A 2-distance set with 277 points in the Euclidean space of dimension 23

Imagine you are hosting a massive party in a room with 23 dimensions. (Don't worry, you can't visualize 23 dimensions, but mathematically, it's just a space with 23 directions you can move in, like up/down, left/right, forward/backward, and 20 other invisible directions).

The goal of this paper is to arrange 277 guests in this room so that the distance between any two guests is strictly one of only two specific numbers: either 2 steps or $\sqrt{6}$ steps (about 2.45 steps).

In the world of math, this is called a 2-distance set.

The Big Challenge: The "Room Size" Limit

For a long time, mathematicians knew a rule of thumb: If you have a room with $d$ dimensions, you can't fit more than a certain number of people if you only allow two distances.

For a 23-dimensional room, the "safe" limit was thought to be 276 people.
This limit comes from a very specific, perfect arrangement: taking the midpoints of the edges of a giant 23-sided shape (a simplex). It's like the mathematical equivalent of a perfect crystal structure.

For decades, no one could find a way to squeeze in one extra person (277) without breaking the "two-distance" rule. It was like trying to fit a 277th guest into a theater that was mathematically proven to only hold 276.

The Breakthrough: Building the "Impossible" Party

The authors of this paper (Ge, Koolen, and Munemasa) managed to do the impossible. They constructed a group of 277 points in 23-dimensional space where every pair is exactly distance 2 or $\sqrt{6}$ apart.

Here is how they did it, using a story analogy:

1. The Foundation: The "Switching" Graph

First, they started with a known structure of 276 points. Think of this as a giant, intricate web of connections.

They used a special mathematical object called a Ternary Golay Code (a type of error-correcting code used in space communication) and a multipartite graph (a network with 11 groups of 3 people each).
They connected these points in a very specific way based on a "Seidel matrix" (a fancy table of numbers that describes who is connected to whom).
This created a perfect, symmetrical arrangement of 276 points in a 24-dimensional space.

2. The "Switching Root": Finding the Hidden Key

In this 24-dimensional space, they discovered a special "magic vector" (a direction and length), which they call the switching root ( $r$ ).

Imagine the 276 guests are standing on a flat floor.
The magic vector $r$ is like a special flashlight beam shining down on them.
The authors found that if you shine this light, every single one of the 276 guests is exactly 1 unit away from the light source in a specific mathematical sense.
This means all 276 guests are actually standing on a flat 23-dimensional sheet (a hyperplane) inside that 24-dimensional room.

3. Adding the 277th Guest

This is the magic trick.

They took a specific group of 3 guests from the original 276 (who were standing in a special triangular formation).
They combined them with the "magic flashlight" vector to create a new point ( $u$ ).
They proved that if you add this new point $u$ to the group, the distance from $u$ to everyone else is still either 2 or $\sqrt{6}$ .
Result: You now have 277 points, all obeying the two-distance rule, living in a 23-dimensional space.

Why is this a Big Deal?

Breaking the Record: Before this, the largest known 2-distance set in 23 dimensions was 276. They found a way to add one more. It's like finding a way to fit one more person into a bus that was thought to be completely full.
Maximality: They didn't just find 277; they proved you cannot add a 278th person. If you try to add anyone else, the distances will break the rules. It is a "maximal" set.
The Connection to 24 Dimensions: Interestingly, if you go back up to 24 dimensions, you can add one more point to make 278, but that point doesn't fit in the 23-dimensional "sheet" where the others live. The 277-point set is the absolute limit for that specific 23-dimensional slice.

The "Magma" Code

At the end of the paper, they include a block of computer code (written in a language called Magma).

Think of this as the blueprint or the recipe.
Because the numbers are so huge and the geometry is so complex, a human couldn't check the math by hand. They used a computer to verify that every single distance between every pair of the 277 points is indeed correct.
The code took about 13 minutes to run on a computer, confirming their discovery.

Summary

In simple terms: Mathematicians built a perfect geometric arrangement of 277 points in a 23-dimensional world where everyone is exactly the same distance from their neighbors (either "short" or "medium" distance). They broke a long-standing record by fitting in one more person than anyone thought possible, and they proved that this is the absolute maximum number of people who can fit in that specific arrangement.

