Structure of kissing arrangements in ${\mathbb{R}}^{12}$ and a place for the $841$st sphere

Abstract

Most currently known kissing arrangements of size $840$ in ${\mathbb{R}}^{12}$ share a common structure. They consist of $60$ vectors supported on ${\mathbb{R}}^6\times \{0\}$, another $60$ vectors supported on $\{0\}\times {\mathbb{R}}^6$, and $720$ additional \emph{bridge vectors}. The bridge vectors encode the interaction between the two six-dimensional factors and are constructed from the unique $1$-factorization of the complete graph $K_6$. In this paper we investigate kissing arrangements of this type while keeping the bridge vectors fixed. We show that each $60$-point block admits substantial flexibility: $12$ of its vectors may be chosen as the signed coordinate vectors $\pm e_i$, while the remaining $48$ vectors may vary within a positive-dimensional family of configurations, which we call $48$-systems. As a consequence, we obtain infinitely many pairwise non-isometric kissing arrangements of size $840$ in ${\mathbb{R}}^{12}$. The geometric freedom revealed by these constructions provides new insight into the local structure of extremal configurations. Exploiting this structure, we develop a specialized initialization scheme for logarithmic Riesz energy optimization. Starting from such structurally informed initial configurations, we numerically construct a kissing arrangement of size $841$ in ${\mathbb{R}}^{12}$.

Added a report on the new spherical codes, added a remark, and corrected several typos