A 325-point two-distance set in dimension 24
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.
Blokhuis proved that a two-distance set in $\mathbb{R}^d$ has at most $\binom{d+2}{2}$ points, which is $325$ for $d = 24$. The midpoints of the edges of a regular $d$-simplex form a two-distance set of $\binom{d+1}{2}$ points. Lisoněk determined the largest size of a two-distance set in each dimension $d \le 8$, and constructed one of $45 = \binom{10}{2}$ points in $\mathbb{R}^8$. Ge, Koolen and Munemasa constructed a two-distance set $Z$ of $277$ points in $\mathbb{R}^{23}$ and showed that the two-distance sets in $\mathbb{R}^{24}$ properly containing $Z$ have at most $279$ points.
Problem. Is there a two-distance set of $325$ points in $\mathbb{R}^{24}$? The two distances are not prescribed, and the points need not lie on a sphere.
Source: Ge, Koolen and Munemasa (2026), Section 3 (arXiv 2025).
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},
note = {Proposition 3 as corrected in arXiv v4 (3 August 2026)},
archivePrefix = {arXiv},
}
@book{blokhuis1984,
author = {Blokhuis, Aart},
title = {Few-Distance Sets},
series = {CWI Tract},
volume = {7},
publisher = {Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica},
address = {Amsterdam},
year = {1984},
}
@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-325-points-dimension24,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A 325-point two-distance set in dimension 24}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/545-two-distance-325-points-dimension24.md},
note = {Entry 545 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-00911, "A 325-point two-distance set in dimension 24", clopen.solutions.