Spherical designs and lattices

TL;DR

This paper proves that certain integral lattices with specific minimal vector properties are extremal, even, and unimodular, and establishes non-existence results for lattices with higher minimums and design strengths.

Contribution

It characterizes extremal lattices with spherical design properties and proves non-existence of certain lattices with higher minimums and design strengths.

Findings

Lattices with minimum <=7 and 9-designs are extremal, even, and unimodular.

Lattices with minimum <=11 do not yield 13-designs.

Provides classification results for lattices based on spherical design properties.

Abstract

In this article we prove that integral lattices with minimum <= 7 (or <= 9) whose set of minimal vectors form spherical 9-designs (or 11-designs respectively) are extremal, even and unimodular. We furthermore show that there does not exist an integral lattice with minimum <=11 which yields a 13-design.