An even unimodular 72-dimensional lattice of minimum 8

TL;DR

This paper constructs a new 72-dimensional even unimodular lattice with minimum 8 using tensor products of known lattices over a quadratic number field, revealing its automorphism group.

Contribution

It introduces a novel construction of a 72-dimensional lattice via tensor products over an imaginary quadratic field, expanding lattice theory.

Findings

Constructed a 72-dimensional lattice with minimum 8

Identified a large automorphism group containing a specific matrix group

Demonstrated the lattice's properties using tensor product methods

Abstract

An even unimodular 72-dimensional lattice $\Gamma $ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant $-7$. The automorphism group of $\Gamma $ contains the absolutely irreducible rational matrix group $(\PSL_2(7) \times \SL _2(25)) : 2$.