On lattices whose minimal vectors form a 6-design

Let L be a lattice of dimension n ≤ 24 such that the minimal vectors of L form a 6-design and generate L . Then L is similar to either the root lattice E 8 , the Barnes–Wall lattice BW 16 , the Leech lattice Λ 24 , or n = 23 . For n = 23 we conjecture that the only possibilities for L are the shorter Leech lattice O 23 or its even sublattice Λ 23 .