Universal polar dual pairs of spherical codes found in $E_8$ and $\Lambda_{24}$

Abstract

We identify universal polar dual pairs of spherical codes $C$ and $D$ such that for a large class of potential functions $h$ the minima of the discrete $h$-potential of $C$ on the sphere occur at the points of $D$ and vice versa. Moreover, the minimal values of their normalized potentials are equal. These codes arise from the known sharp codes embedded in the even unimodular extremal lattices $E_8$ and $\Lambda_{24}$ (Leech lattice). This embedding allows us to use the lattices' properties to find new universal polar dual pairs. In the process we extensively utilize the interplay between the binary Golay codes and the Leech lattice. As a byproduct of our analysis, we identify a new universally optimal (in the sense of energy) code in the projective space $\mathbb{RP}^{21}$ with $1408$ points (lines). Furthermore, we extend the Delsarte-Goethals-Seidel definition of derived codes from their seminal $1977$ paper and generalize their Theorem 8.2 to show that if a $\tau$-design is enclosed in $k\leq \tau$ parallel hyperplanes, then each of the hyperplane's sub-code is a $(\tau+1-k)$-design in the ambient subspace.