Classification of equiangular lines with fixed angle $\arccos(1/(1+2\sqrt2))$

Abstract

We determine the maximum number $N_α(d)$ of equiangular lines with fixed angle $\arccosα$ for $α= 1/(1+2\sqrt2)$ in $d$-dimensional Euclidean space: $2,3,4,6,8,10,14,15,16,17,18,20,22$ for $d \in \{2,\dots,14\}$, and $\max(24, \lfloor 3(d-1)/2 \rfloor)$ for $d \ge 15$. This appears to be the first complete determination of $N_α(d)$ in all dimensions $d$ for a fixed nontrivial $α$, since the work of Lemmens and Seidel for $α= 1/3$ in 1973.

Figures (3)

• Figure 1: Heawood graph.
• Figure 2: Minimal forbidden subgraphs for the switching closure of the cherry graph.
• Figure 3: Minimal forbidden subgraphs for the switching closure of the single-edge graph.

Theorems & Definitions (28)

• Theorem 1.1
• Definition 1.2: Switching equivalence
• Definition 1.3: Switching closure
• Theorem 1.4
• Theorem 2.1: van Lint and Seidel vLS66
• Definition 2.2: Closure
• Proposition 2.3
• proof
• proof : Proof of \ref{['thm:main']} assuming \ref{['thm:stat']}
• Definition 3.1: Minimal forbidden subgraph