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

TL;DR

This paper demonstrates that minimal vectors of extremal even unimodular lattices in 32-dimensional space are universally optimal, form minimal spherical designs, and are maximal codes for certain sets T, advancing understanding of spherical codes and designs.

Contribution

It establishes the universal optimality and design properties of minimal vectors in specific high-dimensional lattices, a novel insight in lattice theory and spherical design research.

Findings

Minimal vectors are T-avoiding universally optimal.

They form minimal T-avoiding spherical designs.

They are maximal T-avoiding codes.

Abstract

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 $T$.