Here is a detailed technical summary of the paper "A 2-distance set with 277 points in the Euclidean space of dimension 23" by Ge, Koolen, and Munemasa.

1. Problem Statement

The paper addresses the classification and construction of $s$ -distance sets in Euclidean space $\mathbb{R}^d$ . Specifically, it focuses on 2-distance sets (sets of points where pairwise distances take only two distinct values).

Background: Blokhuis established an upper bound of $\binom{d+2}{2}$ for the size of a 2-distance set in $\mathbb{R}^d$ .
The Gap: The set of midpoints of the edges of a regular $d$ -simplex forms a 2-distance set of size $\binom{d+1}{2}$ . For dimensions $d \le 8$ , larger sets (exceeding $\binom{d+1}{2}$ ) were known. However, for $d > 8$ , no construction of a 2-distance set with size greater than $\binom{d+1}{2}$ was known.
Specific Context ( $d=23$ ):
- The standard lower bound is $\binom{24}{2} = 276$ .
- Glazyrin and Yu proved that if a 2-distance set lies on the unit sphere $S^{22} \subset \mathbb{R}^{23}$ , its size is bounded by 276.
- The unique regular two-graph on 276 vertices yields a set of 276 equiangular lines, forming a 2-distance set of size 276 in $\mathbb{R}^{23}$ .
Objective: The authors aim to construct a 2-distance set in $\mathbb{R}^{23}$ with 277 points, thereby exceeding the standard lower bound and the spherical bound, and to determine if this set is maximal.

2. Methodology

The construction relies on algebraic graph theory, coding theory, and lattice theory.

A. Graph Construction via Two-Graphs and Codes

Base Graph: The authors utilize the unique regular two-graph on 276 vertices. They construct a specific graph $\Gamma$ in its switching class.
Vertex Set: The vertex set $V(\Gamma)$ $V (Γ)$ is the union of two sets:
- $X$ : A set of 33 vertices representing a complete multipartite graph with 11 parts of size 3 ( $X = \{(a, i) \mid a \in \mathbb{F}_3, 1 \le i \le 11\}$ ).
- $Y$ : A set of 243 vertices derived from the dual of the ternary Golay code $C^\perp$ (where $|C^\perp| = 3^6 = 729$ , but the construction uses a specific subset or structure related to the code's properties; the paper states $|Y|=243$ ).
Adjacency Rules:
- Vertices in $X$ are connected if they belong to different parts.
- A vertex $(a, i) \in X$ is connected to $y \in Y$ if the $i$ -th coordinate of $y$ is not equal to $a$ .
- Two vertices $y, y' \in Y$ are connected if their difference $y - y'$ has a Hamming weight of 6.
Spectral Analysis: The Seidel matrix $S = J - I - 2A(\Gamma)$ is analyzed. The spectrum is found to be $\{[55]^{23}, [-5]^{253}\}$ . This spectral property is crucial for embedding the graph into Euclidean space.

B. Vector Embedding

Gram Matrix Construction: Using the spectrum of the Seidel matrix, the authors consider the matrix $M = A(\Gamma) + 3I$ . This matrix is positive semi-definite with rank 24.
Vector Representation: There exists a set of vectors $V = \{v_u \mid u \in X \cup Y\} \subset \mathbb{R}^{24}$ $V = {v_{u} ∣ u \in X \cup Y} \subset R^{24}$ such that their inner products correspond to the entries of $M$ $M$ .
- $\|v_u\|^2 = 3$ .
- $\langle v_u, v_v \rangle = 1$ if $u, v$ are adjacent.
- $\langle v_u, v_v \rangle = 0$ otherwise.
Distance Calculation: The squared Euclidean distances between these vectors are $\|v_u - v_v\|^2 = \|v_u\|^2 + \|v_v\|^2 - 2\langle v_u, v_v \rangle$ $∥ v_{u} - v_{v} ∥^{2} = ∥ v_{u} ∥^{2} + ∥ v_{v} ∥^{2} - 2 ⟨ v_{u}, v_{v} ⟩$ .
- If adjacent: $3 + 3 - 2(1) = 4$.
- If non-adjacent: $3 + 3 - 2(0) = 6$.
- Thus, $V$ forms a 2-distance set of size 276 in $\mathbb{R}^{24}$ with squared distances $\{4, 6\}$ .

