On $T$-avoiding spherical codes and designs in ${\mathbb{R}}^{32}$

In this article, we show that the minimal vectors of the extremal even unimodular lattices in ${\mathbb{R}}^{32}$ are $T$-avoiding universally optimal for suitable sets $T$. Moreover, they are minimal $T$-avoiding spherical designs and maximal $T$-avoiding codes for appropriate choices of $T$.