Johnson optimality and uniqueness for distance ratio 1:√2
Abstract
We prove that a two-distance set in ℝd with distance ratio 1:√2 has at most d(d+1)/2 points for every d≥9. Equality in full affine dimension characterizes the Johnson construction; for d≥10, the full-dimensional condition is automatic. The proof separates configurations according to the connectivity of the short-distance graph. The connected case uses root lattices and intersection theorems, while the disconnected case uses an orthogonal decomposition into component direction spaces and the span of their center differences. We also prove a relative bound for Euclidean two-distance sets and show that equality forces sphericity. Together with the Larman--Rogers--Seidel theorem, these results give maximum cardinality d(d+1)/2 for 9≤ d≤21 and $275$ for d=22, without exhaustive enumeration.
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BibTeX
@misc{bu2026johnson,
title = {{Johnson optimality and uniqueness for distance ratio $1:\sqrt2$}},
author = {Bryan Bu and Chin-Yen Lee and Yu-Hang Sheng and Meng-Tsung Tsai and Hyunjun Yi and Wei-Hsuan Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40367},
eprint = {2610.40367},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40367},
}
RIS
TY - UNPB
TI - Johnson optimality and uniqueness for distance ratio 1:√2
AU - Bryan Bu
AU - Chin-Yen Lee
AU - Yu-Hang Sheng
AU - Meng-Tsung Tsai
AU - Hyunjun Yi
AU - Wei-Hsuan Yu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40367
UR - https://arxiv.org/abs/2610.40367
AB - We prove that a two-distance set in ℝ^d with distance ratio 1:√2 has at most d(d+1)/2 points for every d≥9. Equality in full affine dimension characterizes the Johnson construction; for d≥10, the full-dimensional condition is automatic. The proof separates configurations according to the connectivity of the short-distance graph. The connected case uses root lattices and intersection theorems, while the disconnected case uses an orthogonal decomposition into component direction spaces and the span of their center differences. We also prove a relative bound for Euclidean two-distance sets and show that equality forces sphericity. Together with the Larman--Rogers--Seidel theorem, these results give maximum cardinality d(d+1)/2 for 9≤ d≤21 and $275$ for d=22, without exhaustive enumeration.
ER -