C. Dimension Reduction and Extension

Switching Root: Following Cao et al., a "switching root" vector $r \in \mathbb{R}^{24}$ is identified such that $\langle r, r \rangle = 2$ and $\langle r, v \rangle = 1$ for all $v \in V$ .
Affine Hyperplane: The set $V$ lies in the affine hyperplane $H = \{v \in \mathbb{R}^{24} \mid \langle v, r \rangle = 1\}$ , which is isomorphic to $\mathbb{R}^{23}$ .
Adding the 277th Point:
- Let $\{x_1, x_2, x_3\} \subset X$ be the three vectors corresponding to one part of the multipartite graph.
- Define a new vector $u = x_1 + x_2 + x_3 - r$ .
- It is shown that $\langle u, u \rangle = 5$ and $\langle u, z \rangle \in \{1, 2\}$ for $z \in V$ .
- The squared distances from $u$ to any $z \in V$ are calculated as $5 + 3 - 2\langle u, z \rangle$, resulting in values of 4 or 6.
- Result: The set $Z = \{u\} \cup X \cup Y$ is a 2-distance set of size 277 in $\mathbb{R}^{23}$ .

3. Key Contributions

Construction of a 277-point Set: The paper provides the first known construction of a 2-distance set in $\mathbb{R}^{23}$ with size 277, exceeding the bound $\binom{24}{2} = 276$ .
Maximality Proof: The authors prove that this set $Z$ $Z$ is maximal within the affine hyperplane $H \cong \mathbb{R}^{23}$ $H ≅ R^{23}$ . It cannot be extended to a larger 2-distance set within $\mathbb{R}^{23}$ $R^{23}$ .
- Note: The set is not maximal in $\mathbb{R}^{24}$ ; it can be extended to 278 points by adding a specific vector in the direction of $r$ , but this extension leaves the 23-dimensional hyperplane.
Lattice Verification: The maximality is established using the Magma computer algebra system. The authors analyze the integral lattice generated by the set and its dual, verifying that no other vector in $\mathbb{R}^{24}$ can be added to the set while maintaining the 2-distance property within the hyperplane.

4. Results

Size: 277 points.
Dimension: 23 (embedded in $\mathbb{R}^{24}$ as an affine hyperplane).
Distances: The set has squared distances $\{4, 6\}$ (distances $2 $and$ \sqrt{6}$).
Uniqueness/Maximality: The set is unique up to isometry in the context of the construction and is proven to be maximal in $\mathbb{R}^{23}$ .
Comparison: This serves as a high-dimensional analogue to Lisoněk's 1997 construction of a 29-point set in $\mathbb{R}^7$ (where $29 = \binom{8}{2} + 1$).

5. Significance

Breaking the Barrier: This work resolves a long-standing open problem regarding the existence of 2-distance sets larger than $\binom{d+1}{2}$ for $d > 8$ . It demonstrates that the bound is not tight for $d=23$ .
Connection to Two-Graphs: It highlights the deep connection between regular two-graphs, equiangular lines, and Euclidean distance sets. The construction leverages the unique regular two-graph on 276 vertices.
Open Questions: The paper raises the question of whether a 2-distance set of size 325 ( $\binom{26}{2}$ ) exists in $\mathbb{R}^{24}$ . While the constructed set cannot be extended to 325 points in $\mathbb{R}^{24}$ , the existence of any such set remains an open and challenging problem.
Methodological Impact: The use of switching roots and lattice theory to prove maximality provides a robust framework for analyzing the extendability of distance sets in high dimensions.

In summary, this paper successfully constructs a 277-point 2-distance set in $\mathbb{R}^{23}$ , proving it is maximal in that dimension, and utilizes advanced algebraic combinatorics and computational verification to establish these results.