Spherical designs from norm-3 shell of integral lattices

TL;DR

This paper classifies integral lattices with shells of norm 3 that form spherical 5-designs, extending the understanding of lattice structures with highly symmetric geometric configurations.

Contribution

It provides a classification of integral lattices whose norm-3 shells are spherical 5-designs, building on previous work on strongly perfect lattices.

Findings

Identifies all integral lattices with norm-3 shells as 5-designs

Extends classification of lattices with symmetric spherical shells

Connects lattice shells to spherical design theory

Abstract

A set of vectors all of which have a constant (non-zero) norm value in an Euclidean lattice is called a shell of the lattice. Venkov classified strongly perfect lattices of minimum 3 (R\'{e}seaux et "designs" sph\'{e}rique, 2001), whose minimal shell is a spherical 5-design. This note considers the classification of integral lattices whose shells of norm 3 are 5-designs.