Certain lattice neighboring from Barnes-Wall lattice and some laminated lattices

TL;DR

This paper classifies specific integral lattices including Barnes-Wall and laminated lattices, explores their neighboring lattices, and investigates related properties like spherical designs.

Contribution

It provides a classification of integral lattices containing Barnes-Wall and laminated lattices and studies their neighboring lattices and properties.

Findings

Classified integral lattices with Barnes-Wall and laminated lattices.

Identified certain neighboring lattices for each classified lattice.

Studied spherical designs and other properties of these lattices.

Abstract

We give classifications of integral lattices which include the Barnes-Wall lattice $BW_{16}$ or laminated lattices of dimension $1 \leqslant d \leqslant 8$ and of minimum 4. Also, we give certain lattice neighboring from each lattice. Furthermore, we study spherical designs and the other properties.