On 3-lattices and spherical designs

TL;DR

This paper classifies 3-lattices up to dimension seven and explores their spherical designs, extending previous classifications and analyzing geometric properties of lattice shells.

Contribution

It extends the classification of 3-lattices from dimension four to dimension seven and investigates the spherical designs derived from their shells.

Findings

Complete classification of 3-lattices up to dimension seven.

Identification of spherical designs within lattice shells.

Enhanced understanding of lattice geometric structures.

Abstract

An integral lattice which is generated by some vectors of norm $q$ is called $q$-lattice. Classification of 3-lattices of dimension at most four is given by Mimura (On 3-lattice, 2006). As a expansion, we give a classification of 3-lattices of dimension at most seven. In addition, we consider the spherical designs from its shells.