Ge 2026 two distance set 277 points 23 dimensions

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intro_midpoint_construction: Verifies the standard lower construction of a two-distance set from the edge midpoints of a regular simplex.

lemma_2: Shows that two vector families with the displayed block Gram matrix have equal total sums.

theorem_1: Constructs a 277-point two-distance set in R^23 with distances 2 and square root of 6.

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Hong-Jun Ge, Jack Koolen, and Akihiro Munemasa, A 2-distance set with 277 points in the Euclidean space of dimension 23, arXiv:2504.18110v4 (3 August 2026), 8 pages. The published reference is DOI https://doi.org/10.1007/s00454-026-00843-9; the source record is https://arxiv.org/abs/2504.18110. The arXiv record (https://arxiv.org/abs/2504.18110, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

The paper constructs a 276-point two-distance set from the regular two-graph on 276 vertices, embeds it in $\mathbb{R}^{24}$ by the positive semidefinite Gram matrix $A+3I$, and uses a switching root to adjoin one more point in an affine hyperplane isometric to $\mathbb{R}^{23}$. Its Theorem 1 gives 277 points with squared distances $4$ and $6$. The introductory simplex-midpoint construction gives the general lower bound $\binom{d+1}{2}$; its complete elementary verification is recorded separately below.

The proof of Theorem 1 states, and does not reprove, the Goethals--Seidel regular-two-graph spectrum, the ternary Golay-code facts used to define the 276-vertex graph, and the Cao--Koolen--Munemasa--Yoshino switching-root lemma. The result page records these inputs precisely and gives the rest of the construction and distance calculation. Lemma 2 is also transcribed because it shows that the added vector is independent of the selected part of the graph. Section 3's maximality proposition is a lesser computational result: its proof uses Magma to test 16,689,170 dual-lattice vectors and is not promoted to a full proof here.

Source verification. The retained v4 PDF is the canonical 220,784-byte, 8-page artifact. All eight pages were rendered at 180 dpi and visually inspected. The theorem, graph spectrum, switching-root formula, Lemma 2, and the page labels and external references were checked against the images.

Bears on. #502.

Results.

• [[distance_problems/ge_2026_two_distance_set_277_points_23_dimensions/intro_midpoint_construction|Introductory simplex-midpoint construction]]: $\binom{d+1}{2}$ points in $\mathbb{R}^d$, with the two distances checked explicitly for $d\geq3$.
• [[distance_problems/ge_2026_two_distance_set_277_points_23_dimensions/theorem_1|Theorem 1]]: a 277-point two-distance set in $\mathbb{R}^{23}$ with distances $2$ and $\sqrt6$.
• Lemma 2: the Gram-matrix identity equating two vector sums.