# A $40$-dimensional extremal Type II lattice with no $4$-frames

## TL;DR

This paper constructs a unique 40-dimensional extremal Type II lattice lacking 40 orthogonal minimal vectors and analyzes its automorphism group, providing a novel example beyond classical binary code-based lattices.

## Contribution

It introduces a new 40-dimensional extremal Type II lattice without 4-frames and determines its automorphism group, expanding the known examples of such lattices.

## Key findings

- Constructed a 40-dimensional extremal Type II lattice without 4-frames.
- Determined the automorphism group of the lattice.
- Provided an example different from classical binary code-based lattices.

## Abstract

We construct a $40$-dimensional extremal Type II lattice not having any subsets consisting of $40$ orthogonal minimal vectors, and determine the automorphism group. This lattice gives an example different from the $16470$ lattices constructed from binary codes by classical constructions.