The Two-Distance Sets in Dimension Four

Abstract

A finite set of vectors $\mathcal {X}$ in the d-dimensional Euclidean space $\mathbb {R}^d$ is called a 2-distance set, if the set of mutual distances between distinct elements of $\mathcal {X}$ has cardinality exactly 2. In this note we report, among other things, the results of a computer-aided enumeration of the 2-distance sets in $\mathbb {R}^4$.

A Tables of Data

In the tables the vectorization (i.e., row-wise concatenation) of the lower triangular part of a graph adjacency matrix of order n is denoted by a string of letters a and b of length $n(n-1)/2$, where letter a indicates adjacent vertices. The ordered pair $(a^*,b^*)$ indicates the values for which the matrix $G(a^*,b^*)$ is positive semidefinite.

Table 2. Spherical 2-distance sets on $n\in \{8,9,10\}$ points in $\mathbb {R}^4$

Table 3. Spherical 2-distance sets on $n=7$ points in $\mathbb {R}^4$

Table 4. Spherical 2-distance sets on $n=6$ points in $\mathbb {R}^4$

Table 5. Examples of spherical 2-distance sets on $n=5$ points in $\mathbb {R}^4$

Table 6. General (nonspherical) 2-distance sets on $n\in \{7,8,9\}$ points in $\mathbb {R}^4$