An even unimodular 72-dimensional lattice of minimum 8

Abstract

An even unimodular 72-dimensional lattice $Γ$ 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 $Γ$ contains the absolutely irreducible rational matrix group $(\PSL_2(7) \times \SL _2(25)) : 2$.

New version: Replaced Griess' offender construction by a shorter and more sophisticated description of the vectors of norm 6 that also allows to compute the vectors of norm 8 and hence the kissing configuration of Gamma. I thank Noam Elkies for asking this question. Also included remark that Mark Watkins independently verified extremality of the lattice