack and Munemasa, Akihiro},
title = {A 2-distance set with 277 points in the {E}uclidean space of dimension 23},
journal = {Discrete Comput. Geom.},
volume = {76},
number = {3},
year = {2026},
pages = {1679--1686},
doi = {10.1007/s00454-026-00843-9},
eprint = {2504.18110},
archivePrefix = {arXiv},
}
@article{lisonek1997,
author = {Lison{\v{e}}k, Petr},
title = {New maximal two-distance sets},
journal = {J. Combin. Theory Ser. A},
volume = {77},
number = {2},
year = {1997},
pages = {318--338},
doi = {10.1006/jcta.1997.2749},
}
@misc{aim-two-distance-277-point-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of 277-point two-distance sets in dimension 23}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/546-two-distance-277-point-uniqueness.md},
note = {Entry 546 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00910, "Uniqueness of 277-point two-distance sets in dimension 23", clopen.solutions.