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

TL;DR

The paper addresses constructing and optimizing $T$-avoiding spherical codes in $\mathbb{R}^{32}$ by leveraging minimal vectors from extremal even unimodular lattices. It uses a linear programming framework with Gegenbauer polynomials to derive tight bounds and identifies a concrete $146{,}880$-point code on $\mathbb{S}^{31}$ that is simultaneously a spherical $7$-design, a maximal $T$-avoiding $(1/2)$-code for a chosen $T$, and universally optimal for $T$-avoiding energy with respect to absolutely monotone potentials. The authors prove universality and tight-design properties through carefully constructed polynomial interpolants and quadrature formulas, and they connect these codes to lattices arising from self-dual binary codes, including $L(CP_i)$ lattices, whose automorphism groups provide distinct lattice symmetries. Collectively, the work links extremal lattice theory, spherical design theory, and energy optimization to produce high-dimensional, highly structured $T$-avoiding codes with strong optimality guarantees. This advances understanding of how lattice-derived codes can attain universal optimality and maximality under distance-avoidance constraints, with potential implications for energy minimization and dense packing in high dimensions.

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 $T$.

Theorems & Definitions (11)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Remark 4.1
• Theorem 4.2
• proof
• Theorem 5.1
• proof
• Theorem 6.1